Boundary Stabilization of the Wave Equation with Time-Varying and Nonlinear Feedback

Research ArticleBoundary Stabilization of the Wave Equation withTime-Varying and Nonlinear Feedback

Jian-Sheng Tian12 Wei Wang12 Fei Xue12 and Pei-Yong Cong12

1 College of Computer Science Beijing University of Technology Beijing 100124 China2 State Engineering Laboratory of Information System Classified Protection Key Technologies Beijing University of TechnologyBeijing 100124 China

Correspondence should be addressed to Jian-Sheng Tian jian-shengtianoutlookcom

Received 24 April 2014 Accepted 7 June 2014 Published 29 June 2014

Academic Editor Yoshinori Hayafuji

Copyright copy 2014 Jian-Sheng Tian et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying and nonlinearterm By the Riemannian geometry methods and a suitable assumption of nonlinearity and the time-varying term we obtain theuniform decay of the energy of the system

1 Introduction

There aremany results concerning the boundary stabilizationof classical wave equations See [1ndash6] for linear cases and [7ndash12] for nonlinear onesThe stability of the wave equation withvariable coefficients has attracted much attention See [13ndash23] and many others In [20] by the methods in [11 24]the authors study the stability of the wave equation withnonlinear term and time-varying term However under thecondition the nonlinear term has upper bound and the time-varying term has lower bound the stability of the waveequation was not studied in [20] In this paper our purpose isto study the stability of thewave equation under the conditionthe nonlinear term has upper bound and the time-varyingterm has lower bound

LetΩ be a bounded domain inR119899 with smooth boundaryΓ It is assumed that Γ consists of two parts Γ

1and Γ2(Γ =

Γ1cup Γ2) with Γ

2= 0 Γ1cap Γ2= 0 Define

A119906 = minus div119860 (119909) nabla119906 for 119906 isin 1198671 (Ω) (1)

where div is the divergence operator of the standardmetric ofR119899 119860(119909) = (119886

119894119895(119909)) is symmetric positively definite matrices

for each 119909 isin R119899 and 119886119894119895(119909) are smooth functions on R119899

We consider the stabilization of the wave equations withvariable coefficients and time-varying delay in the dissipativeboundary feedback

119906119905119905+A119906 = 0 (119909 119905) isin Ω times (0 +infin)

119906 (119909 119905)|Γ2

= 0 119905 isin (0 +infin)

120597119906 (119909 119905)

120597]A+ 120601 (119905) 119892

1119906119905(119909 119905) = 0 (119909 119905) isin Γ

1timesisin (0 +infin)

119906 (119909 0) = 1199060(119909) 119906

119905(119909 0) = 119906

1(119909) 119909 isin Ω

(2)

1198921isin 119862(R) and there exists a positive constant 119888

1such that

1198921(0) = 0 119904119892

1(119904) le |119904|

2 for 119904 isin R10038161003816100381610038161198921 (119904)

1003816100381610038161003816 ge 1198881 |119904| for |119904| gt 1(3)

and 120601(119905) isin 119862([0 +infin)) satisfies

120601 (119905) ge 1206010

forall119905 ge 0 (4)

lim119905minusgt+infin

119865 (119905)

119905= 0 (5)

where 1206010is a positive constant and 119865(119905) = max

0le120588le119905120601(120588)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 176583 5 pageshttpdxdoiorg1011552014176583

2 Mathematical Problems in Engineering

120597119906120597]A is the conormal derivative120597119906

120597]A= ⟨119860 (119909) 119906 ]⟩ (6)

where ⟨sdot sdot⟩ denotes the standard metric of the Euclideanspace R119899 and ](119909) is the outside unit normal vector for each119909 isin Γ Moreover the initial data (119906

0 1199061) belongs to a suitable

spaceDefine the energy of the system (2) by

119864 (119905) =1

2intΩ

(1199062

119905+

119899

sum

119894119895=1

119886119894119895119906119909119894

119906119909119895

)119889119909 (7)

We define

119892 = 119860minus1

(119909) for 119909 isin R119899 (8)

as a Riemannian metric on R119899 and consider the couple(R119899 119892) as a Riemannian manifold with an inner product

⟨119883 119884⟩119892= ⟨119860minus1

(119909)119883 119884⟩ |119883|2

119892= ⟨119883119883⟩

119892119883119884 isin R

119899

119909

(9)

Let 119863119892denote the Levi-Civita connection of the metric

119892 For the variable coefficients the main assumptions are asfollows

Assumption A There exists a vector field119867 on Ω and a con-stant 120588

0gt 0 such that

119863119892119867(119883119883) ge 120588

0|119883|2

119892for 119883 isin R

119899

119909 119909 isin Ω (10)

