A generalized fractional Pohozaev identity and applications
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Sidy Moctar Djitte
Abstract
We prove a fractional Pohozaev-type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to the nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.
Funding source: Deutscher Akademischer Austauschdienst
Award Identifier / Grant number: 57385104
Funding source: Bundesministerium für Bildung und Forschung
Award Identifier / Grant number: 57385104
Funding statement: This work is supported by DAAD and BMBF (Germany) within the project 57385104.
References
[1] M. Cozzi, Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes, J. Funct. Anal. 272 (2017), no. 11, 4762–4837. 10.1016/j.jfa.2017.02.016Search in Google Scholar
[2] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization, Adv. Des. Control 4, Society for Industrial and Applied Mathematics, Philadelphia, 2001. Search in Google Scholar
[3] S. Dipierro, X. Ros-Oton, J. Serra and E. Valdinoci, Non-symmetric stable operators: regularity theory and integration by parts, Adv. Math. 401 (2022), no. 108321. 10.1016/j.aim.2022.108321Search in Google Scholar
[4] S. M. Djitte, M. M. Fall and T. Weth, A fractional Hadamard formula and applications, Calc. Var. Partial Differential Equations 60 (2021), no. 6, Paper No. 231. 10.1007/s00526-021-02094-3Search in Google Scholar
[5] M. M. Fall, P. A. Feulefack, R. Y. Temgoua and T. Weth, Morse index versus radial symmetry for fractional Dirichlet problems, Adv. Math. 384 (2021), Paper No. 107728. 10.1016/j.aim.2021.107728Search in Google Scholar
[6] M. M. Fall and S. Jarohs, Gradient estimates in fractional Dirichlet problems, Potential Anal. 54 (2021), no. 4, 627–636. 10.1007/s11118-020-09842-8Search in Google Scholar
[7] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726. 10.1002/cpa.21591Search in Google Scholar
[8] G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N. S.) 5 (2014), no. 2, 373–386. Search in Google Scholar
[9] J. McGough and J. Mortensen, Pohozaev obstructions on non-starlike domains, Calc. Var. Partial Differential Equations 18 (2003), no. 2, 189–205. 10.1007/s00526-002-0188-3Search in Google Scholar
[10] W. Reichel, Uniqueness Theorems for Variational Problems by the Method of Transformation Groups, Lecture Notes in Math. 1841, Springer, Berlin, 2004. 10.1007/b96984Search in Google Scholar
[11] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302. 10.1016/j.matpur.2013.06.003Search in Google Scholar
[12] X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), no. 2, 587–628. 10.1007/s00205-014-0740-2Search in Google Scholar
[13] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67–112. 10.1002/cpa.20153Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- On the Hölder regularity of all extrema in Hilbert’s 19th Problem
- A partially overdetermined problem in domains with partial umbilical boundary in space forms
- On the existence of canonical multi-phase Brakke flows
- Concavity properties for solutions to p-Laplace equations with concave nonlinearities
- On the Almgren minimality of the product of a paired calibrated set and a calibrated manifold of codimension 1
- The Neumann and Dirichlet problems for the total variation flow in metric measure spaces
- No Lavrentiev gap for some double phase integrals
- Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity
- Properties of the free boundaries for the obstacle problem of the porous medium equations
- A generalized fractional Pohozaev identity and applications
Articles in the same Issue
- Frontmatter
- On the Hölder regularity of all extrema in Hilbert’s 19th Problem
- A partially overdetermined problem in domains with partial umbilical boundary in space forms
- On the existence of canonical multi-phase Brakke flows
- Concavity properties for solutions to p-Laplace equations with concave nonlinearities
- On the Almgren minimality of the product of a paired calibrated set and a calibrated manifold of codimension 1
- The Neumann and Dirichlet problems for the total variation flow in metric measure spaces
- No Lavrentiev gap for some double phase integrals
- Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity
- Properties of the free boundaries for the obstacle problem of the porous medium equations
- A generalized fractional Pohozaev identity and applications
