Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients

COMMUNICATIONS ON Website: http://AIMsciences.orgPURE AND APPLIED ANALYSISVolume 5, Number 1, 2006 pp. 213–240

UNIQUENESS RESULTS FOR FULLY NONLINEAR DEGENERATEELLIPTIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

Pierpaolo Soravia

Universita degli Studi di PadovaDipartimento di Matematica Pura e Applicata

via Belzoni, 7, 35131 Padova, Italy

(Communicated by Martino Bardi)

Abstract. In this paper we prove the comparison principle for viscosity solutions of secondorder, degenerate elliptic pdes with a discontinuous, inhomogeneous term having discon-tinuities on Lipschitz surfaces. It is shown that appropriate sub and supersolutions u, v

of a Dirichlet type boundary value problem satisfy u ≤ v in Ω. In particular, continuousviscosity solutions are unique. We also give examples of existence results and apply thecomparison principle to prove convergence of approximations.

1. Introduction. In this paper, we consider the pde

F (x, u(x), Du(x), D2u(x)) = f(x), in Ω, (1.1)

where Ω ⊂ RN is an open and bounded domain. Here F : Ω × R × R

N × SN → R,is a continuous function, while f : R

N → R is piecewise continuous as describedbelow in the main body of the paper. The notation SN indicates the set of N ×Nsymmetric matrices. Equation (1.1) is not the most general structure that we candeal with and extensions are described in Section 4 of the paper, see e.g. (5.1).The unknown function in the equation is u : Ω → R and Du indicates its gradient,while D2u denotes the matrix of second partial derivatives. Since the solution uof the fully nonlinear pde above may be nonsmooth, in general, we adopt viscositysolutions as the concept of solution, as we recall below. We couple equation (1.1)with a Dirichlet boundary condition and we plan to investigate the uniqueness ofsolutions. We will discuss the comparison principle for viscosity solutions of thefully nonlinear, degenerate elliptic, partial differential equation (1.1). Due to thediscontinuous coefficient f , the class of equations that we consider in not includedin the standard viscosity solutions theory, see Crandall, Ishii, Lions [11]. Indeedfor degenerate equations with Borel measurable f , uniqueness of solutions is not tobe expected, in general. For first order equations, in [33] we found representationformulas for the minimal and maximal viscosity solutions of the Dirichlet problemand showed that they are different functions unless, ideally, characteristic curvesof the equation are transversal to the set of discontinuities Γ of the discontinuouscoefficient f . This is impossible when Γ has nonempty interior and gives us a reasonto limit ourselves to a piecewise continuous coefficient.

The systematic study of viscosity solutions for equations with discontinuous in-gredients starts with the paper by Caffarelli, Crandall, Kocan and Swiech [7], which

2000 Mathematics Subject Classification. Primary: 35B50, 35A05; Secondary: 35J60, 35J70.Key words and phrases. Comparison principle, degenerate elliptic equations, viscosity solutions,

discontinuous coefficients.This research was partially supported by Miur-Cofin project “Metodi di viscosita, metrici e di

teoria del controllo in equazioni alle derivate parziali nonlineari”.

213

214 PIERPAOLO SORAVIA

builds upon earlier work by Caffarelli on the maximum principle for Pucci opera-tors, see also [12] and [20]. For more general uniqueness results see Swiech [35] andthe references therein. The paper [7] is devoted to uniformly elliptic pdes and theyprove existence and uniqueness results with f ∈ Lp, p ≥ N − ε. Their theory alsoextends to fully parabolic equations. Even for linear, uniformly elliptic equationshowever, the uniqueness problem is subtle and it is not completely understood. In-deed the papers by Nadirashvili [28] and Safonov [30] prove that uniqueness of theDirichlet problem fails, in general, if the coefficients of the second order derivativesare discontinuous. The more recent paper by Camilli, Siconolfi [8], on first ordereikonal-type equations, exploits instead coercivity of the Hamiltonian and provesuniqueness for f ∈ L∞. Both papers [7] and [8] however use a notion of solutionwhich is stronger than the usual one of viscosity solution, and its definition is heavilydetermined by the class of equations considered. The key point seems to be that, ifwe deal with classes of equations with some strong structure, by suitably reinforcingthe notion of solution we can obtain uniqueness results for a more general coefficientf , other than piecewise continuous.

In order to allow general fully nonlinear degenerate elliptic equations, addressedhere for the first time in the theory, we stick instead to the standard viscositysolution notion, see [11] and [17]. This however has some advantages: being thenotion of solution less restrictive, it is easier to check and it is more natural whendealing with limits of approximations. The price we have to pay is the restrictionon the type of discontinuity allowed on f .

When the set Γ is the union of Lipschitz hypersurfaces, we can prove the com-parison principle of viscosity solutions. The difficulty that we face is the fact thatusually a byproduct of the comparison principle is the continuity of solutions. How-ever equation (1.1) does have discontinuous solutions, in general, as we show in anexample. We prove that discontinuous super and subsolutions can be comparedprovided at least one of the two functions satisfies suitable directional continuityin a transversal direction at the points of the boundary of the domain and on thediscontinuity set Γ, see below for the precise statements. The former is sometimesknown as ”the cone condition”, the latter is introduced here and is similar looking.However it is required also at points in Ω but it is one directional. Thus our com-parison theorem is not a complete ”strong comparison principle”, i.e. it does notapply to any pair of discontinuous super and subsolutions satisfying the appropriateboundary condition.

We prove several applications of our comparison theorem related to the study ofthe Dirichlet problem for (1.1). On one hand we show that if a discontinuous solutionhas the appropriate directional continuity on ∂Ω ∪ Γ, it is necessarily continuous,making it an indirect and simple way to prove existence of continuous solutions, inaddition to Perron’s method for instance. On the other hand continuous viscositysolutions turn out to be unique in the class of discontinuous solutions. Thirdly,whenever we can find a continuous solution of the Dirichlet problem, then it turnsout to be the uniform limit of the vanishing viscosity approximations. The latterconvergence result can be also extended to other kinds of approximations.

We leave largely open the question of existence of continuous solutions in a gen-eral framework, but we plan to come back to this issue in a future paper. Continuoussolutions are however quite natural to arise, their existence depending only on thementioned transversality conditions. As examples of existence of continuous solu-tions we quote from Caffarelli, Crandall, Kocan and Swiech [7] uniformly ellipticpdes, therefore putting their theory into the new context of discontinuous solutions,

ELLIPTIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 215

at least in the case of piecewise continuous coefficients. Moreover we can also dealwith linear subelliptic operators, by using uniform interior a-priori estimates on themodulus of continuity of the vanishing viscosity approximations proved by Krylov[24], and first order equations of optimal control, where the transversality condi-tions on Γ ∪ ∂Ω can be well expressed in terms of the Lie algebra generated bythe vector field of the underlying control system. For convex first order equationsthe existence and uniqueness results that we can obtain are rather complete, seealso a detailed discussion in the other paper on the degenerate eikonal equation [34]whose structure is not directly contained in the framework of the present paper.The proof we propose for the comparison principle follows some ideas contained inour previous paper [32] and avoids the so called method of doubling the variables,although this technique is somewhat incorporated in the use of the nonlinear con-volution regularization. It also couples key technical ingredients adopted from theclassical paper on state constraints by Soner [31], as later developed for Dirichlettype problems by Ishii [15], see also [9], [4] and [23], and by Barles-Buredeau [6].The proof of the comparison principle for (1.1) is as self-contained as possible.

We finally want to mention that previous existence and uniqueness results forpiecewise continuous, uniformly elliptic operators were obtained by Kutev and Lions[25]. The connection between viscosity solutions and the notion of good solutionintroduced by Fabes was studied by Jensen [20], Crandall, Kocan, Soravia, Swiech[12] and by Jensen, Kocan, Swiech [22]. Uniqueness of viscosity solutions of secondorder, degenerate elliptic equations, of the form (1.1) but with continuous f , wasfirst established by Lions [26] for equations of stochastic optimal control, by controltheoretic arguments. Next Jensen [18] gave the first proof for general pdes. Hisapproach was simplified and extended in Jensen, Lions, Souganidis [21] and Jensen[19], see also Ishii [16], Crandall, Ishii [10], Ishii, Lions [17]. Different argumentswere proposed in Lions, Souganidis [27] and the author [32].

The plan of the paper in as follows. In Section 1 we discuss definitions, assump-tions and some properties of nonlinear convolution. Section 2 proves the comparisonprinciple and its main consequences to the Dirichlet problem for (1.1). Section 3 isdevoted to vanishing viscosity approximations and to the question of existence ofcontinuous solutions for (1.1). In Section 4 we extend the framework of our results.Finally in the Appendix we give some auxiliary results in particular on other notionsof solution.

2. Preliminaries. In this section we discuss the notion of viscosity solution andpresent the main assumptions and some preliminary statements. We start withtwo general definitions. Our equation (1.1) is set in an open and bounded domainΩ ⊂ R

N . In some statements below the boundary ∂Ω will be required to be aLipschitz hypersurface, which is equivalent to the following, as shown for instancein the appendix of Bardi-Soravia [3]. The notation B(x, r) indicates the open ballof center x ∈ R

N and radius r > 0.

Definition 2.1. The set Γ ⊂ RN is said to be a Lipschitz hypersurface if for all

x ∈ Γ one of its neighborhoods is partitioned into two connected nonempty open setsΩ+, Ω− and Γ itself, and we can find a transversal unit vector η ∈ R

N , |η| = 1, with

the following property: there are c, r > 0 such that and if x ∈ B(x, r) ∩ Ω± thenB(x ± tη, ct) ⊂ Ω± for all 0 < t ≤ c, respectively.We say that an open set Ω is a Lipschitz domain if ∂Ω is a Lipschitz hypersurface.In this case if for x ∈ ∂Ω and transversal unit vector η we have Ω+ ⊂ Ω, then wecall η = ηΩ an inward unit vector.


