Our concern is the computation of optimal shapes in problems involving (-Δ)1/2. We focus on the energy J(Ω) associated to the solution uΩ of the basic Dirichlet problem ( - Δ)1/2uΩ = 1 in Ω, u = 0 in Ωc. We show that regular minimizers Ω 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.
Mots-clés : fractional laplacian, fhape optimization, shape derivative, moving plane method
@article{COCV_2013__19_4_976_0,
author = {Dalibard, Anne-Laure and G\'erard-Varet, David},
title = {On shape optimization problems involving the fractional laplacian},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {976--1013},
publisher = {EDP-Sciences},
volume = {19},
number = {4},
year = {2013},
doi = {10.1051/cocv/2012041},
mrnumber = {3182677},
zbl = {1283.49049},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2012041/}
}
TY - JOUR AU - Dalibard, Anne-Laure AU - Gérard-Varet, David TI - On shape optimization problems involving the fractional laplacian JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 976 EP - 1013 VL - 19 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2012041/ DO - 10.1051/cocv/2012041 LA - en ID - COCV_2013__19_4_976_0 ER -
%0 Journal Article %A Dalibard, Anne-Laure %A Gérard-Varet, David %T On shape optimization problems involving the fractional laplacian %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 976-1013 %V 19 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2012041/ %R 10.1051/cocv/2012041 %G en %F COCV_2013__19_4_976_0
Dalibard, Anne-Laure; Gérard-Varet, David. On shape optimization problems involving the fractional laplacian. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 976-1013. doi : 10.1051/cocv/2012041. http://archive.numdam.org/articles/10.1051/cocv/2012041/
[1] , and , Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math. 12 623-727. | MR | Zbl
[2] , and , Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17 (1964) 35-92. | MR | Zbl
[3] , , and , Comparison results and steady states for the Fujita equation with fractional Laplacian. Ann. Inst. Henri Poincaré Anal. Non Linéaire 22 (2005) 83-97. | Numdam | MR | Zbl
[4] , The boundary Harnack principle for the fractional Laplacian. Studia Math. 123 (1997) 43-80. | MR | Zbl
[5] and , Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224 (2010) 2052-2093. | MR | Zbl
[6] , and , Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. (JEMS) 12 (2010) 1151-1179. | MR | Zbl
[7] , and , Asymptotics without logarithmic terms for crack problems. Commun. Partial Diff. Eq. 28 (2003) 869-926. | MR | Zbl
[8] , Elliptic Boundary Value Problems on Corner Domains, in Lect. Notes Math., vol. 1341, Smoothness and asymptotics of solutions. Springer-Verlag, Berlin (1988). | MR | Zbl
[9] and , Regularity in a one-phase free boundary problem for the fractional laplacian. Ann. Inst. Henri Poincaré, Anal. Non Linéaire, à paraître (2011). | Numdam | Zbl
[10] and , Variation et optimisation de formes. Math. Appl., vol. 48, Une analyse géométrique. Springer, Berlin (2005). | MR | Zbl
[11] , and , Microfluidics: The no-slip boundary condition (2007).
[12] and , Symmetry of minimizers for some nonlocal variational problems. J. Funct. Anal. 254 (2008) 535-592. | MR | Zbl
[13] and , An overdetermined problem in riesz potential and fractional laplacian, Preprint Arxiv: 1101.1649v2 (2011). | Zbl
[14] , A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971) 304-318. | MR | Zbl
[15] , Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60 (2007) 67-112. | MR | Zbl
[16] and , Surface roughness and hydrodynamic boundary conditions. Phys. Rev. E 73 (1986) 479-487.
Cité par Sources :
