On overdetermined problems for a general class of nonlocal operators

Biswas, Anup; Jarohs, Sven

doi:10.1016/j.jde.2019.09.010

Cited by 14 publications

(12 citation statements)

References 25 publications

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“…then Ω must be a ball. Serrin's work has been generalized for a vast class of operators, see for instance, [8,9,13,16,18,19,34]. In this article, we follow the method of [8,16] to establish our result on overdetermined problem concerning (1.6).…”

Section: Introduction and Resultsmentioning

confidence: 95%

Boundary regularity of mixed local-nonlocal operators and its application

Biswas¹,

Modasiya²,

Sen³

2022

Preprint

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Section: Introduction and Resultsmentioning

confidence: 95%

Boundary regularity of mixed local-nonlocal operators and its application

Biswas¹,

Modasiya²,

Sen³

2022

Preprint

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“…This modification can be adjusted by following through the proof of [34, Th. 1.1] (see claim 1 there); see also [7,33].…”

Section: Rotational Symmetry Of Positive Solutionsmentioning

confidence: 95%

“…In Theorem 4.2 we discuss the torsion equation in our context, over-determined by a constraint imposed on the domain boundary, which is a non-local development of a classical inverse problem by Serrin and Weinberger [51,53]. Some recent works considering overdetermined problems for nonlocal operators include [26,30,52] for the fractional Laplacian, and [7] for the operators Ψ(-∆) considered in the present paper.…”

Section: Introductionmentioning

confidence: 97%

Hopf’s lemma for viscosity solutions to a class of non-local equations with applications

Biswas

Lőrinczi

2021

Nonlinear Analysis

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We consider a large family of integro-differential equations and establish a non-local counterpart of Hopf's lemma, directly expressed in terms of the symbol of the operator. As closely related problems, we also obtain a variety of maximum principles for viscosity solutions. In our approach we combine direct analysis with functional integration, allowing a robust control around the boundary of the domain, and make use of the related ascending ladder height-processes. We then apply these results to a study of principal eigenvalue problems, the radial symmetry of the positive solutions, and the overdetermined non-local torsion equation.

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“…On the other hand, the PDE approaches enable us to extend the regularity theory to nonlinear integro-differential equations. Especially, the class of integro-differential operators mentioned above has been studied recently [3,9,4]. In this work let us focus on the Evan-sKrylov theory and the Schauder theory, which have been studied only for equations of fixed order.…”

Section: Introductionmentioning

confidence: 99%

Generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders

Kim

Lee

2020

Preprint

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We establish the generalized EvansKrylov and Schauder type estimates for nonlocal fully nonlinear elliptic equations with rough kernels of variable orders. In contrast to the fractional Laplacian type operators having a fixed order of differentiability σ ∈ (0, 2), the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function ϕ describing the variable orders of differentiability, which is allowed to oscillate between two functions r σ 1 and r σ 2 for some 0 < σ1 ≤ σ2 < 2. By introducing the generalized Hölder spaces, we provide C ϕψ estimates that generalizes the standard EvansKrylov and Schauder type C σ+α estimates.

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On overdetermined problems for a general class of nonlocal operators

Cited by 14 publications

References 25 publications

Boundary regularity of mixed local-nonlocal operators and its application

Boundary regularity of mixed local-nonlocal operators and its application

Hopf’s lemma for viscosity solutions to a class of non-local equations with applications

Generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders

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