The following regularity property for functions will be a key request in our state-ments.

Definition 2.2. Given a unit vector η, we say that a function u : Ω → R, iscontinuous at x ∈ Ω in the direction of η if there are sequences tn → 0+, andpn → 0, pn ∈ R

N , such that

limn→+∞

u(x + tnη + tnpn) = u(x).

Given a Lipschitz hypersurface Γ ⊂ RN and a transversal unit vector η, it will be

particularly relevant for us the case of nontangential continuity of a given functionu at points of Γ, namely the continuity in the direction of η. We will also deal withthe continuity of u at points of ∂Ω in an inward direction.

We proceed by recalling the definition of viscosity solution for equations of theform (1.1). This definition follows the one proposed by Ishii [15]. In order tointroduce it, we need the notion of upper and lower semicontinuous envelope of alocally bounded function v : D → R. They are, respectively,

v∗(x) = limr→0+

sup|y−x|≤r, y∈D

v(y), v∗(x) = limr→0+

inf|y−x|≤r, y∈D

v(y).

The definition of viscosity solution for equation (1.1) is as follows.

Definition 2.3. A lower (resp. upper) semicontinuous function u : Ω → R isa viscosity super- (resp. sub-) solution of (1.1) if for all ϕ ∈ C2(Ω) and x ∈argminx∈Ω(u − ϕ), (resp. x ∈ argmaxx∈Ω(u − ϕ)), we have

F (x, u(x), Dϕ(x), D2ϕ(x)) ≥ f∗(x), (resp. F (x, u(x), Dϕ(x), D2ϕ(x)) ≤ f∗(x)).

We also say in this case that (Dϕ(x), D2ϕ(x)) ∈ D2,−u(x) the subjet of u at x (resp.(Dϕ(x), D2ϕ(x)) ∈ D2,+u(x) the superjet). A locally bounded function u : Ω → R

is a discontinuous viscosity solution of (1.1) if u∗ is a subsolution and u∗ is asupersolution.

We postpone to the appendix some discussion about the definition of solution,but we refer also to [15], [33] and the references therein.

We will couple equation (1.1) with a Dirichlet boundary condition, allowingenough generality so that solutions may not necessarily satisfy the boundary con-dition in a pointwise sense. Let g : ∂Ω → R be a continuous function.

Definition 2.4. We say that an upper semicontinuous function u : Ω → R, subso-lution of (1.1), satisfies the Dirichlet type boundary condition in the viscosity sense

u ≤ g or F (x, u, Du, D2u) ≤ f∗(x), on ∂Ω

if for all ϕ ∈ C2(RN ) and x ∈ ∂Ω, x ∈ argmaxx∈Ω(u − ϕ) such that u(x) > g(x),then we have

F (x, u(x), Dϕ(x), D2ϕ(x)) ≤ f∗(x).

Lower semicontinuous functions that satisfy

u ≥ g or F (x, u, Du, D2u) ≥ f∗(x), on ∂Ω,


or discontinuous solutions of the Dirichlet-type boundary condition

u = g or F (x, u, Du, D2u) = f(x), on ∂Ω, (2.0)

are defined accordingly.A locally bounded function u : Ω → R is a discontinuous solution of the Dirichlettype problem for (1.1) with boundary datum g if it is a discontinuous solution of(1.1) and of (2.0).

Remark 1.5. Dirichlet type boundary conditions in the viscosity sense are known torepresent a weak version of the more classical Dirichlet boundary condition. Theyappear naturally when proving existence of solutions via deterministic or stochasticoptimal control or when taking limits of solutions of approximate regularized prob-lems, while for instance existence of solutions to classical Dirichlet problems mayfail in general without appropriate assumptions. This fact is standard and appearsin many references of the viscosity solutions literature, see e.g. [2], [6] and [11].

Example. We give an example of solvability of an equation with the notion ofviscosity solution. The equation

−εu” + 2|u′| + u = f(x), x ∈ (−1, 1),

where ε > 0 and f is the characteristic function of the positive real numbers, issolved in the viscosity sense by the function

uε(x) =

√1+ε−1

2√

1+εe

1+√

1+εε

x, x < 0;

1 −√

1+ε+12√

1+εe

1−√

1+εε

x, x ≥ 0.

The reason is that uε is a strong solution, as explained in the appendix. The functionuε turns out to be the unique viscosity solution of the corresponding Dirichletproblem. Notice that uε(x) → u(x) = (1− e−

x2 )f(x), uniformly for x ∈ [−1, 1], and

that as we show below (but see also [33]) u is the unique solution of the first orderproblem

u + 2|u′| = f(x), x ∈ (−1, 1),

u(−1) = 0, u(1) = (1 − e−12 ).

Therefore convergence of the vanishing viscosity approximations to the correct so-lution of the limiting problem holds in this example. This is a general fact and willbe justified in Section 3.

We now explain the main regularity assumptions on the Hamiltonian F thatappears in (1.1). In particular, in order to call (1.1) degenerate elliptic we supposethat for all compact subsets O ⊂ Ω we can find modulus ωO, i.e. ωO : R+ → R+

nondecreasing, continuous and vanishing at the origin, such that F satisfies

−ωO(|p − q|) ≤ F (x, r, p, X) − F (x, r, q, Y ) ≤ ωO(|p − q| + Tr(Y − X)), (2.1)

for Y ≥ X, x ∈ O, r ∈ R p, q ∈ RN , where we used the usual ordering on SN .

Moreover we suppose that there is a constant ν > 0 such that r → F (x, r, p, X)−νris nondecreasing, for all (x, p, X) ∈ Ω × R

N × SN . We will refer below to thesetwo properties together by saying that F is proper. Note that condition (2.1) is a


little stronger than the corresponding condition required in the standard theory forcontinuous Hamiltonians, see [11].

We now introduce the key assumption on the discontinuities allowed in the Hamil-tonian. The following assumption has several motivations. From one side, the studyof first order equations pointed out that, with our notion of solution, uniquenesscannot occur if the set of discontinuities for f has nonempty interior. On the otherside, in order to implement some ideas of the constrained viscosity solutions theory,we need some regularity of the boundary of the domain and of the set of disconti-nuities of f .

Assumption (DC). The set

Γ = x ∈ RN : f is discontinuous at x

is the disjoint union of a finite family of connected Lipschitz hypersurfaces and f ispiecewise continuous across Γ. In particular, for x ∈ Γ we can find c, r > 0, open,nonempty, connected sets Ω+, Ω− and inward unit vectors η+, η− = −η+ to Ω±

respectively, such that if x ∈ B(x, r), then one and only one of the following occurs

x ∈ Ω+, x ∈ Ω−, x ∈ Γ.

Moreover if x ∈ B(x, R) ∩ (Ω± ∪ Γ), respectively, then B(x + tη±, ct) ⊂ Ω± for all0 < t ≤ c. We also suppose that Ω± ⊂ Ω if x ∈ Ω, assume that the discontinuouscoefficient f is continuous in each component Ω± with a continuous extension inΩ±, and that if x ∈ Γ

f(x) ∈ [ limΩ−∋y→x

f(y), limΩ+∋y→x

f(y)],

where it is assumed that the above limits exist and the notation ± is introduced insuch a way that the interval is well defined.

If x ∈ Γ ∩ ∂Ω we assume that in the above we can choose c, r, η+, η− in such away that η+, η− are also inward for Ω, i.e. for instance

B(x + tη+, ct) ⊂ Ω ∩ Ω+, B(x + tη−, ct) ⊂ Ω ∩ Ω−

for all x ∈ B(x, R) ∩ Ω ∩ Ω± and 0 < t ≤ c, respectively. To this end we allowη+ 6= −η−, in general. However we suppose also that −η+,−η− are inward vectorsfor Ω−, Ω+, respectively.

One of the technical difficulties of the problem comes from considering boundarypoints where the inhomogeneous term in the equation is discontinuous. In order tocope with this fact we require that ∂Ω and Γ meet transversally, which is expressedin the above assumption for x ∈ ∂Ω ∩ Γ.

In addition to properness, we require the Hamiltonian to satisfy the followingregularity condition which refers to previous condition (DC). For all x ∈ Ω andsufficiently small s > 0, relative to B(x, s) ∩ Ω there is a modulus ωx such that

F (x, r, α(x − y) −√αη, (I + 1

αX)−1X) − F (y, r, α(x − y) −√

αη, X)

≥ −ωx(|x − y| + α|x − y − 1√αη|2),

for all x, y ∈ B(x, s) ∩ Ω, (r, X) ∈ R × SN , −αI < X ≤ 2αI, α > 0,

η = 0 or η = δη− if x ∈ Γ, δ ∈ [0, 1].

F (x, r, α(x − y) +√

αη, X) − F (y, r, α(x − y) +√

αη, (I − 1αX)−1X)

≥ −ωx(|x − y| + α|x − y + 1√αη|2),

for all x, y ∈ B(x, s) ∩ Ω, (r, X) ∈ R × SN , −2αI ≤ X < αI, α > 0,

η = 0 or η = δη+ if x ∈ Γ, δ ∈ [0, 1].

(2.2)


Remark 2.6. Assumption (2.2) is a little stronger than the corresponding standardassumption in [11], but it is general enough to be satisfied by Hamiltonians havingthe structure of Isaacs equations of stochastic differential games, i.e.

F (x, r, p, X) = supa∈A

infb∈B

− TrA(x, a, b)X − f(x, a, b) · p + c(x, a, b)r − l(x, a, b),

with coefficients A, f, c, l continuous and uniformly bounded in W 1,∞(RN ) for all(a, b) ∈ A × B, c strictly positive. Condition (2.2) and the more standard (4.14) in[11] look different, ours follows more closely Jensen [19]. About comparing the twodifferent looking assumptions one may check e.g.. Remark 1.1 in [32].

