TY  - JOUR
A1  - Djitte, Sidy Moctar
A1  - Fall, Mouhamed Moustapha
A1  - Weth, Tobias
T1  - A fractional Hadamard formula and applications
T2  - Calculus of variations and partial differential equations
N2  - We derive a shape derivative formula for the family of principal Dirichlet eigenvalues λs(Ω) of the fractional Laplacian (−Δ)s associated with bounded open sets Ω⊂RN of class C1,1. This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case s=12. As an application, we consider the maximization problem for λs(Ω) among annular-shaped domains of fixed volume of the type B∖B¯¯¯¯′, where B is a fixed ball and B′ is ball whose position is varied within B. We prove that λs(B∖B¯¯¯¯′) is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings.
Y1  - 2021
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/63485
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-634852
SN  - 1432-0835
N1  - This work is supported by DAAD and BMBF (Germany) within the project 57385104.
N1  - Open Access funding enabled and organized by Projekt DEAL.
N1  - Mathematics Subject Classification: Primary 49Q10; Secondary 35S15; 35S05
VL  - 60
IS  - art. 231
PB  - Springer
CY  - Berlin ; Heidelberg
ER  -