ON SHAPE OPTIMIZATION PROBLEMS INVOLVING THE FRACTIONAL LAPLACIAN

被引：28

作者：

Dalibard, Anne-Laure ^{[1
]}

Gerard-Varet, David ^{[2
,3
]}

机构：

[1] Ecole Normale Super, CNRS, DMA, F-75005 Paris, France

[2] IMJ, F-75013 Paris, France

[3] Univ Paris 07, F-75013 Paris, France

来源：

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS | 2013年 / 19卷 / 04期

关键词：

Fractional laplacian; fhape optimization; shape derivative; moving plane method; BOUNDARY;

D O I：

10.1051/cocv/2012041

中图分类号：

TP [自动化技术、计算机技术];

学科分类号：

0812 ;

摘要：

Our concern is the computation of optimal shapes in problems involving (-Delta)(1/2). We focus on the energy J(Omega) associated to the solution u(Omega) of the basic Dirichlet problem (-Delta)(1/2) u(Omega) = 1 in Omega, u = 0 in Omega(c). We show that regular minimizers Omega of this energy under a volume constraint are disks. Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the moving plane method.

引用

页码：976 / 1013

页数：38

共 16 条

[1] ESTIMATES NEAR BOUNDARY FOR SOLUTIONS OF ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS SATISFYING GENERAL BOUNDARY CONDITIONS .2. [J].

AGMON, S ;

DOUGLIS, A ;

NIRENBERG, L .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1964, 17 (01) :35-&

[2]

Agmon S., COMMUN PURE APPL MAT, V12, P623

[3] Comparison results and steady states for the Fujita equation with fractional Laplacian [J].

Birkner, M ;

López-Mimbela, JA ;

Wakolbinger, A .

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, 2005, 22 (01) :83-97

[4]

Bogdan K, 1997, STUD MATH, V123, P43

[5] Positive solutions of nonlinear problems involving the square root of the Laplacian [J].

Cabre, Xavier ;

Tan, Jinggang .

ADVANCES IN MATHEMATICS, 2010, 224 (05) :2052-2093

[6] Variational problems with free boundaries for the fractional Laplacian [J].

Caffarelli, Luis A. ;

Roquejoffre, Jean-Michel ;

Sire, Yannick .

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 2010, 12 (05) :1151-1179

[7] Asymptotics without logarithmic terms for crack problems [J].

Costabel, M ;

Dauge, M ;

Duduchava, R .

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2003, 28 (5-6) :869-926

[8]

DAUGE M, 1988, LECT NOTES MATH, V1341, P1

[9]

DeSilva D., 2011, ANN I H POINCARE ANA

[10]

HENROT A, 2005, MATH APPL, V48

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