A regularization tool for functions, which we employ below is nonlinear convo-lution that we now recall. Part of its basic properties is collected in PropositionA.1 in the Appendix, here we anticipate some of them to point out the role ofthe directional continuity. If O ⊂ R

N is a closed set and w : O → R is a lowersemicontinuous function satisfying,

lim infx∈O, x→+∞

w(x)

|x|2 > − 1

2κ2, for some κ > 0, (2.3)

then, for ε ∈ (0, κ] and η ∈ RN , we define its inf-convolution as

wε(x) = infy∈O

w(y) +1

2

∣

∣

∣

∣

x − y

ε+ η

∣

∣

∣

∣

2

, (2.4)

for x ∈ RN . In a similar way, we can define the sup-convolution as well, namely

if v : O → R is an upper semicontinuous function and −v satisfies (2.3), we putvε = −(−v)ε. If η = 0 the above definition is well known. We extend it for η 6= 0 toimplement ideas used in the theory of constrained viscosity solutions, see e.g. Soner[31].

We will always think below that a lower semicontinuous function w : O → R,O ⊂ R

N closed, is extended by +∞ off O and write directly

wε(x) = infy∈RN

w(y) +1

2

∣

∣

∣

∣

x − y

ε+ η

∣

∣

∣

∣

2

,

instead of (2.4). Notice that for nonempty O wε : RN → R. The following result

illustrates the role of directional continuity of Definition 1.2. It also shows themeaning of the assumption on the regularity of the boundary of the domain.

Proposition 2.7. Let w : Ω → R be a lower semicontinuous, bounded function.Let O ⊂⊂ Ω be such that for some η ∈ R

N , and c > 0, we have B(x + εη, εc) ⊂ Ωfor all x ∈ O, 0 < ε ≤ c. For x ∈ R

N , let Tεx(∈ Ω) be a point such that

wε(x) = w(Tεx) +1

2

∣

∣

∣

∣

x − Tεx

ε+ η

∣

∣

∣

∣

2

.

For x ∈ O given, and a sequence pεn, |pεn

| ≤ c, pεn→ 0 thus

x + εn(η + pεn) ∈ Ω.


Then, at least for the sequence (εn)n,

lim infε→0+, O∋y→x

wε(y) ≥ w(x). (2.5)

If moreoverlim

εn→0+w(x + εn(η + pεn

)) = w(x),

then

limε→0+

wε(x) = w(x), limε→0+

|x − Tεx + εη|ε

= 0, limε→0+

w(Tεx) = w(x). (2.6)

In particular Tεx ∈ Ω for ε sufficiently small (but depending on x ∈ O).

Proof. We will outline this proof which is more or less standard for the reader’sconvenience.

Let x ∈ O ⊂⊂ Ω. Observe that, by definition and the assumption,

w(Tεx) ≤ wε(x) ≤ w(x + εη + εpε) +1

2|pε|2 < +∞, (2.7)

hence if w is bounded in Ω, then for ε small enough we obtain that wε are uniformlybounded in O by the bounds of w. In particular by definition of Tεx we have that∣

∣

x−Tεxε

+ η∣

∣ is bounded, uniformly for x ∈ O and hence, for ε small enough

|x − Tεx| ≤ Cε, |x − Tεx + εη| ≤ Cε, x ∈ O, (2.8)

where C does not depend on ε and O. Going back to (2.7) and taking limits, thenwe prove the first equation in (2.6) by lower semicontinuity of w and directionalcontinuity.

From (2.7), (2.8) it follows that

inf|y−x|≤Cr+r

w(y) ≤ infO∋y, ε≤r, |y−x|≤r

wε(y),

and thus (2.5) follows from the lower semicontinuity of w. By definition of Tεx, wealso obtain

∣

∣

∣

∣

x − Tεx

ε+ η

∣

∣

∣

∣

2

≤ 2

(

w(x + εη + εpε) − w(Tεx) +|pε|2

2

)

and therefore by lower semicontinuity of w and the assumption we obtain |x−Tεx+εη| = ox(ε) as ε → 0, at least for the sequence (εn). The third equation in (2.6) isnow a corollary.

Remark 2.8. The fact that the rate of the second limit in (2.6) may depend on x isquite inconvenient and will be adapted later in cases needed in the proof of Theorem2.3. Notice that (2.6) is a consequence of the directional continuity assumptionon the function w, which is always satisfied if η = 0, by choosing pε ≡ 0. Ofcourse Proposition 1.7 above can be restated for sup-convolutions as well, in anobvious way. Proposition 1.7 points out that the modified inf-convolution withη 6= 0 is an approximation of w (for instance at boundary points), provided w iscontinuous in the direction of η (and η is an inward vector for Ω). We will use the


modified inf-convolution with appropriate η also at the points of Γ to take care ofthe discontinuities in the equation.

3. Comparison Principle and Uniqueness. In this section we study the bound-ary value problem

F (x, u(x), Du(x), D2u(x)) = f(x), in Ω,

u(x) = g(x), in ∂Ω,(3.1)

and discuss the main uniqueness results and some consequences. Below, by sayingthat w is twice differentiable at a point x, we mean that it has a second order Taylorexpansion

w(y) = w(x) + (p, y − x) +1

2(X(y − x), y − x) + o(|y − x|2), as y → x,

for some (p, X) ∈ RN × SN . In this case, with a slight abuse of notation, we say

that (p, X) = (Du(x), D2u(x)). We recall that if u is twice differentiable at x then(Du(x), D2u(x)) ∈ D2,+u(x) ∩ D2,−u(x).

We start by stating the following Lemma, which is a refinement of Lemma 2.1 in[32]. It is one of the crucial technical steps of the proof of the comparison theorem.

Lemma 3.1. Let Ω ⊂ RN be a bounded domain and w : Ω → R be an upper

semicontinuous function. We will set w(x) = −∞ off Ω. Let wε : RN → R, ε > 0,

be a family of semiconvex functions such that

lim supε→0+, y→x

wε(y) ≤ w(x), for x ∈ RN .

Let K ⊂ RN be a set of Lebesgue measure zero and suppose that x ∈ arg maxΩ(w)

be such that for some η ∈ RN , pε → 0

limε→0+

wε(x + ε(η + pε)) = w(x).

Then for r > 0 fixed and ρ > 0 there are points zε,ρ ∈ RN\K such that wε is twice

differentiable at zε,ρ and

limρ→0+

zε,ρ = xε, limε→0+

xε = x, D2wε(zε,ρ) ≤ (ρ + r)I,

limε→0+

(

limρ→0+

wε(zε,ρ)

)

= w(x), limρ→0+

|Dwε(zε,ρ) − r(zε,ρ − x)| = 0.(3.2)

Proof. Let x ∈ arg maxΩ(w), we are going to localize the argument around x. For

r > 0, let xε be a maximum point of wε(·) − r2 | · −x|2 in B(x, 1). At least along a

subsequence εn → 0+ we may suppose that xε → y. Thus we get

wε(xε) ≥ wε(x + ε(η + pε)) +r

2|xε − x|2 − rε2

2|η + pε|2,

and then by the assumptions

w(y) ≥ lim supε→0+, y→y

wε(y) ≥ lim supε→0+

wε(xε) ≥ lim infε→0+

wε(xε) ≥ w(x) +r

2|y − x|2.


Therefore we conclude that y ∈ Ω, y = x and then

xε → x, wε(xε) → w(x).

Given ε sufficiently small, for any σ > 0, the function

wε(·) − r

2| · −x|2 − σ

2| · −xε|2

then attains at xε a strict local maximum. Moreover, as this function is semiconvex,we can apply Jensen’s Lemma and Aleksandrov’s Theorem in the Appendix, andperturb such a point to gain in regularity. Given α, s > 0 sufficiently small, we canfind points z ∈ B(xε, s) ∩ (RN\K), p ∈ R

N , |p| ≤ α, such that the function

µ(·) = wε(·) − r

2| · −x|2 − σ

2| · −xε|2 − (p, ·),

attains a local maximum at z, and moreover wε is twice differentiable at z. Inparticular we can state that

0 = Dµ(z) = Dwε(z) − r(z − x) − σ(z − xε) − p,

and0 ≥ D2µ(z) = D2wε(z) − (r + σ)I.

Given ρ > 0, we can then choose the parameters α, s, σ and points zε,ρ ∈ RN\K in

such a way that

|zε,ρ − xε| ≤ ρ, |Dwε(zε,ρ) − r(zε,ρ − x)| ≤ ρ, D2wε(zε,ρ) ≤ (r + ρ)I.

The conclusions then follow.

Remark 3.2. Observe that if in Lemma 2.1 we know that x ∈ Ω, then we maysuppose xε, zε,ρ ∈ Ω for ε and then ρ sufficiently small. It is important to notice

that if x ∈ ∂Ω, then we might need to choose the points xε, zε,ρ outside Ω. Howeverone can always avoid the fixed in advance null set K.

The first statement of a comparison theorem relates a super and a subsolutionof a Dirichlet type boundary value problem.

Theorem 3.3. Let Ω be a bounded domain with Lipschitz boundary. Assume thatthe Hamiltonian F is proper and satisfies condition (2.2). Let us suppose that theassumption (DC) is satisfied. Let u, v : Ω → R be respectively an upper and a lower-semicontinuous function, respectively a subsolution and a supersolution of

F (x, u, Du, D2u) = f(x), in Ω.

Let us assume that u, v satisfy the Dirichlet type boundary conditions in the viscositysense

u ≤ g or F (x, u, Du, D2u) ≤f∗(x), on ∂Ω,

v ≥ g or F (x, v, Dv, D2v) ≥f∗(x), on ∂Ω.