Moreover we assume thatsup119909isinΩ

div119867 lt inf119909isinΩ

div119867 + 21205880 (11)

119867 sdot ] le 0 119909 isin Γ2

119867 sdot ] ge 120575 119909 isin Γ1

(12)

where 120575 is a positive constant

Assumption (10) was introduced by [13] as a checkableassumption for the exact controllability of the wave equationwith variable coefficients For examples on the condition see[13 14]

Based on Assumption (10) Assumption A was given by[19] to study the stabilization of the wave equation with vari-able coefficients and boundary condition of memory type

Define

1198671

Γ2

(Ω) = 119906 isin 1198671

(Ω) | 119906|Γ2

= 0 (13)

To obtain the stabilization of the system (2) we assume thesystem (2) is well-posed such that

119906 isin 1198621

([0 +infin) 1198712

(Ω)) cap 119862 ([0 +infin) 1198671

Γ2

(Ω)) (14)

The main result of this paper is stated as follows

Theorem 1 Let Assumption A holds trueThen there exist pos-itive constants 119862 119862

2 such that

119864 (119905) le 1198621ℎ(

1198622119864 (0)

119905) +

1198621119865 (119905)

119905119864 (0) 119905 gt 0 (15)

2 Basic Inequality of the System

In this section we work on Ω with two metrics at the sametime the standard dotmetric ⟨sdot sdot⟩ and theRiemannianmetric119892 = ⟨sdot sdot⟩

119892given by (8)

If 119891 isin 1198621

(R119899) we define the gradient nabla119892119891 of 119891 in the

Riemannian metric 119892 via the Riesz representation theoremby

119883(119891) = ⟨nabla119892119891119883⟩119892

(16)

where 119883 is any vector field on (R119899 119892) The following lemmaprovides further relations between the two metrics see [13]in Lemma 21

Lemma 2 Let 119909 = (1199091 119909

119899) be the natural coordinate

system inR119899 Let119891 ℎ be functions and letH119883 be vector fieldsThen

(a)

⟨119867(119909) 119860 (119909)119883 (119909)⟩119892= ⟨119867 (119909) 119883 (119909)⟩ 119909 isin R

119899

(17)

(b)

nabla119892119891 =

119899

sum

119894=1

(

119899

sum

119895=1

119886119894119895(119909) 119891119909119895

)120597

120597119909119894

= 119860 (119909) nabla119891 119909 isin R119899

(18)

where nabla119891 is the gradient of 119891 in the standard metric(c)

nabla119892119891 (ℎ) = ⟨nabla

119892119891 nabla119892ℎ⟩119892

= ⟨nabla119891119860 (119909) nablaℎ⟩ 119909 isin R119899

(19)

where the matrix 119860(119909) is given in formula (1)

To proveTheorem 1 we still need several lemmas furtherDefine

1198640(119905) =

1

2intΩ

(1199062

119905+10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889119909 (20)

Then we have

119864 (119905) = 1198640(119905) + 120585int

119905

119905minus120591(119905)

intΓ1

1199062

119905(119909 120588) 119889Γ 119889120588 (21)

Lemma 3 Let (119906) be the solution of system (2) Then thereexists a constant 119862

1such that

119864 (0) minus 119864 (119879) = 1198621int

119879

0

intΓ1

120601 (119905) 119906119905(119909 119905) 119892

1(119906119905(119909 119905)) 119889Γ 119889119905

(22)

where 119879 ge 0 The assertion (22) implies that 119864(119905) is decreasing

Proof Differentiating (7) we obtain

1198641015840

(119905) = intΩ

(119906119905119906119905119905+ nabla119892119906 sdot nabla119906

119905) 119889119909

= intΓ1

120601 (119905) 119906119905(119909 119905) 119892

1(119906119905(119909 119905)) 119889Γ

(23)

Then the inequality (22) holds true

Mathematical Problems in Engineering 3

3 Proofs of Theorem 1

From Proposition 21 in [13] we have the following identities

Lemma 4 Suppose that 119906(119909 119905) solves equation 119906119905119905+A119906 = 0

(119909 119905) isin Ωtimes (0 +infin) and thatH is a vector field defined onΩThen for 119879 ge 0

int

119879

0

intΓ

120597119906

120597]AH (119906) 119889Γ 119889119905

+1

2int

119879

0

intΓ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)H sdot ]119889Γ 119889119905

= (119906119905H (119906))

1003816100381610038161003816

119879

0+ int

119879

0

intΩ

119863119892H (nabla119892119906 nabla119892119906) 119889119909 119889119905

+1

2int

119879

0

intΩ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) divH119889119909119889119905

