Principal Eigenvalue of Mixed Problem for the Fractional Laplacian:\n Moving the Boundary Conditions

2017/02/23 by Tommaso Leonori, María Medina, Leonori, Tommaso +8 · 1 citation
Computer Science · Mathematics · #31B10 #35J20 #35J25 #35P20 #60J75 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1702.07644

openalex publication_date 2017/02/23 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

We analyze the behavior of the eigenvalues of the following non local mixed\nproblem \ \(-
Delta)^s u · amp;= · amp;
lambda₁(D)
u\n · amp;
inn
Omega,

u · amp;= · amp;0 · amp;
inn D,


mathcalN_su · amp;= · amp;0 · amp;
inn N. \nOur goal is to construct different sequences of problems by modifying the\nconfiguration of the sets D and N, and to provide sufficient and necessary\nconditions on the size and the location of these sets in order to obtain\nsequences of eigenvalues that in the limit recover the eigenvalues of the\nDirichlet or Neumann problem. We will see that the non locality plays a crucial\nrole here, since the sets D and N can have infinite measure, a phenomenon\nthat does not appear in the local case (see for example citeD,D2,CP).\n

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• Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations

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