Suppose that u, v are nontangentially continuous on ∂Ω\Γ in the inward directionηΩ, and on Γ ∩ ∂Ω in the directions η−, η+ respectively. Assume moreover that at


each point of Γ\∂Ω, either u is nontangentially continuous in the direction of η−

or v is nontangentially continuous in the direction of η+. Then u ≤ v in Ω.

Remark 3.4. It is important to notice that, as the proof below will show, the non-tangential continuity of u, v at x ∈ ∂Ω is only needed if we allow the Dirichlettype boundary condition, but can be avoided if the super and subsolutions satisfythe usual Dirichlet condition. The same goes for the Lipschitz regularity of theboundary ∂Ω, which is not needed if u ≤ g, v ≥ g on ∂Ω. We notice that theboundary ∂Ω and the discontinuity set Γ require similar assumptions on the superand subsolutions. However while on ∂Ω both functions u, v need some tangentialcontinuity, on Γ ∩ Ω either u or v is required to satisfy it, provided the direction isappropriate, i.e. it is η+ for v or η− for u. We will need to be a little more carefulin Γ ∩ ∂Ω where the two conditions combine. The role of nontangential continuityin viscosity solutions theory is not new. It was pointed out by Ishii [15] for firstorder equations and used by Katsoulakis [23] in the study of constrained viscositysolutions, but see also the book by Barles [5] and the paper by Barles-Burdeau [6]for an extensive study of the condition.

Proof. In the argument below we always suppose that u, v are extended as u(x) =−∞ and v(x) = +∞ for x ∈ R

N\Ω. We assume by contradiction that there isx ∈ Ω such that

u(x) − v(x) = maxx∈Ω

u(x) − v(x) = 2γ > 0. (3.3)

The proof below will be local in a neighborhood of the point x, therefore we will omitthe subscripts in the moduli ω that we will encounter. The localization argument isalready incorporated in Lemma 2.1. We will leave undetermined at this point theposition of x, because the differences in the proof will come up in few places. Themain new parts in the argument are of course when x ∈ Γ.

We need to set the two following facts. If x ∈ ∂Ω, we start by checking theboundary condition and observe that it holds

either u(x) > g(x) or v(x) < g(x). (3.4)

We will suppose below, just to fix the ideas, that the former is attained. For thesame reason, if instead x ∈ Γ∩Ω we will suppose that v is nontangentially continuousin the direction η+.Step 1: smoothing up. We start regularizing and, for ε > 0, we introduce thenonlinear convolutions uε and vε. More precisely, in both definitions we have tochoose carefully the vector η to be used. We will choose for u, v respectively, byapplying assumption (DC):

ηu =

0, if x ∈ Ω\Γ,

−η+, if x ∈ Γ,ηv =

0, if x ∈ Ω\Γ,

ηΩ, if x ∈ ∂Ω\Γ,

η+, if x ∈ Γ.

We recall that on Γ ∩ Ω we have η− = −η+, while this fact may not be true on∂Ω ∩ Γ. Observe that, by Proposition 1.7 and the regularity of ∂Ω ∪ Γ

lim supε→0+,y→x

uε(y) − vε(y) ≤ u(x) − v(x), for x ∈ Ω.


The same is also true in RN\Ω, as easily checked. By the nontangential continuity

in the direction of ηv, at least along a subsequence εn → 0 we can find RN ∋ pn → 0

such thatlim

n→+∞v(x + 2εn(ηv + pn)) = v(x).

Thus, by definition of nonlinear convolution:- if x ∈ Γ, so ηu = −ηv, we get

uεn(x − εnηu) − vεn(x + εnηv) ≥ u(x) − v(x + 2εn(ηv + pn)) − 2|pn|2;

- if x ∈ ∂Ω\Γ, so ηu = 0,

u2εn(x) − v2εn(x) ≥ u(x) − v(x + 2εn(ηv + pn)) − |pn|2/2;

- if x ∈ Ω\Γ we have uε(x) − vε(x) ≥ u(x) − v(x).Taking limits, in all cases we can find η ∈ R

N such that

lim infn→+∞

uεn(x + εnη) − vεn(x + εnη) ≥ u(x) − v(x),

at least along a subsequence. Notice that no continuity of v is needed to obtain theprevious inequality, if ηv = 0, by choosing pε = 0. We will avoid below the notationfor the subsequence εn.

The assumptions of Lemma 2.1 are thus satisfied with w = u−v and wε = uε−vε,and with the choice

K = Γ ∪n x : either uεn or vεnis not twice differentiable at x.

Therefore by that result, for a fixed r > 0 (that we need as small as required at theend of step 3) and given ε, ρ > 0, we construct points xε ∈ R

N , zε,ρ ∈ RN\Γ, such

that both uε and vε are twice differentiable at zε,ρ and satisfy (3.2). In particular,if we choose ε and then ρ sufficiently small, by the first equation of the second lineof (3.2) and (3.3), we may suppose that

uε(zε,ρ) − vε(zε,ρ) ≥ γ, uε(xε) − vε(xε) ≥ γ.

Step 2: nonlinear convolution. We will now use the properties of nonlinear convo-lution. Notice that for ρ > 0 we have

(Duε(zε,ρ), D2uε(zε,ρ) + ρI) ∈ D2,+uε(zε,ρ),

(Dvε(zε,ρ), D2vε(zε,ρ) − ρI) ∈ D2,−vε(zε,ρ).

(3.5)

We will denote below Xρ = D2uε(zε,ρ) + ρI and Xρ = D2vε(zε,ρ) − ρI. By apply-ing Propositions 1.7 and A.1, and Remark 1.8, for sufficiently small values of theparameters ε, ρ, we can find a unique point T εzε,ρ ∈ Ω such that

uε(zε,ρ) = u(T εzε,ρ) −1

2

∣

∣

∣

∣

zε,ρ − T εzε,ρ

ε+ ηu

∣

∣

∣

∣

2

,

Duε(zε,ρ) =(T εzε,ρ − zε,ρ − εηu)

ε2, D2uε(zε,ρ) ≥ − 1

ε2I, (3.6)

(Duε(zε,ρ), (I + ε2Xρ)−1Xρ) ∈ D2,+u(T εzε,ρ).


We can also find a unique point Tεzε,ρ ∈ Ω such that

vε(zε,ρ) = v(Tεzε,ρ) +1

2

∣

∣

∣

∣

zε,ρ − Tεzε,ρ

ε+ ηv

∣

∣

∣

∣

2

,

Dvε(zε,ρ) =(zε,ρ − Tεzε,ρ + εηv)

ε2, D2vε(zε,ρ) ≤

1

ε2I, (3.7)

(Dvε(zε,ρ), (I − ε2Xρ)−1Xρ) ∈ D2,−v(Tεzε,ρ).

As ρ → 0+, at least along a subsequence, by (3.2) of Lemma 2.1, (3.6) and (3.7),we may suppose below that, for ε sufficiently small,

T εzε,ρ → zε ∈ Ω, Tεzε,ρ → zε ∈ Ω, D2vε(zε,ρ) → Yε, −(r +1

ε2)I ≤ Yε ≤ 1

ε2I.

Observe also that from the second equation of the second line in (3.2) and (3.6),(3.7) it follows that

zε − xε − εηu = xε − zε + εηv + ε2r(xε − x). (3.8)

In order to improve the second equation in (2.6) we do the following. Notice thatby the first equations in (3.6), (3.7) we obtain

1

2

∣

∣

∣

∣

xε − zε

ε+ ηu

∣

∣

∣

∣

2

+1

2

∣

∣

∣

∣

xε − zε

ε+ ηv

∣

∣

∣

∣

2

= limρ→0+

1

2

∣

∣

∣

∣


ε+ ηu

∣

∣

∣

∣

2

+1

2

∣

∣

∣

∣


ε+ ηv

∣

∣

∣

∣

2

≤ u(zε) − uε(xε) + vε(xε) − v(zε) = u(zε) − v(zε) − wε(xε).

(3.9)

Since the right hand side is bounded, as x, zε, zε ∈ Ω, then we get that zε → xand zε → x. Using this fact again in (3.9), the semicontinuity of u, v and the firstequation in the second line of (3.2) we conclude that

lim supε→0+

limρ→0+

1

2

∣

∣

∣

∣


ε+ ηu

∣

∣

∣

∣

2

+1

2

∣

∣

∣

∣


ε+ ηv

∣

∣

∣

∣

2

= lim supε→0+

1

2

∣

∣

∣

∣

xε − zε

ε+ ηu

∣

∣

∣

∣

2

+1

2

∣

∣

∣

∣

xε − zε

ε+ ηv

∣

∣

∣

∣

2

= 0.

(3.10)

An important consequence of this estimate is the fact that given c > 0 of As-sumption (DC), for ε and then ρ sufficiently small we can suppose that

|zε,ρ − T εzε,ρ + εηu| ≤ cε/2, |zε,ρ − Tεzε,ρ + εηv| ≤ cε/2. (3.11)

Notice that (3.11) also gives us

|T εzε,ρ + ε(ηv − ηu) − Tεzε,ρ| ≤ cε, (3.12)

which will be useful if x ∈ ∂Ω.


Another consequence of (3.6) (3.7), Lemma 2.1 and the semicontinuity propertiesof u, v is that

v(x) ≤ lim supε→0+

limρ→0+

v(Tεzε,ρ) = lim supε→0+

vε(xε) = lim supε→0+

uε(xε) − wε(xε) ≤ v(x)

and similarly for u. Thus

limε→0+

limρ→0+

v(Tεzε,ρ) = v(x), limε→0+

limρ→0+

u(T εzε,ρ) = u(x). (3.13)

Step 3: estimate of the left hand side of the equation. We now want to applythe definition of viscosity sub and supersolution at the points T εzε,ρ, Tεzε,ρ, viaProposition A.1. We have to consider a few separate cases.