(24)

Moreover assume that 119875 isin 1198621(Ω) Then

int

119879

0

intΩ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119875119889119909 119889119905

= (119906119905 119906119875)

1003816100381610038161003816

119879

0+1

2int

119879

0

intΩ

nabla119892119875 (1199062

) 119889119909 119889119905

minus int

119879

0

intΓ

119875119906120597119906

120597]A119889Γ 119889119905

(25)

Lemma 5 Suppose that all assumptions in Theorem 1 holdtrue Let 119906 be the solution of the system (2) Then there existpositive constants 119862 119879

0for which

119864 (119879) le119862

119879int

119879

0

intΓ1

(1199062

119905+ (

120597119906

120597]A)

2

)119889Γ119889119905 (26)

where 119879 ge 1198790

Proof We let 120579 be a positive constant satisfying

1

2sup119909isinΩ

div 119867 lt 120579 lt1

2inf119909isinΩ

div 119867 + 1205880 (27)

Set

H = 119867 119875 = 120579 minus 1205880 (28)

Substituting the identity (25) into the identity (24) we obtain

ΠΓ= (119906119905 119867 (119906) + 119875119906)

1003816100381610038161003816

119879

0

+ int

119879

0

intΩ

(119863119892119867(nabla119892119906 nabla119892119906) minus 120588

0

10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889119909 119889119905

+ int

119879

0

intΩ

((1

2div 119867 + 120588

0minus 120579) 119906

2

119905

+(120579 minus1

2div 119867) 10038161003816100381610038161003816nabla119892119906

10038161003816100381610038161003816

2

119892

) 119889119909 119889119905

(29)

where

ΠΓ= int

119879

0

intΓ

120597119906

120597]A(119867 (119906) + 119906119875) 119889Γ 119889119905

+1

2int

119879

0

intΓ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119867 sdot ] 119889Γ 119889119905

(30)

Decompose ΠΓas

ΠΓ= ΠΓ1

+ ΠΓ2

(31)

Since 119906|Γ2

= 0 we obtain nablaΓ119906|Γ2

= 0 that is

nabla119892119906 =

120597119906

120597]A

]A1003816100381610038161003816]A

1003816100381610038161003816

2

119892

for 119909 isin Γ2 (32)

Similarly we have

119867(119906) = ⟨119867 nabla119892119906⟩119892

=120597119906

120597]A

119867 sdot ]1003816100381610038161003816]A

1003816100381610038161003816

2

119892

for 119909 isin Γ2 (33)

Using the formulas (32) and (33) in the formula (30) on theportion Γ

2 with (12) we obtain

ΠΓ2

=1

2int

119879

0

intΓ2

(120597119906

120597]A)

2

119867 sdot ]1003816100381610038161003816]A

1003816100381610038161003816

2

119892

119889Γ 119889119905 le 0 (34)

From (12) we have

ΠΓ1

= int

119879

0

intΓ1

120597119906

120597]A(119867 (119906) + 119906119875) 119889Γ 119889119905

+1

2int

119879

0

intΓ1

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119867 sdot ] 119889Γ 119889119905

le 119862120576int

119879

0

intΓ1

(120597119906

120597]A)

2

119889Γ 119889119905

+ 120576int

119879

0

intΓ1

(1199062

+10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889Γ 119889119905

+ int

119879

0

intΓ1

(1198621199062

119905minus 120575

10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889Γ 119889119905

le 119862int

119879

0

intΓ1

(120597119906

120597]A)

2

119889Γ 119889119905 + 120576119864 (119905)

+ 119862int

119879

0

intΓ1

1199062

119905119889Γ 119889119905

(35)

Substituting the formulas (34) and (35) into the formula(29) with (27) we obtain

int

119879

0

119864 (119905) 119889119905

le 119862 (119864 (0) + 119864 (119879)) + 119862int

119879

0

intΓ1

(1199062

119905+ (

120597119906

120597]A)

2

)119889Γ119889119905

(36)


It follows from (22) that

int

119879

0

119864 (119905) 119889119905 ge 119879119864 (119879) (37)

Substituting the formulas (22) and (37) into the formula(36) the inequality (26) holds

Proof of Theorem 1 Since 119864(119905) is decreasing with (4) and(26) for sufficiently large 119879 we have

