On the mixed local-nonlocal Hénon equation

Salort, Ariel; Vecchi, Eugenio

doi:10.48550/arxiv.2107.09520

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…In the case of a bounded function f , Biagi-Dipierro-Valdinoci-Vecchi [5] has obtained local Hölder continuity for globally bounded solutions and Garain-Ukhlov [30] studied existence, uniqueness, local boundedness and further qualitative properties of solutions. Moreover, for more general inhomogeneites, local boundedness is proved in Salort-Vecchi [33]. Very recently, Hölder and gradient regularity were proved in [17], where very general mixed nonlinear problems are considered.…”

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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“…The case of mixed operators. The study of mixed local/nonlocal operators has been recently received an increasing level of attention, both in view of their intriguing mathematical structure, which combines the classical setting and the features typical of nonlocal operators in a framework that is not scale-invariant [40,45,46,5,32,10,21,4,20,24,23,22,39,7,1,18,30,27,28,35,36,37,38,19,9,6,54], and of their importance in practical applications such as the animal foraging hypothesis [29,51].…”

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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On the mixed local-nonlocal Hénon equation

Abstract: In this paper we consider a Hénon-type equation driven by a nonlinear operator obtained as a combination of a local and nonlocal term. We prove existence and non-existence akin to the classical result by Ni, and a stability result as the fractional parameter s → 1.

Cited by 2 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

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