- If x ∈ Ω, by the first two equations in (3.2) and (3.11), we may suppose that, forε, ρ sufficiently small T εzε,ρ, Tεzε,ρ ∈ Ω;

- if x ∈ ∂Ω\Γ, then by our choice of ηv = ηΩ and ηu = 0, and (3.12) we have thatTεzε,ρ ∈ Ω while T εzε,ρ ∈ Ω;

- if x ∈ (Γ ∩ ∂Ω), then by (3.12), −ηu = ηv and ηv inward Ω, |T εzε,ρ + 2εηv −Tεzε,ρ| ≤ cε and then Tεzε,ρ ∈ Ω.

Thus we might need to use the boundary condition for u at T εzε,ρ. To this end,observe that by (3.4), (3.13) and continuity of g, we may suppose that, for ε andthen ρ sufficiently small, if T εzε,ρ ∈ ∂Ω then we have

u(T εzε,ρ) > g(T εzε,ρ).

Our discussion then always allows us to use the equation at the points Tεzε,ρ,T εzε,ρ for v, u, respectively. By definition of viscosity solution, the properness ofF , assumption (2.2) and Lemma 2.1, we calculate

f∗(T εzε,ρ) ≥ F

(

T εzε,ρ, uε(zε,ρ) +

1

2

∣

∣

∣

∣


ε+ ηu

∣

∣

∣

∣

2

,

T εzε,ρ − zε,ρ − εηu

ε2, (I + ε2Xρ)−1Xρ

)

≥ F

(

T εzε,ρ, vε(zε,ρ) + γ,T εzε,ρ − zε,ρ − εηu

ε2, (I + ε2Xρ)−1Xρ

)

≥ F

(

zε,ρ, vε(zε,ρ) + γ,T εzε,ρ − zε,ρ − εηu

ε2, Xρ

)

− ω

(

|T εzε,ρ − zε,ρ| +∣

∣

∣

∣

T εzε,ρ − zε,ρ

ε− ηu

∣

∣

∣

∣

2)

≥ νγ + F

(

zε,ρ, vε(zε,ρ),T εzε,ρ − zε,ρ − εηu

ε2, D2vε(zε,ρ) + (r + 2ρ)I

)

− ω

(

|T εzε,ρ − zε,ρ| +∣

∣

∣

∣

T εzε,ρ − zε,ρ

ε− ηu

∣

∣

∣

∣

2)

.

(3.14)


Moreover

f∗(Tεzε,ρ) ≤ F

(

Tεzε,ρ, vε(zε,ρ) −1

2

∣

∣

∣

∣


ε+ ηv

∣

∣

∣

∣

2

,

zε,ρ − Tεzε,ρ + εηv

ε2, (I − ε2Xρ)

−1Xρ

)

≤ F

(

Tεzε,ρ, vε(zε,ρ),zε,ρ − Tεzε,ρ + εηv

ε2, (I − ε2Xρ)

−1Xρ

)

≤ F

(

zε,ρ, vε(zε,ρ),zε,ρ − Tεzε,ρ + εηv

ε2, D2vε(zε,ρ) − ρI

)

+ ω

(

|zε,ρ − Tεzε,ρ| +∣

∣

∣

∣


ε+ ηv

∣

∣

∣

∣

2)

.

(3.15)

Hence, subtracting (3.15) from (3.14), taking the limit as ρ → 0+ and then using(2.1), (3.10) and (3.8), we get

lim supρ→0+

f∗(T εzε,ρ) − f∗(Tεzε,ρ)

≥ νγ + F (xε, vε(xε),zε − xε − εηu

ε2, Yε + rI) − F (xε, vε(xε),

xε − zε + εηv

ε2, Yε)

− ω

(

|zε − xε| +∣

∣

∣

∣

zε − xε

ε− ηu

∣

∣

∣

∣

2)

− ω

(

|zε − xε| +∣

∣

∣

∣

zε − xε

ε− ηv

∣

∣

∣

∣

2)

≥ νγ − ω(r|xε − x| + rN) + o(1),

as ε → 0+. Thus

lim supε→0+

lim supρ→0+

f∗(T εzε,ρ) − f∗(Tεzε,ρ) ≥ νγ − ω(rN) > 0, (3.16)

when r > 0 is fixed sufficiently small.Step 4: the inhomogeneous term. In order to handle the discontinuous term, whichis the final brick of the construction, we only need to discuss the case x ∈ Γ, wheref is discontinuous. Notice that as we argued at the end of Step 1, zε,ρ /∈ K as theredefined. Hence in particular zε,ρ /∈ Γ. Thus for any ε fixed we may restrict ourselvesto subsequences ρn → 0 such that it always happens either zε,ρ ∈ Ω− or zε,ρ ∈ Ω+

for ρ sufficiently small. We just pause to recall that zε,ρ may be not in Ω but this willnot affect the estimate below. At this point we may choose a subsequence εn → 0such that always zε,ρ ∈ Ω− or zε,ρ ∈ Ω+ for any ε, ρ sufficiently small. In order tosimplify notations below, the choice of the subsequences will not appear explicitly.Therefore (3.11) implies that either T εzε,ρ ∈ Ω− or Tεzε,ρ ∈ Ω+, respectively. Thuswe obtain in the limit that zε ∈ Ω− ∪ Γ or zε ∈ Ω+ ∪ Γ, respectively. In either caseby assumption (DC) we can estimate the limit as follows

lim supε→0+

lim supρ→0+

f∗(T εzε,ρ) − f∗(Tεzε,ρ)

= lim supε→0+

lim supρ→0+

f(T εzε,ρ) − f∗(Tεzε,ρ), if zε,ρ ∈ Ω−

f∗(T εzε,ρ) − f(Tεzε,ρ), if zε,ρ ∈ Ω+

≤ lim supε→0+

f∗(zε) − f∗(zε), if zε,ρ ∈ Ω−

f∗(zε) − f∗(zε), if zε,ρ ∈ Ω+

≤

f∗(x) − f∗(x), if zε,ρ ∈ Ω−

f∗(x) − f∗(x), if zε,ρ ∈ Ω+= 0.

(3.17)


Finally (3.17), and (3.16) provide the required contradiction.

Remark 3.5. The choice of the vectors ηv, ηu in the proof above can be explainedas follows. We only need them to be non null vectors on ∂Ω and on Γ in orderto ”push” the variable inside Ω and/or inside Ω± respectively. The main difficultycomes on ∂Ω ∩ Γ where one needs to reach both these effects. Luckily enough, atsuch points by (3.4) only one of the two functions needs to have its variable pushedinto Ω.

A consequence of Theorem 2.3 is the following Corollary which gives us a set ofsufficient conditions for the existence of a continuous viscosity solution of a boundaryvalue problem.

Corollary 3.6. Let Ω be bounded and u : Ω → R be a bounded viscosity solution of

F (x, u, Du, D2u) = f(x), Ω,

where F is proper and satisfies condition (2.2). Let us suppose that the assumption(DC) is satisfied. If u is continuous on ∂Ω and either u∗ or u∗ is nontangentiallycontinuous on Γ\∂Ω in the direction of η+, η−, respectively, then u ∈ C(Ω).

Proof. Apply Theorem 3.3 to the supersolution u∗ and the subsolution u∗ and g = uon ∂Ω.

The following variant of the comparison theorem also holds. It does not refer toa Dirichlet-type problem, but compares directly the super and subsolutions. Noticethat only the nontangential continuity of one of the two functions to be comparedis now needed at the boundary.

Theorem 3.7. Let Ω be a bounded domain with Lipschitz boundary. Assume thatthe Hamiltonian F is proper and satisfies condition (2.2). Let us suppose that theassumption (DC) is satisfied. Let u, v : Ω → R be respectively an upper and a lower-semicontinuous function, respectively a subsolution and a supersolution of

F (x, u, Du, D2u) = f(x), in Ω.

Let us assume that v restricted on ∂Ω is continuous and that u satisfies the Dirichlettype boundary condition in the viscosity sense

u ≤ v or F (x, u, Du, D2u) ≤f∗(x), on ∂Ω.

Suppose moreover that v is nontangentially continuous on ∂Ω\Γ in the inward di-rection ηΩ and on Γ in the direction of η+. Then u ≤ v in Ω.

Proof. We can apply the same proof of Theorem 2.3 with the choice of g = v on∂Ω. The reason why the continuity of u is not needed on ∂Ω is that v is dealt withas a boundary condition and thus for x ∈ ∂Ω we may always use the equation for uat Tεzε,ρ .

The following result is an obvious corollary of the previous statement.

Corollary 3.8. Let Ω be bounded and u : Ω → R be a continuous viscosity solutionof the Dirichlet boundary value problem

F (x, u, Du, D2u) = f(x), Ω,

u = g, ∂Ω,


where F is proper and satisfies condition (2.2). Let us assume condition (DC). Thenu is unique in the class of discontinuous solutions of the corresponding Dirichlet typeproblem.

Remark 3.9. For degenerate elliptic equations the above comparison principle isnew. Even when F is uniformly elliptic and condition (DC) holds, it somewhatstrengthens the uniqueness results in Caffarelli, Crandall, Kocan, Swiech [7], Swiech[35] and the references therein where uniqueness is found in the class of continuousLp-viscosity solutions.

The following example shows that discontinuous solutions may exist after all, thusuniqueness is not always ensured by Corollary 2.8. Let us consider the boundaryvalue problem

u − xu′ = f(x), ] − 1, 1[,

u(±1) = 0,

where f(x) = 1, for x > 0, and f(x) = 0 for x ≤ 0. It is easy to verify that thefunction

u(x) =

0, x ≤ 0,

1 − x, x > 0

is a viscosity solution of the problem. However we can still use Corollary 2.8 tostate that the problem has no continuous solution. For a more detailed discussionabout existence of discontinuous solutions to boundary value problems, we refer thereader to the papers by Garavello and the author [13], [33], showing for instancethat u above is the unique lower semicontinuous solution.