119864 (119879)

le119862

119879int

119879

0

intΓ1

(1206012

(119905) 1198922

(119906119905) + 1199062

119905) 119889Γ 119889119905

le119862

119879int

119879

0

intΓ1

(1199062

119905+ 1198922

(119906119905)) 119889Γ 119889119905

+119865 (119879)int

119879

0

intΓ1

119906119905120601 (119905) 119892 (119906

119905) 119889Γ 119889119905

le119862

119879int

119879

0

int119909isinΓ1|119906119905|le1

ℎ (119906119905119892 (119906119905)) 119889Γ 119889119905 + 119862119865 (119879) 119864 (0)

le119862

119879int

119879

0

intΓ1

ℎ (119906119905119892 (119906119905)) 119889Γ 119889119905 +

119862119865 (119879)

119879119864 (0)

le 119862meas (Γ1) ℎ(

int119879

0

intΓ1

119906119905119892 (119906119905) 119889Γ 119889119905

119879 sdotmeas (Γ1)

) +119862119865 (119879)

119879119864 (0)

le 1198621ℎ(

1198622119864 (0)

119879) +

1198621119865 (119879)

119879119864 (0)

(38)

Note that 119864(119905) is decreasing the estimate (15) holds

4 Application of the System (2)Nonlinear feedback describes a property of a physical systemthat is the response by the physical system to an appliedforce is nonlinear in its effect One of the applications ofthe system (2) is in sound waves where the system (2)describes the reflection of sound in heterogeneous materialsat surfaces of some materials with nonlinearity of interest inengineering practice Theorem 1 indicates that the energy ofthe sound waves with the reflection of sound at surfaces inheterogeneous materials at surfaces of some materials withnonlinearity is uniform decay under a suitable assumption ofthe nonlinearity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J E Lagnese ldquoNote on boundary stabilization of wave equa-tionsrdquo SIAM Journal on Control and Optimization vol 26 no5 pp 1250ndash1256 1988

[2] D L Russell ldquoControllability and stabilizability theory forlinear partial differential equations recent progress and openquestionsrdquo SIAM Review vol 20 no 4 pp 639ndash739 1978

[3] R Triggiani ldquoWave equation on a bounded domain withboundary dissipation an operator approachrdquo Journal of Math-ematical Analysis and Applications vol 137 no 2 pp 438ndash4611989

[4] Y You ldquoEnergy decay and exact controllability for the Petrovskyequation in a boundeddomainrdquoAdvances inAppliedMathemat-ics vol 11 no 3 pp 372ndash388 1990

[5] M Aassila M M Cavalcanti and V N D Cavalcanti ldquoExis-tence and uniform decay of the wave equation with nonlinearboundary damping and boundary memory source termrdquo Cal-culus of Variations and Partial Differential Equations vol 15 no2 pp 155ndash180 2002

[6] M M Cavalcanti V N Domingos Cavalcanti and P MartinezldquoExistence and decay rate estimates for the wave equationwith nonlinear boundary damping and source termrdquo Journal ofDifferential Equations vol 203 no 1 pp 119ndash158 2004

[7] F Conrad and B Rao ldquoDecay of solutions of the wave equationin a star-shaped domain with nonlinear boundary feedbackrdquoAsymptotic Analysis vol 7 no 3 pp 159ndash177 1993

[8] V Komornik ldquoOn the nonlinear boundary stabilization of thewave equationrdquo Chinese Annals of Mathematics B vol 14 no 2pp 153ndash164 1993

[9] V Komornik Exact Controllability and Stabilization The Mul-tiplier Method John Wiley amp Sons Chichester UK 1994

[10] V Komornik and E Zuazua ldquoA direct method for the boundarystabilization of the wave equationrdquo Journal de MathematiquesPures et Appliquees vol 69 no 1 pp 33ndash54 1990

[11] I Lasiecka and D Tataru ldquoUniform boundary stabilization ofsemilinear wave equations with nonlinear boundary dampingrdquoDifferential and Integral Equations vol 6 no 3 pp 507ndash5331993

[12] E Zuazua ldquoUniform stabilization of the wave equation by non-linear boundary feedbackrdquo SIAM Journal on Control and Opti-mization vol 28 no 2 pp 466ndash477 1990

[13] P F Yao ldquoOn the observability inequalities for exact con-trollability of wave equations with variable coefficientsrdquo SIAMJournal on Control and Optimization vol 37 no 5 pp 1568ndash1599 1999

[14] P-F YaoModeling andControl inVibrational and StructuralDy-namics A Differential Geometric Approach Chapman amp HallCRC Applied Mathematics and Nonlinear Science Series CRCPress Boca Raton Fla USA 2011

[15] I Lasiecka R Triggiani and P F Yao ldquoInverseobservabilityestimates for second-order hyperbolic equations with variablecoefficientsrdquo Journal of Mathematical Analysis and Applicationsvol 235 no 1 pp 13ndash57 1999

