Partial Differential Equations

http://dx.doi.org/10.1090/pspum/004

Proceedings of Symposia in

PURE MATHEMATICS

Volume 4

Partial Differential Equations Charles B. Morrey, Jr. Editor

Yfe American Mathematical Society £ i i i u | | ii it ji ^ Mm <*

%\ II11IIII / ? Providence, Rhode Island

PROCEEDINGS OF THE FOURTH SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT THE UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA APRIL 21-22, 1960

with the Support of the UNITED STATES AIR FORCE

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CONTENTS

Extensions and Applications of the De Giorgi-Nash Results. By CHARLES B. MORREY, JR 1

Dirichlefs Principle in the Calculus of Variations. By JAMES SERRIN 17

Associated Spaces, Interpolation Theorems and the Regularity of Solutions of Differential Problems.

By N. ARONSZAJN 23

Lebesgue Spaces of Differentiate Functions and Distributions. By A. P. CALDERON 33

The Majorant Method. By P. C. ROSENBLOOM . 5 1

A Priori Estimates for Elliptic and Parabolic Equations. By FELIX E. BROWDER 73

Differential Equations in Hilbert Spaces. By FRANCOIS TREVES 83

A Maximum Property of Cauchy's Problem in Three-Dimensional Space-Time.

By H. F. WEINBERGER 91

Comments on Elliptic Partial Differential Equations. By Louis NIRENBERG 101

Some Unusual Boundary Value Problems. By MARTIN SCHECHTER 109

A New Proof and Generalizations of the Cauchy-Kowalewski Theorem to Nonanalytic and to Non-normal Systems.

By AVNER FRIEDMAN . . . . . 1 1 5

Regularity of Continuations of Solutions. By FRITZ JOHN 121

Some Local Properties of Elliptic Equations. By DAVID GILBARG 127

Estimates at Infinity for Steady State Solutions of the Navier-Stokes Equations.

By ROBERT FINN 143

VI CONTENTS

Interior Estimates for Solutions of Elliptic Monge-Ampere Equations. By ERHARD HEINZ 149

Zero Order A Priori Estimates for Solutions of Elliptic Differential Equations. By H. O. CORDES . . . 157

INDEX 167

INDEX

A priori estimates, 73-74, 138 for elliptic and parabolic boundary value

problems, 73 for mixed initial-boundary value prob­

lems for general parabolic equations, 73

A priori Holder estimate, 127, 132 Adjoint

boundary problem, 76 pairing, 26

Admissible (norm), 26 Associated

norms,^23, 26-27 space F , 23 spaces, 23, 26-27

Asymptotic behavior, 133, 139 expansions, 133, 139

Bernstein's inequality, 58 Bessel potentials, 23, 29, 33

class Pa of of order a in B", 29 Boundary conditions, 111

overdetermined, 112 underdetermined, 112

Boundary value problems, 109 adjoint of, 76 estimates for, 103 exterior, 144, 147 Leray's, 144 mixed initial-, 78 regular, 74 (See also Elliptic)

Calculus of variations, existence theorems in, 20

Canonical pairing [£*, B, <&*, &>], 28 self-pairing, 28

Cauchy initial value problem, 101 -Kowalewski system, 115 problem, 51

Characteristics, 102

Class

Pa of Bessel potentials of order a in B", 29

Comparison arguments, 128 function, 130

Compatible (norms), 25 Compensating function, 31 Compensation method, 24, 31 Conditionally coercive (quadratic forms),

24 Conjugate norm, 26 Correct operators, 86 Curvature, 96

V<p, 5 Derivative of an operator, 87 Dini continuous, 128 Dirichlet

problem, 84, 104, 137 system, 112

Dirichlet Integral, 144, 147 of order j3 in D, 30

Distribution, 109 Domain

G of class C1, 4 of polyhedral type and class Cy**\ 30

Elliptic equations, 109 operator, 102, 109 systems of second order, 149 uniformly, 128

Elliptic boundary value problems existence of solutions to — in

operator-theoretic terms, 76 regularity of solutions of, 73

Ellipticity, non-uniform, 134 Energy inequalities, 86 Equation

of the type d/dt + A(t), 89 of waves, 121 second order of the type d2/dfi +

B(t)d/dt + A{t), 89

168 INDEX

Equivalent norms (J ||i ~ || ||2>» 2 9

Estimates for boundary value problems, 103 of Di Giorgi-Nash type, 73 zero order, 158 (See also A priori)

Existence of solutions to elliptic boundary value

problems in operator-theoretic terms, 76

theorems in the calculus of variations, 20

Extended maximum principle, 128 Extension theorem, 24, 30 Exterior boundary value problem, 144,

147 Extremal, 8

Force, 146 Frechet derivative, 57 Fundamental solution tensor, 144, 146

General restriction theorem, 31 Generalized solutions, 109 Goursat problem, 51, 71, 116

Harnack inequality, 132, 134 Hyperbolic, 101

Initial value problems, 85 Integral

equation, 93 singular operators, 102

Interpolated norms, 25 Interpolation of norms, 23 Isolated singular points, 131 Isomorphism {St T) of the pairing

[F, W, (v, wy] into (or onto) [V, W, <v', «;'>'], 28

K€- [K'€-] condition, 159 with respect to the symmetric positive

definite n x n matrix C = ((ctVfc)), 161 Kowalewski-Cauchy system, 115

Leray exterior boundary value problem of, 144 -Schauder fixed point theorem, 138

Limit theorem, 132 Liouville, 147

theorem, 134-135

Majorant method, 51 Majorants, 62 Maximum

principle, 21, 106, 128 property, 91, 95

Minimal surface equation, 136, 140 Mixed

initial-boundary value problem, 78 problems in cylindrical domains, 83

Monge-Ampere equations, 149

Navier-Stokes equations, 143 Neighborhood of infinity, 143 Neumann problem, 84 Newton's method, 54 Nonlinear equations—removable singu­

larities, 135 Non-normal systems, 116 Non-uniform ellipticity, 134 Norms

associated, 23, 26 conjugate, 26 equivalent, 29 interpolated, 23, 25 standard IMU,D» 30

Normal (boundary conditions), 109

Operator correct, 86 derivative of, 87 elliptic, 109 singular integral, 102

Pairing, 23, 26 adjoint, 26 of vector spaces, 23 proper, 26 (See also Canonical and Isomorphism)

Perturbation methods, 54 Properly elliptic, 109

Quadratic interpolation, 24-26 theorem, 26

Quasi-linear parabolic equation, 79

INDEX 169

Regular boundary value problem, 74 Regularity

of continuations (of solutions), 121 of solutions of elliptic boundary \

problems, 73 Removable singularities, 130 Restriction theorem, converse to, 31

Singular integral operators, 102 Space

associated, 23, 26-27 associated V, 23 G\(G)y 1 Cj>h(G)y 74 @l(E), 89 H\ 85 Hl(D), 2

LUEn), 35 AS, 37 Lipc (G), 4 P * ^ " ) , 30 P&(^n), 30 W'*[G), 74

Standard norm ||w||afz>, 30

Stokes, Navier- equations, 143 Stress tensor, 145 Stronger (norm) (>-), 28 Strongly hyperbolic, 101 Subsonic flows, 139 Surface area, 19

Telescopic series, 125

Variational integral definition of, 17-18 lower semicontinuity of, 18

Variational problems, 17 Velocity field, 143

Wake, 146 Wave equation, 92, 121 W7eak solution, 128 Weaker (norm) (<), 28 Well posed problem, 121

Zero order estimate, 158