Remark 3.10. Notice that the example above does not contradict the regularityCorollary 2.6. Indeed what follows from that statement is that, if Ω is split into tworegions Ω± separated by a Lipschitz hypersurface Γ, a solution which is piecewisecontinuous across Γ of a Dirichlet boundary value problem, and which is continuouson ∂Ω, is required to jump in the same direction as f , like in the example above. Ajump in the opposite direction would force continuity in view of Corollary 2.6, seealso the two dimensional example in [34].

4. Approximations and Existence. In this section we deal with two differentapplications of the comparison principle, namely the construction of approximationsand the existence of continuous solutions for which the uniqueness Corollary 2.8applies. Concerning existence, we will give here only a few examples for certainclasses of equations, but we will outline a general method based on Corollary 2.6which we plan to pursue for more general existence results in a future paper. Werefer the reader also to the paper [34] where this idea is used in the case of the eikonalequation. The main examples that we describe below are the cases of uniformlyelliptic pdes, linear subelliptic operators, and of first order equations.

As an example of approximations, we will consider the classical vanishing viscos-ity method, i.e. we study the equation

−ε∆u(x) + F (x, u(x), Du(x), D2u(x)) = f(x), x ∈ Ω (4.1)

which is uniformly elliptic. We couple it with a boundary condition of the form

u(x) = gε(x), x ∈ ∂Ω, (4.2)

(we notice that in this section subscripts have nothing to do with inf-convolution).The Dirichlet boundary value problem for a uniformly elliptic operator can be solvedalso with a discontinuous inhomogeneous term. The theory starts with [7], see also[12] and Swiech [35]. In particular a general existence result is the following.


Theorem 4.1. (Swiech [35]) Suppose that Ω is open, bounded with Lipschitz con-tinuous boundary and let gε ∈ C(∂Ω). Assume that the Hamiltonian F is proper andsatisfies condition (2.2). Let f ∈ L∞(Ω). Then there exists a unique L∞-viscositysolution uε ∈ C(Ω) of the Dirichlet problem (4.1-2).

We notice that Theorem 3.1 as formulated in [35] applies to more general uni-formly elliptic operators than (4.1). When assumption (DC) applies to f our resultCorollary 2.8 extends uniqueness to the Dirichlet-type boundary value problem andto discontinuous viscosity solutions (not necessarily L∞-solutions). For the notionof L∞ viscosity solution that appears in Theorem 3.1 and which is a stronger no-tion than viscosity solution used in this paper, we refer the reader to the mentionedreferences and to the appendix.

The main question we are interested in concerns the convergence of the familyof solutions uε. In the presence of L∞ estimates on the family uε, one canconstruct bounded functions (half relaxed limits)

v(x) = lim infε→0+,y→x

uε(y), w(x) = lim supε→0+,y→x

uε(y). (4.3)

Their use is contained in the following statement.

Proposition 4.2. Let Ω be open, bounded with Lipschitz continuous boundary,assume that the Hamiltonian F is proper and satisfies condition (2.2), and that(DC) holds. Let us suppose that the Dirichlet boundary value problem (3.1) hasviscosity super and subsolutions u, u ∈ C(Ω) with u = u = g on ∂Ω and that gε → guniformly in ∂Ω. If the family of solutions uε of the approximating problems(4.1-2) satisfies ‖uε‖ ≤ C for all ε > 0, then, with the notation in (4.3),

w ≤ u, u ≤ v, in Ω.

Proof. We first notice that v, w are respectively viscosity super and subsolutions ofthe Dirichlet-type boundary value problem, see Definition 1.4. The proof of thisfact is virtually identical to that of Theorem 1.7 and Proposition 4.7 of Chapter 5in [2], see also [11]. At this point we apply Theorem 2.7 and conclude.

We have several ways to use Proposition 3.2. The first classical method seeksexistence of solutions to (3.1) by using further estimates on the modulus of con-tinuity for the family uε in order to apply Ascoli-Arzela Theorem. When f isdiscontinuous and F is degenerate, finding such estimates is largely open, but theyare proved for instance by Krylov [24] for linear subelliptic operators. Namely heshows that when gε(x) ≡ 0 and

F (x, u(x), Du(x), D2u(x)) = u(x) − σik(x)(σjk(x)uxj (x))xi − bi(x)uxi(x) (4.4)

(here we use the summation convention on the repeated indices), where σk =(σik)i, b = (bi)i are C∞ vector fields, k = 1, . . . , M , (M ≤ N ,) and the Lie al-gebra generated by σk, b : k = 1 . . . , M has dimension N at every point in R

N ,one can estimate for a given f ∈ L∞, for some α ∈ (0, 1) small enough,

supΩ

uε ≤ C, supx,y∈Ω1

|uε(x) − uε(y)| ≤ C|x − y|α

for any Ω1 ⊂⊂ Ω, where uε is the family of solutions of (4.1-2). One knows thatuε ∈ W 1,p(Ω)for any p ∈ (1,∞), hence uε is a strong solution of (4.1) and a viscosity


solution, see Proposition A.4 and Remark A.5. Thus, at least along a subsequence,one can find u = u = u ∈ C(Ω) such that uε → u locally uniformly in Ω and in factu is locally Holder continuous. By using Proposition 3.2 we therefore obtain thatu ∈ C(Ω) is a viscosity solution (thus unique in view of Corollary 2.8) of (3.1). Weobtain the following.

Proposition 4.3. In the assumptions of Proposition 3.2, suppose moreover that Fis the subelliptic operator (4.4). Then problem (3.1) has a unique continuous vis-cosity solution which is the uniform limit of the vanishing viscosity approximations.

We can use Proposition 3.2 in an alternative way if we know in advance by othermethods that (3.1) has a unique continuous viscosity solution. For instance byapplying Corollary 2.6 and 2.8 as we will do below for first order equations. Thenwe can apply Proposition 3.2 by choosing u = u = u. The conclusion is that v = w,that is the following holds.

Proposition 4.4. In the assumptions of Proposition 3.2, suppose moreover thatproblem (3.1) has a (unique) viscosity solution u ∈ C(Ω). Then u is the uniformlimit of the family of vanishing viscosity approximations uε.

We discuss now the case of first order convex equations of the form

F (x, r, p, X) = r + maxa∈A

−c(x, a) · p − h(x, a),

where we assume throughout the rest of this section that A is a compact subset ofa metric space, c : Ω × A → R

N is continuous and c(·, a) is Lipschitz continuous,uniformly in a, h : Ω×A → R is continuous and h(·, a) is uniformly continuous witha uniform modulus of continuity with respect to a. We also suppose that f, h arebounded. Thus our equation is the Bellman equation of an optimal control problemfor the dynamical system

y = c(y, a), y(0) = x,

where a : [0, +∞[→ A is measurable. The control problem has as payoff functional

J(x, a) =

∫ τx(a)

0

e−t(h(y, a) + f(y)) dt + e−τx(a)g(y(τx(a))) → min,

where g : ∂Ω → R is continuous and τx(a) = inft ≥ 0 : y(t) /∈ Ω. We will indicateas y(·, a) the trajectory of the dynamical system.

If we consider the Dirichlet-type problem

u(x) + maxa∈A−c(x, a) · Du(x) − h(x, a) = f(x), x ∈ Ω;

u(x) = g(x) or u(x) + maxa∈A−c(x, a) · Du(x) − h(x, a) = f(x), x ∈ ∂Ω,

then a standard application of viscosity solutions theory to optimal control, see [2],produces that the value functions

VM (x) = infa(·)

∫ τx(a)

0

e−t(h(y, a) + f∗(y)) dt + eτx(a)g(y(τx(a))),

Vm(x) = infar(·)

∫ τx(ar)

0

e−t(hr(y, ar) + f∗(y)) dt + eτx(ar)g(y(τx(ar)))


are both discontinuous solutions of such a boundary value problem, see also [33].In the case of function Vm, instead of measurable a(·), we are using the so calledrelaxed controls ar : [0, +∞[→ Ar , ar ∈ L∞, where Ar is the set of Radon proba-bility measures on A and hr(x, ρ) =

∫

Ah(x, a)dρ(a). In the considered assumptions

these two functions are respectively upper semicontinuous and maximal, lower semi-continuous and minimal viscosity solutions of the boundary value problem, see e.g.[33]. By using the comparison theorems of Section 2, we want to give new sufficientconditions for existence of a unique continuous viscosity solution. Such conditions,based on Corollary 2.6, widely extend our previous results in [33]. The first state-ment is the following, where to simplify the arguments we will suppose that Γ is asmooth hypersurface.

Proposition 4.5. Let us suppose that the assumption (DC) holds and that Γ is asmooth hypersurface. At x ∈ Γ∩Ω let n(x) be the unit inward normal vector to Ω+

and let us suppose that:

there is a sequence of control functions an(·), positive numbers tn → 0+,

k > 0 and vector v ∈ RN such that v · n(x) < 0 and

xn = y(tn, an) = x + (tn)kv + o((tn)k), as n → +∞.

(3.5)

Then the value function VM is continuous at x in the direction of v (thus from Ω−).

Proof. By the Dynamic Programming Principle, see [2], for n sufficiently large, wemay write

VM (x) ≤∫ tn

0

e−t(h(y(t, an), an) + f∗(y(t, an))) dt + e−tnVM (y(tn, an))

≤ Mtn + e−tnVM (xn),

where M is a bound for |h| and |f |. As n → +∞ we obtain

VM (x) ≤ lim infn→+∞

VM (xn) ≤ lim supn→+∞

VM (xn) ≤ VM (x),

since VM is upper semicontinuous.The conclusion comes by construction of the sequence xn.