[16] Z H Ning and Q X Yan ldquoStabilization of the wave equationwith variable coefficients and a delay in dissipative boundaryfeedbackrdquo Journal of Mathematical Analysis and Applicationsvol 367 no 1 pp 167ndash173 2010

[17] Z H Ning C X Shen and X P Zhao ldquoStabilization ofthe wave equation with variable coefficients and a delay indissipative internal feedbackrdquo Journal of Mathematical Analysisand Applications vol 405 no 1 pp 148ndash155 2013


[18] Z H Ning C X Shen X Zhao H Li C Lin and Y M ZhangldquoNonlinear boundary stabilization of the wave equations withvariable coefficients and time dependent delayrdquo Applied Mathe-matics and Computation vol 232 pp 511ndash520 2014

[19] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith variable coefficients and boundary condition of memorytyperdquo Asymptotic Analysis vol 50 no 1-2 pp 31ndash67 2006

[20] B Gong and X Zhao ldquoBoundary stabilization of a semilinearwave equationwith variable coefficients under the time-varyingand nonlinear feedbackrdquo Abstract and Applied Analysis vol2014 Article ID 728760 6 pages 2014

[21] H Li C S Lin S PWang andYM Zhang ldquoStabilization of thewave equation with boundary time-varying delayrdquo Advances inMathematical Physics vol 2014 Article ID 735341 6 pages 2014

[22] B-Z Guo and Z-C Shao ldquoOn exponential stability of a semi-linear wave equation with variable coefficients under the non-linear boundary feedbackrdquoNonlinear AnalysisTheory Methodsamp Applications vol 71 no 12 pp 5961ndash5978 2009

[23] Z H Ning C X Shen and X P Zhao ldquostabilization of the waveequationwith variable coefficients and a internalmemory typerdquoNonlinear Analysis Real World Applications In press

[24] M M Cavalcanti V N D Cavalcanti and I Lasiecka ldquoWell-posedness and optimal decay rates for the wave equation withnonlinear boundary dampingmdashsource interactionrdquo Journal ofDifferential Equations vol 236 no 2 pp 407ndash459 2007

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


120597119906120597]A is the conormal derivative120597119906

120597]A= ⟨119860 (119909) 119906 ]⟩ (6)

where ⟨sdot sdot⟩ denotes the standard metric of the Euclideanspace R119899 and ](119909) is the outside unit normal vector for each119909 isin Γ Moreover the initial data (119906

0 1199061) belongs to a suitable

spaceDefine the energy of the system (2) by

119864 (119905) =1

2intΩ

(1199062

119905+

119899

sum

119894119895=1

119886119894119895119906119909119894

119906119909119895

)119889119909 (7)

We define

119892 = 119860minus1

(119909) for 119909 isin R119899 (8)

as a Riemannian metric on R119899 and consider the couple(R119899 119892) as a Riemannian manifold with an inner product

⟨119883 119884⟩119892= ⟨119860minus1

(119909)119883 119884⟩ |119883|2

119892= ⟨119883119883⟩

119892119883119884 isin R

119899

119909

(9)

Let 119863119892denote the Levi-Civita connection of the metric

119892 For the variable coefficients the main assumptions are asfollows

Assumption A There exists a vector field119867 on Ω and a con-stant 120588

0gt 0 such that

119863119892119867(119883119883) ge 120588

0|119883|2

119892for 119883 isin R

119899

119909 119909 isin Ω (10)

Moreover we assume thatsup119909isinΩ

div119867 lt inf119909isinΩ

div119867 + 21205880 (11)

119867 sdot ] le 0 119909 isin Γ2

119867 sdot ] ge 120575 119909 isin Γ1

(12)

where 120575 is a positive constant

Assumption (10) was introduced by [13] as a checkableassumption for the exact controllability of the wave equationwith variable coefficients For examples on the condition see[13 14]

Based on Assumption (10) Assumption A was given by[19] to study the stabilization of the wave equation with vari-able coefficients and boundary condition of memory type

Define

1198671

Γ2

(Ω) = 119906 isin 1198671

(Ω) | 119906|Γ2

= 0 (13)

To obtain the stabilization of the system (2) we assume thesystem (2) is well-posed such that

119906 isin 1198621

([0 +infin) 1198712

(Ω)) cap 119862 ([0 +infin) 1198671

Γ2

(Ω)) (14)

The main result of this paper is stated as follows

Theorem 1 Let Assumption A holds trueThen there exist pos-itive constants 119862 119862

2 such that

119864 (119905) le 1198621ℎ(

1198622119864 (0)

119905) +

1198621119865 (119905)

119905119864 (0) 119905 gt 0 (15)

