Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T12:56:17.953Z Has data issue: false hasContentIssue false
  • English
  • Français

On shape optimization problems involving the fractional laplacian

Published online by Cambridge University Press:  01 August 2013

Anne-Laure Dalibard  and
David Gérard-Varet
Anne-Laure Dalibard
Affiliation:
DMA/CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France
David Gérard-Varet
Affiliation:
IMJ and University Paris 7, 175 rue du Chevaleret, 75013 Paris France. gerard-varet@math.jussieu.fr
Get access

Abstract

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.

Keywords

Fractional laplacianfhape optimizationshape derivativemoving plane method
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 976 - 1013
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math. 12 623–727.
Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17 (1964) 35–92. Google Scholar
Birkner, M., Alfredo López-Mimbela, J., and Wakolbinger, A., Comparison results and steady states for the Fujita equation with fractional Laplacian. Ann. Inst. Henri Poincaré Anal. Non Linéaire 22 (2005) 83–97. Google Scholar
Bogdan, K., The boundary Harnack principle for the fractional Laplacian. Studia Math. 123 (1997) 43–80. Google Scholar
Cabré, X. and Tan, J., Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224 (2010) 2052–2093. Google Scholar
Caffarelli, L.A., Roquejoffre, J.-M. and Sire, Y., Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. (JEMS) 12 (2010) 1151–1179. Google Scholar
Costabel, M., Dauge, M. and Duduchava, R., Asymptotics without logarithmic terms for crack problems. Commun. Partial Diff. Eq. 28 (2003) 869–926. Google Scholar
M. Dauge, Elliptic Boundary Value Problems on Corner Domains, in Lect. Notes Math., vol. 1341, Smoothness and asymptotics of solutions. Springer-Verlag, Berlin (1988).
D. DeSilva and J.-M. Roquejoffre, Regularity in a one-phase free boundary problem for the fractional laplacian. Ann. Inst. Henri Poincaré, Anal. Non Linéaire, à paraître (2011).
A. Henrot and M. Pierre, Variation et optimisation de formes. Math. Appl., vol. 48, Une analyse géométrique. Springer, Berlin (2005).
E. Lauga, M.P. Brenner and H.A. Stone, Microfluidics: The no-slip boundary condition (2007).
Lopes, O. and Mariş, M., Symmetry of minimizers for some nonlocal variational problems. J. Funct. Anal. 254 (2008) 535–592. Google Scholar
G. Lu and J. Zu, An overdetermined problem in riesz potential and fractional laplacian, Preprint Arxiv: 1101.1649v2 (2011).
Serrin, J., A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971) 304–318. Google Scholar
Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60 (2007) 67–112. Google Scholar
Vinogradova, O. and Yakubov, G., Surface roughness and hydrodynamic boundary conditions. Phys. Rev. E 73 (1986) 479–487. Google ScholarPubMed