To discuss assumption (4.5), we notice that if at x we can find a control a suchthat f(x, a) · n(x) < 0, then by choosing v = f(x, a), an(t) ≡ a, tn = 1

nwe obtain

xn = x +1

nv + o(

1

n).

Thus (4.5) is satisfied with k = 1.To obtain a version of (4.5) with larger exponent k, one can use the Lie algebra

generated by the vector field c. Let us suppose for instance that c is symmetric,i.e. A ⊂ R

K is symmetric and c(x,−a) = −c(x, a) for all x, a and that there aretwo controls a1, a2 such that the Lie bracket [c(·, a1), c(·, a2)](x) · n(x) < 0. Thenby setting v = [c(·, a1), c(·, a2)](x), k = 2, we can find a sequence of controls an(·)such that (4.5) is satisfied. This is well known by setting an(t) = a1 for t ∈ [0, 1

n],

an(t) = a2 for t ∈ [ 1n, 2

n], an(t) = −a1 for t ∈ [ 2

n, 3

n], an(t) = −a2 for t ∈ [ 3

n, 4

n].

In general, assumption (4.5) is exactly what one can obtain by choosing as v asabove an appropriate Lie bracket of order k. For this fact the reader can consult thepaper by Haynes-Hermes [14], or for a more up to date discussion and refinementsthe work by Rampazzo-Sussmann [29]. We now prove the following statement.


Proposition 4.6. Let us suppose that the assumption (DC) holds and that Γ is asmooth hypersurface. At x ∈ Γ∩Ω let n(x) be the unit inward vector to Ω+ and letus suppose that:

there is an optimal (relaxed) control ar(·), for Vm(x),

positive numbers tn → 0+, and vector v ∈ RN such that v · n(x) > 0 and

xn = y(tn, ar) = x + tnv + o(tn), as n → +∞.

(3.6)

Then the value function Vm is continuous at x in the direction of v (thus from Ω+).

Proof. We use again the Dynamic Programming Principle for Vm and deduce that,for n sufficiently large

Vm(x) =

∫ tn

0

e−t(h(y(t, ar), ar)+ f∗(y(t, ar))) dt+ e−tnVm(xn) ≥ −Mtn +Vm(xn),

where M is a bound for h and f . As n → +∞ we obtain

Vm(x) ≥ lim supn→+∞

Vm(xn) ≥ lim infn→+∞

Vm(xn) ≥ Vm(x),

since Vm is lower semicontinuous.

To comment on the assumptions of Proposition 3.6, observe that

xn − x

tn=

1

tn

∫ tn

0

c(y, ar)dt =1

tn

∫ tn

0

c(x, ar)dt + o(1) = vn + o(1),

as n → +∞, where vn ∈ co c(x, A). This implies that vn → v ∈ co c(x, A)and therefore a necessary condition for (4.6) is that there exists a ∈ A such thatc(x, a)·n(x) > 0. A clearly sufficient condition is instead the existence of 1 > c, d > 0such that

c(x, A) ⊂ x : |x − x − εn(x)| < cε, d ≤ ε < c.We conclude this example with the following existence Theorem.

Theorem 4.7. Let us suppose that the assumption (DC) holds. Assume that theDirichlet boundary value problem

u(x) + maxa∈A−c(x, a) · Du(x) − h(x, a) = f(x), x ∈ Ω;

u(x) = g(x), x ∈ ∂Ω(3.7)

has continuous super and subsolutions, respectively u, u : Ω → R such that u(x) =u(x) = g(x) for x ∈ ∂Ω. If at any x ∈ (Γ ∩ Ω) either (4.5) or (4.6) is satisfied,then (4.7) has a unique continuous solution (either VM or Vm) and it is the uniformlimit of vanishing viscosity approximations.

Proof. It is known that VM , Vm are discontinuous solutions of the Dirichlet typeproblem version of (4.7), see [33]. We first use Theorem 2.7 to compare u with(VM )∗ and VM with u. This gives u ≤ VM ≤ u, thus VM continuously extendsto the points of ∂Ω taking up the boundary condition. Similarly this is also thecase for Vm. At this point the other assumption allows us to use Theorem 2.3 andcompare the subsolution VM with the supersolution Vm which turn out to be equal,


and thus continuous solutions of (4.7). Finally Corollary 2.7 gives uniqueness in theclass of discontinuous solutions and Proposition 3.4 the limit of vanishing viscosityapproximations.

The existence of a continuous subsolution of the Dirichlet problem (4.7), whichis of course necessary to solve the problem, is a standard assumption and it is calledcompatibility of the boundary datum. The existence of a continuous supersolutionis also well understood and it can be obtained for instance, when ∂Ω is a smoothhypersurface, with the request that at each point x ∈ ∂Ω there is an outward vectorfield, that is a control a ∈ A such that nΩ(x) · c(x, a) > 0, where nΩ is the outwardnormal to Ω at x, or a similar condition on a Lie bracket. For this standard fact werefer the reader for instance to the book [2]. Here we prefer to concentrate on thenew issues raised by the presence of the discontinuous coefficient.

5. Extensions to More General Hamiltonians. In this section we want toextend the comparison principles to a more general class of equations, namely

F (x, u(x), Du(x), D2u(x)) = f(x)H(x, Du(x)). (5.1)

The statement requires more regularity on the discontinuous coefficient f and, ac-cording to that, it is divided into cases.

Theorem 5.1. Let Ω be a bounded domain with Lipschitz boundary. Assume thatthe Hamiltonian F is proper and satisfies condition (2.2). Let us suppose that theassumption (DC) is satisfied by f and that H : Ω×R

N → [0, +∞[ is continuous andsatisfies assumptions (2.1), (2.2). Moreover we suppose that H(x, ·) is positively 1-homogeneous for all x ∈ Ω. We also assume that f is α-Holder continuous in Ω+

and Ω− (following the notation of (DC)). Let u, v : Ω → R be respectively an uppersemicontinuous subsolution and a lower semicontinuous supersolution of

F (x, u, Du, D2u) = f(x)H(x, Du(x)), in Ω.

Suppose moreover either one of the following:1. α = 1 and u,v satisfy boundary conditions and regularity on Γ as in the statements

of Theorem 2.3 or 2.6;

2. α ∈ (12 , 1), u is β-Holder continuous with

β > 2(1 − α) (5.2)

and v satisfies the Dirichlet type boundary condition in the viscosity sense

v ≥ u or F (x, v, Dv, D2v) ≥f∗(x)H(x, Du(x)), on ∂Ω.

Then u ≤ v in Ω.

Remark 5.2. In case 2 of the previous statement, the roles of subsolution and super-solution can be exchanged as far as the required regularity is concerned. A slightlybetter looking assumption can be made if β = 1: in this case we do not need thehomogeneity of H and we can allow any α > 0. We will leave the straightforwardadaptations to the reader.


Sketch of the proof. The proof goes along as before in Theorem 2.3. The onlydifferences can be found in step 4 when we need to estimate, instead of (3.17), theterm

f∗(T εzε,ρ)H(T εzε,ρ,T εzε,ρ − zε,ρ − εηu

ε2)

− f∗(Tεzε,ρ)H(Tεzε,ρ,zε,ρ − Tεzε,ρ + εηv

ε2)

=[f∗(T εzε,ρ) − f∗(Tεzε,ρ)]H(T εzε,ρ,T εzε,ρ − zε,ρ − εηu

ε2)

+ f∗(Tεzε,ρ)[H(T εzε,ρ,T εzε,ρ − zε,ρ − εηu

ε2) − H(Tεzε,ρ,


ε2)].

(5.3)We proceed by estimating the two terms in the right hand side separately. Theestimate of the second term in (5.3) goes as follows, using (2.1), (2.2) and the lastequation in (3.2)

f∗(Tεzε,ρ)[H(T εzε,ρ,T εzε,ρ − zε,ρ − εηu

ε2) − H(Tεzε,ρ,


ε2)]

≤‖f‖∞[|H(zε,ρ,T εzε,ρ − zε,ρ − εηu

ε2) − H(zε,ρ,


ε2)|

+ ω(|T εzε,ρ − zε,ρ| + |Tεzε,ρ − zε,ρ − εηu

ε2|)

+ ω(|Tεzε,ρ − zε,ρ| + |zε,ρ − Tεzε,ρ + εηv

ε2)]

≤‖f‖∞[ω(r(zε,ρ − x) + oρ(1)) + ω(|T εzε,ρ − zε,ρ| + |Tεzε,ρ − zε,ρ − εηu

ε2|)

+ ω(|Tεzε,ρ − zε,ρ| + |zε,ρ − Tεzε,ρ + εηv

ε2)],

(5.4)where oρ(1) → 0 as ρ → 0+. At this point, the right hand side of (5.4) goes to zerowhen we take the lim sup as ρ → 0+ first and ε → 0+ next.

To estimate the first term in (5.3) we need the assumptions. Following thediscussion in step 4 of the proof of Theorem 2.3, we may suppose for instance thatalways zε,ρ ∈ Ω−, thus T εzε,ρ ∈ Ω− along appropriate subsequences, the other casebeing similar. Having established this fact, we either find appropriate subsequencessuch that Tεzε,ρ ∈ Ω+, or we may suppose that always Tεzε,ρ ∈ Ω− ∩ Γ. In theformer case we have that

f∗(T εzε,ρ) − f∗(Tεzε,ρ) = f(T εzε,ρ) − f(Tεzε,ρ) < 0

for ε, ρ sufficiently small by assumption (DC), and then the first term in the righthand side of (5.3) is also negative. In the latter case by the assumption on Holdercontinuity of f we can estimate

[f∗(T εzε,ρ) − f∗(Tεzε,ρ)]H(T εzε,ρ,T εzε,ρ − zε,ρ − εηu

ε2)

≤ Cf |T εzε,ρ − Tεzε,ρ|αH(T εzε,ρ,T εzε,ρ − zε,ρ − εηu

ε2)

= CfH(T εzε,ρ, |T εzε,ρ − Tεzε,ρ|αT εzε,ρ − zε,ρ − εηu

ε2).