2 Basic Inequality of the System

In this section we work on Ω with two metrics at the sametime the standard dotmetric ⟨sdot sdot⟩ and theRiemannianmetric119892 = ⟨sdot sdot⟩

119892given by (8)

If 119891 isin 1198621

(R119899) we define the gradient nabla119892119891 of 119891 in the

Riemannian metric 119892 via the Riesz representation theoremby

119883(119891) = ⟨nabla119892119891119883⟩119892

(16)

where 119883 is any vector field on (R119899 119892) The following lemmaprovides further relations between the two metrics see [13]in Lemma 21

Lemma 2 Let 119909 = (1199091 119909

119899) be the natural coordinate

system inR119899 Let119891 ℎ be functions and letH119883 be vector fieldsThen

(a)

⟨119867(119909) 119860 (119909)119883 (119909)⟩119892= ⟨119867 (119909) 119883 (119909)⟩ 119909 isin R

119899

(17)

(b)

nabla119892119891 =

119899

sum

119894=1

(

119899

sum

119895=1

119886119894119895(119909) 119891119909119895

)120597

120597119909119894

= 119860 (119909) nabla119891 119909 isin R119899

(18)

where nabla119891 is the gradient of 119891 in the standard metric(c)

nabla119892119891 (ℎ) = ⟨nabla

119892119891 nabla119892ℎ⟩119892

= ⟨nabla119891119860 (119909) nablaℎ⟩ 119909 isin R119899

(19)

where the matrix 119860(119909) is given in formula (1)

To proveTheorem 1 we still need several lemmas furtherDefine

1198640(119905) =

1

2intΩ

(1199062

119905+10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889119909 (20)

Then we have

119864 (119905) = 1198640(119905) + 120585int

119905

119905minus120591(119905)

intΓ1

1199062

119905(119909 120588) 119889Γ 119889120588 (21)

Lemma 3 Let (119906) be the solution of system (2) Then thereexists a constant 119862

1such that

119864 (0) minus 119864 (119879) = 1198621int

119879

0

intΓ1

120601 (119905) 119906119905(119909 119905) 119892

1(119906119905(119909 119905)) 119889Γ 119889119905

(22)

where 119879 ge 0 The assertion (22) implies that 119864(119905) is decreasing

Proof Differentiating (7) we obtain

1198641015840

(119905) = intΩ

(119906119905119906119905119905+ nabla119892119906 sdot nabla119906

119905) 119889119909

= intΓ1

120601 (119905) 119906119905(119909 119905) 119892

1(119906119905(119909 119905)) 119889Γ

(23)

Then the inequality (22) holds true






int

119879

0

intΓ

120597119906

120597]AH (119906) 119889Γ 119889119905

+1

2int

119879

0

intΓ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)H sdot ]119889Γ 119889119905

= (119906119905H (119906))

1003816100381610038161003816

119879

0+ int

119879

0

intΩ

119863119892H (nabla119892119906 nabla119892119906) 119889119909 119889119905

+1

2int

119879

0

intΩ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) divH119889119909119889119905

(24)


int

119879

0

intΩ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119875119889119909 119889119905

= (119906119905 119906119875)

1003816100381610038161003816

119879

0+1

2int

119879

0

intΩ

nabla119892119875 (1199062

) 119889119909 119889119905

minus int

119879

0

intΓ

119875119906120597119906

120597]A119889Γ 119889119905

(25)


0for which

119864 (119879) le119862

119879int

119879

0

intΓ1

(1199062

119905+ (

120597119906

120597]A)

2

)119889Γ119889119905 (26)

where 119879 ge 1198790


1

2sup119909isinΩ

div 119867 lt 120579 lt1

2inf119909isinΩ

div 119867 + 1205880 (27)

Set

H = 119867 119875 = 120579 minus 1205880 (28)


ΠΓ= (119906119905 119867 (119906) + 119875119906)

1003816100381610038161003816

119879

0

+ int

119879

0

intΩ


0

10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889119909 119889119905

+ int

119879

0

intΩ

((1

2div 119867 + 120588

0minus 120579) 119906

2

119905

+(120579 minus1

2div 119867) 10038161003816100381610038161003816nabla119892119906

10038161003816100381610038161003816

2

119892

) 119889119909 119889119905

(29)

where

ΠΓ= int

119879

0

intΓ

120597119906

120597]A(119867 (119906) + 119906119875) 119889Γ 119889119905

+1

2int

119879

0

intΓ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119867 sdot ] 119889Γ 119889119905

(30)

Decompose ΠΓas

ΠΓ= ΠΓ1

+ ΠΓ2

(31)

Since 119906|Γ2


= 0 that is

nabla119892119906 =

120597119906

120597]A

]A1003816100381610038161003816]A

1003816100381610038161003816

2

119892

for 119909 isin Γ2 (32)

Similarly we have

119867(119906) = ⟨119867 nabla119892119906⟩119892

=120597119906

120597]A

119867 sdot ]1003816100381610038161003816]A

1003816100381610038161003816

2

119892

for 119909 isin Γ2 (33)



ΠΓ2

=1

2int

119879

0

intΓ2

(120597119906

120597]A)

2

119867 sdot ]1003816100381610038161003816]A

1003816100381610038161003816

2

119892

119889Γ 119889119905 le 0 (34)

From (12) we have

ΠΓ1

= int

119879

0

intΓ1

120597119906

120597]A(119867 (119906) + 119906119875) 119889Γ 119889119905

+1

2int

119879

0

intΓ1

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119867 sdot ] 119889Γ 119889119905

le 119862120576int

119879

0

intΓ1

(120597119906

120597]A)

2

119889Γ 119889119905

+ 120576int

119879

0

intΓ1

(1199062

+10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889Γ 119889119905

+ int

119879

0

intΓ1

(1198621199062

119905minus 120575

10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889Γ 119889119905

le 119862int

119879

0

intΓ1

(120597119906

120597]A)

2

119889Γ 119889119905 + 120576119864 (119905)

+ 119862int

119879

0

intΓ1

1199062

119905119889Γ 119889119905

(35)


int

119879

0

119864 (119905) 119889119905

le 119862 (119864 (0) + 119864 (119879)) + 119862int

119879

0

intΓ1

(1199062

119905+ (

120597119906

120597]A)

2

)119889Γ119889119905

(36)



int

119879

0

119864 (119905) 119889119905 ge 119879119864 (119879) (37)



119864 (119879)

le119862

119879int

119879

0

intΓ1

(1206012

(119905) 1198922

(119906119905) + 1199062

119905) 119889Γ 119889119905

le119862

119879int

119879

0

intΓ1

(1199062

119905+ 1198922

(119906119905)) 119889Γ 119889119905

+119865 (119879)int

119879

0

intΓ1

119906119905120601 (119905) 119892 (119906

119905) 119889Γ 119889119905

le119862

119879int

119879

0

int119909isinΓ1|119906119905|le1

ℎ (119906119905119892 (119906119905)) 119889Γ 119889119905 + 119862119865 (119879) 119864 (0)

le119862

119879int

119879

0

intΓ1

ℎ (119906119905119892 (119906119905)) 119889Γ 119889119905 +

119862119865 (119879)

119879119864 (0)

le 119862meas (Γ1) ℎ(

int119879

0

intΓ1

119906119905119892 (119906119905) 119889Γ 119889119905


) +119862119865 (119879)

119879119864 (0)

le 1198621ℎ(

1198622119864 (0)

119879) +

1198621119865 (119879)

119879119864 (0)

(38)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














int

119879

0

intΓ

120597119906

120597]AH (119906) 119889Γ 119889119905

+1

2int

119879

0

intΓ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)H sdot ]119889Γ 119889119905

= (119906119905H (119906))

1003816100381610038161003816

119879

0+ int

119879

0

intΩ

119863119892H (nabla119892119906 nabla119892119906) 119889119909 119889119905

+1

2int

119879

0

intΩ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) divH119889119909119889119905

(24)


int

119879

0

intΩ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119875119889119909 119889119905

= (119906119905 119906119875)

1003816100381610038161003816

119879

0+1

2int

119879

0

intΩ

nabla119892119875 (1199062

) 119889119909 119889119905

minus int

119879

0

intΓ

119875119906120597119906

120597]A119889Γ 119889119905

(25)


0for which

119864 (119879) le119862

119879int

119879

0

intΓ1

(1199062

119905+ (

120597119906

120597]A)

2

)119889Γ119889119905 (26)

where 119879 ge 1198790


1

2sup119909isinΩ

div 119867 lt 120579 lt1

2inf119909isinΩ

div 119867 + 1205880 (27)

Set

H = 119867 119875 = 120579 minus 1205880 (28)


ΠΓ= (119906119905 119867 (119906) + 119875119906)

1003816100381610038161003816

119879

0

+ int

119879

0

intΩ


0

10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889119909 119889119905

+ int

119879

0

intΩ

((1

2div 119867 + 120588

0minus 120579) 119906

2

119905

+(120579 minus1

2div 119867) 10038161003816100381610038161003816nabla119892119906

10038161003816100381610038161003816

2

119892

) 119889119909 119889119905

(29)

where

ΠΓ= int

119879

0

intΓ

120597119906

120597]A(119867 (119906) + 119906119875) 119889Γ 119889119905

+1

2int

119879

0

intΓ

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119867 sdot ] 119889Γ 119889119905

(30)

Decompose ΠΓas

ΠΓ= ΠΓ1

+ ΠΓ2

(31)

Since 119906|Γ2


= 0 that is

nabla119892119906 =

120597119906

120597]A

]A1003816100381610038161003816]A

1003816100381610038161003816

2

119892

for 119909 isin Γ2 (32)

Similarly we have

119867(119906) = ⟨119867 nabla119892119906⟩119892

=120597119906

120597]A

119867 sdot ]1003816100381610038161003816]A

1003816100381610038161003816

2

119892

for 119909 isin Γ2 (33)



ΠΓ2

=1

2int

119879

0

intΓ2

(120597119906

120597]A)

2

119867 sdot ]1003816100381610038161003816]A

1003816100381610038161003816

2

119892

119889Γ 119889119905 le 0 (34)

From (12) we have

ΠΓ1

= int

119879

0

intΓ1

120597119906

120597]A(119867 (119906) + 119906119875) 119889Γ 119889119905

+1

2int

119879

0

intΓ1

(1199062

119905minus10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

)119867 sdot ] 119889Γ 119889119905

le 119862120576int

119879

0

intΓ1

(120597119906

120597]A)

2

119889Γ 119889119905

+ 120576int

119879

0

intΓ1

(1199062

+10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889Γ 119889119905

+ int

119879

0

intΓ1

(1198621199062

119905minus 120575

10038161003816100381610038161003816nabla11989211990610038161003816100381610038161003816

2

119892

) 119889Γ 119889119905

le 119862int

119879

0

intΓ1

(120597119906

120597]A)

2

119889Γ 119889119905 + 120576119864 (119905)

+ 119862int

119879

0

intΓ1

1199062

119905119889Γ 119889119905

(35)


int

119879

0

119864 (119905) 119889119905

le 119862 (119864 (0) + 119864 (119879)) + 119862int

119879

0

intΓ1

(1199062

119905+ (

120597119906

120597]A)

2

)119889Γ119889119905

(36)



int

119879

0

119864 (119905) 119889119905 ge 119879119864 (119879) (37)



119864 (119879)

le119862

119879int

119879

0

intΓ1

(1206012

(119905) 1198922

(119906119905) + 1199062

119905) 119889Γ 119889119905

le119862

119879int

119879

0

intΓ1

(1199062

119905+ 1198922

(119906119905)) 119889Γ 119889119905

+119865 (119879)int

119879

0

intΓ1

119906119905120601 (119905) 119892 (119906

119905) 119889Γ 119889119905

le119862

119879int

119879

0

int119909isinΓ1|119906119905|le1

ℎ (119906119905119892 (119906119905)) 119889Γ 119889119905 + 119862119865 (119879) 119864 (0)

le119862

119879int

119879

0

intΓ1

ℎ (119906119905119892 (119906119905)) 119889Γ 119889119905 +

119862119865 (119879)

119879119864 (0)

le 119862meas (Γ1) ℎ(

int119879

0

intΓ1

119906119905119892 (119906119905) 119889Γ 119889119905


) +119862119865 (119879)

119879119864 (0)

le 1198621ℎ(

1198622119864 (0)

119879) +

1198621119865 (119879)

119879119864 (0)

(38)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int

119879

0

119864 (119905) 119889119905 ge 119879119864 (119879) (37)



119864 (119879)

le119862

119879int

119879

0

intΓ1

(1206012

(119905) 1198922

(119906119905) + 1199062

119905) 119889Γ 119889119905

le119862

119879int

119879

0

intΓ1

(1199062

119905+ 1198922

(119906119905)) 119889Γ 119889119905

+119865 (119879)int

119879

0

intΓ1

119906119905120601 (119905) 119892 (119906

119905) 119889Γ 119889119905

le119862

119879int

119879

0

int119909isinΓ1|119906119905|le1

ℎ (119906119905119892 (119906119905)) 119889Γ 119889119905 + 119862119865 (119879) 119864 (0)

le119862

119879int

119879

0

intΓ1

ℎ (119906119905119892 (119906119905)) 119889Γ 119889119905 +

119862119865 (119879)

119879119864 (0)

le 119862meas (Γ1) ℎ(

int119879

0

intΓ1

119906119905119892 (119906119905) 119889Γ 119889119905


) +119862119865 (119879)

119879119864 (0)

le 1198621ℎ(

1198622119864 (0)

119879) +

1198621119865 (119879)

119879119864 (0)

(38)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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