A Faber-Krahn inequality for mixed local and nonlocal operators

Biagi, Stefano; Dipierro, Serena; Valdinoci, Enrico; Vecchi, Eugenio

doi:10.48550/arxiv.2104.00830

Cited by 7 publications

(8 citation statements)

References 26 publications

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“…For existence and nonexistence results, we refer to Abatangelo-Cozzi [1]. We also refer to Biagi-Dipierro-Valdinoci-Vecchi [4,8], Dipierro-Ros-Oton-Serra-Valdinoci [22], Dipierro-Proietti Lippi-Valdinoci [21], Dipierro-Valdinoci [23] and the references therein.…”

Section: Known Resultsmentioning

confidence: 99%

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations

Garain¹,

Lindgren²

2022

Preprint

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We consider equations involving a combination of local and nonlocal degenerate p-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.

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Section: Known Resultsmentioning

confidence: 99%

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations

Garain¹,

Lindgren²

2022

Preprint

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“…Proof. The proof is similar to that in the linear case, see [8,Theorem 1.1]; however, we present it here in all the details for the sake of completeness.…”

Section: The Hong-krahn-szegö Inequality For L Psmentioning

confidence: 86%

“…In regard to the shape optimization problem, a Faber-Krahn inequality for mixed local and nonlocal linear operators when p = 2 has been established in [8], showing the optimality of the ball in the minimization of the first eigenvalue. The corresponding inequality for the nonlinear setting presented in (1.1) will be given here in the forthcoming Theorem 4.1.…”

Section: Introductionmentioning

confidence: 99%

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

Biagi¹,

Dipierro²,

Valdinoci³

et al. 2021

Preprint

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Given a bounded open set Ω ⊆ R n , we consider the eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of Ω.We prove that the second eigenvalue λ2(Ω) is always strictly larger than the first eigenvalue λ1(B) of a ball B with volume half of that of Ω.This bound is proven to be sharp, by comparing to the limit case in which Ω consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.

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“…where J : R N → R + is a nonnegative, nonsingular, radially symmetric and compactly supported kernel. For related results, see also da Silva-Salort [29], Biagi-Dipierro-Valdinoci-Vecchi [14] and the references therein.…”

Section: Introductionmentioning

confidence: 93%

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

Garain¹,

Ukhlov²

2021

Preprint

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In this article, we consider mixed local and nonlocal Sobolev (q, p)-inequalities with extremal in the case 0 < q < 1 < p < ∞. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal p-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular p-Laplace and mixed local and nonlocal p-Laplace equation is established. Finally, we investigate the existence, uniqueness, regularity and symmetry properties of weak solutions for such problems.

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A Faber-Krahn inequality for mixed local and nonlocal operators

Cited by 7 publications

References 26 publications

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

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