We succeed if we can show that |T εzε,ρ − Tεzε,ρ|α T εzε,ρ−zε,ρ−εηu

ε2 → 0, as we letρ → 0 and then ε → 0.

In case 1 with α = 1 we know that |T εzε,ρ − Tεzε,ρ|/ε is bounded by (3.12) and

thatT εzε,ρ−zε,ρ−εηu

ε= o(1) by (3.10) so we conclude.

In case 2 from one hand we have, by the last equation in (3.2)

|T εzε,ρ − Tεzε,ρ|α

≤ |2(T εzε,ρ − zε,ρ − εηu) + ε(ηu − ηv) − rε2(zε,ρ − x) + ε2oρ(1)|α

≤ 2α|T εzε,ρ − zε,ρ − εηu|α + εα[(ηu − ηv) − rε(zε,ρ − x) + εoρ(1)]α,

where oρ(1) → 0 as ρ → 0. Here, by (3.10), the right hand side is of the order O(εα)when taking ρ → 0 and then ε → 0. On the other hand, by the first equation in(3.6) and the assumed regularity of u,

|T εzε,ρ − zε,ρ − εηu|ε2

≤1

ε

√

2Cu|T εzε,ρ − zε,ρ|β2

≤ 1

ε1− β2

√

2Cu

[

∣

∣

∣

∣

T εzε,ρ − zε,ρ − εηu

ε

∣

∣

∣

∣

β2

+ 1

]

,

where Cu is the Holder constant of u. Thus the right hand side is of the order

O(εβ2−1). We win as soon as α > 1 − β

2 .

Appendix. We collect here standard basic facts that we used in the paper, andsome auxiliary results to compare viscosity solutions with other notions in the liter-ature. We start with some known properties of semiconvex (or semiconcave) func-tions, that are helpful when dealing with nonlinear convolution. The first proposi-tion exhibits a key property of semiconvex regularization to be used in the study ofsecond order pdes. Initially due to Jensen [18], a simple proof of this fact can befound in [12].

Proposition A.1. Let w : Ω → R be upper semicontinuous, −w satisfying (2.3).Then the function wε is 1

2ε2 semiconvex, hence, if at x the function wε is twicedifferentiable, then

D2wε(x) ≥ 1

ε2I.

If moreover x ∈ RN and (p, X) ∈ D2,+wε(x), then for every n × n real matrix T

we have

(p,1

ε2(I − T t)(I − T ) + T tXT ) ∈ D2,+w(x),

where x = x − εη + ε2p is the unique point at which wε(x) = w(x) − 12 |x−x

ε+ η|2.

If, in particular, X > − 1ε2 I, then by choosing T = (I + ε2X)−1 we get

(p, (I + ε2X)−1X) ∈ D2,+w(x).

The next result is a classical regularity result, for its proof see also [11].

Theorem A.2. (Aleksandrov [1]) Let w : RN → R be a convex (semiconvex)

function. Then w is twice differentiable almost everywhere.

The last result describes a perturbation property of local maxima of semiconvexfunctions.


Lemma A.3. (Jensen [18]) Let ϕ : RN → R be semiconvex and let x be a strict lo-

cal maximum point of ϕ. Then ϕ satisfies the following upper perturbation property:for r, k > 0 the set

Γ+(ϕ, x) = x ∈ B(x, r) : x is a local max for ϕ(·)− < p, · > for some |p| ≤ k

has positive measure.

We next relate the notion of viscosity solution with other notions in the literaturefor (1.1). We say that a function u ∈ C1,1(Ω) is a strong solution of

F (x, u, Du, D2u) = f(x), (A.1)

if it satisfies the equation almost everywhere.We also say that an upper semicontinuous function u : Ω → R is an L∞-viscosity

subsolution of (A.1) (see [7]) if for any test function ϕ ∈ C1,1(Ω) and a localmaximum point xo of u − ϕ, (u − ϕ)(xo) = 0, we obtain

ess − lim infx→xo

[

F (x, ϕ(x), Dϕ(x), D2ϕ(x)) − f(x)]

≤ 0,

where we are using the essential lim inf in the sense of Lebesgue measure. Thereare some immediate facts. It is clear that when F is continuous, an L∞-viscositysubsolution of (A.1) is also a viscosity subsolution in the sense of Section 2, sincethe former uses a bigger family of test functions and does not take into account setsof measure zero when constructing the envelopes of the discontinuous term f , i.e.

ess- lim supx→xo

f(x) ≥ f∗(xo).

Moreover if u ∈ C1,1 is a viscosity subsolution of (A.1), and the jump set Γ of fhas measure zero, then u is also a strong solution. If Γ has positive measure, theprevious property may not be necessarily true. The following result completes thepicture.

Proposition A.4. If F is proper, any strong subsolution of (A.1) is an L∞-viscosity subsolution, thus also a viscosity subsolution.

Proof. Suppose not. Then we may find ϕ ∈ C1,1(Ω) and a strict local maximumpoint xo of u − ϕ, (u − ϕ)(xo) = 0 such that

ess − lim infx→xo

[

F (x, ϕ(x), Dϕ(x), D2ϕ(x)) − f(x)]

> 0.

Thus we may find ε > 0 such that

F (x, ϕ(x), Dϕ(x), D2ϕ(x)) ≥ f(x) + ε, a.e. in B(xo, ε) ⊂ Ω. (A.2)

As u−ϕ is locally semiconvex, we may apply Lemma A.3 and for all σ > 0 we findout that the set

Γ+(u − ϕ, xo)

= x ∈ B(xo, ε) :u(·) − ϕ(·)− < p, · > has a local max at x for some |p| ≤ σ


has positive measure. We now pick any point x ∈ Γ+(u − ϕ, xo) where both u, ϕare twice differentiable and the equation is satisfied pointwise by u. Then we havethat Du(x) = Dϕ(x)+p, for some p, |p| ≤ σ, D2u(x) ≤ D2ϕ(x), and by degenerateellipticity

F (x, u(x), Dϕ(x), D2ϕ(x)) − ω(σ) ≤ F (x, u(x), Dϕ(x) + p, D2ϕ(x)) ≤ f(x)

on a positive measure set of B(xo, ε). This contradicts (A.2) by taking σ smallenough.

Remark A.5. It is very important to notice that the definition of strong solution andthe result of Proposition A.4 can be extended to functions in W 2,p(Ω) for p > Nbecause this is a class of continuous, almost everywhere twice differentiable functionswhich satisfy the upper and lower perturbation properties, similar to Lemma A.3.These are however deep facts, see [7] and the references therein.

Let us suppose that the discontinuous term f is piecewise continuous across asmooth hypersurface Γ in the sense of assumption (DC), that u is a strong solutionof (A.1) and it is piecewise C2 across Γ in the sense that u ∈ C1(Ω) ∪ C2(Ω+ ∪Γ) ∪ C2(Ω− ∪ Γ), where the notation Ω± comes from assumption (DC). We startobserving the following.

Lemma A.6. Let u be piecewise C2 across a hypersurface Γ, then at xo ∈ Γ ∩ Ω,denoted by D2u(x+

o ), D2u(x−o ) the Hessians of u from the two sides of the hyper-

surface, they are comparable in the order of symmetric matrices.

Proof. Let v ∈ TxoΓ the tangent space, and γ : (−1, 1) → Γ be a smooth curve with

γ(0) = xo, γ(0) = v. Then

d

dtDu(γ(t))|t=0 = v · D2u(x+

o ) = v · D2u(x−o ).

Thus if we choose as a base for RN the set v1, . . . , vn−1, n(xo) where the first n−1

vectors are a base of TxoΓ and n(xo) is a normal unit vector of Γ at xo, then the

only possible unequal element of the two matrices D2u(x+o ), D2u(x−

o ) is the elementof place n × n, i.e. n(xo) · D2u(x+

o )n(xo), or n(xo) · D2u(x−o )n(xo), respectively.

Thus the two matrices are comparable.

Piecewise C2 strong solutions satisfy the equation in a much stronger sense.

Proposition A.7. Let f be piecewise continuous across a smooth hypersurface Γand let u be a strong solution which is piecewise C2 across Γ. Then u satisfies

F (x, u(x), Du(x), D2u(x)) ≤ f∗(x),

F (x, u(x), Du(x), D2u(x)) ≥ f∗(x),

in the viscosity sense.

Proof. Since u is a strong solution and it is piecewise C2, then at xo ∈ Γ, by takinglimits from the sides we get

F (xo, u(xo), Du(xo), D2u(x+

o )) = f∗(x),

F (xo, u(xo), Du(xo), D2u(x−

o )) = f∗(x).


By Lemma A.6, and by ellipticity of the operator F , we then get

D2u(x+o ) < D2u(x−

o ).

Then it is clear that super and superjets of u are given by

D2,+u(xo) = (Du(xo), X) : X ≥ D2u(x−o ),

D2,−u(xo) = (Du(xo), Y ) : Y ≤ D2u(x+o )

from which the conclusion by ellipticity of the operator F .

The situation of Proposition A.7 is that, for instance, of uniformly elliptic piece-wise continuous equations as in Kutev-Lions [25]. Our framework is however sim-plified with respect to that paper. Notice that from the proof of Proposition A.7 itfollows that, if at xo ∈ Γ, F is degenerate in the direction of n(xo), namely

F (xo, u(xo), Du(xo), X + cn(xo) ⊗ n(xo)) = F (xo, u(xo), Du(xo), X),

for all X ∈ SN , c ∈ R, then the solution of equation (A.1) cannot be piecewise C2

across Γ.

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E-mail address: [email protected]

Received January 2005; revised November 2005.

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Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients