Regularity of spectral fractional Dirichlet and Neumann problems

Grubb, Gerd

doi:10.1002/mana.201500041

Cited by 69 publications

(56 citation statements)

References 57 publications

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“…We also note that it is possible to obtain a nonexistence result to (1.7) for large (see Remark 3.2). Finally, we point out that, by the second part of Corollary 3.5 in [27], one can infer that the solutions of (1.1) are indeed 2 −0 (…”

Section: Introductionmentioning

confidence: 81%

“…We also note that it is possible to obtain a nonexistence result to for t large (see Remark ). Finally, we point out that, by the second part of Corollary 3.5 in , one can infer that the solutions of are indeed

C^{2 s - 0} (\bar{normalΩ}) = \cap_{ε > 0} C^{2 s - ε} (\bar{normalΩ})

. Anyway, we do not discuss this point here but we refer the reader to in which a deeper analysis on the regularity of some spectral fractional problems is given.…”

Section: Introductionmentioning

confidence: 99%

“…. Anyway, we do not discuss this point here but we refer the reader to [17,27] in which a deeper analysis on the regularity of some spectral fractional problems is given.…”

Section: Introductionmentioning

confidence: 99%

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An Ambrosetti–Prodi type result for fractional spectral problems

Ambrosio

2019

Mathematische Nachrichten

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We consider the following class of fractional parametric problems lefttrue(−ΔDirtrue)su=f(x,u)+tφ1+hleft4.ptin4.ptnormalΩ,leftu=0left4.pton4.pt∂normalΩ,where Ω⊂double-struckRN is a smooth bounded domain, s∈(0,1), N>2s, true(−ΔDirtrue)s is the fractional Dirichlet Laplacian, f:normalΩ¯×R→R is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti–Prodi type assumptions, t∈R, φ1 is the first eigenfunction of the Laplacian with homogenous boundary conditions, and h:Ω→R is a bounded function. Using variational methods, we prove that there exists a t0∈R such that the above problem admits at least two distinct solutions for any t≤t0. We also discuss the existence of solutions for a fractional periodic Ambrosetti–Prodi type problem.

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Section: Introductionmentioning

confidence: 81%

C^{2 s - 0} (\bar{normalΩ}) = \cap_{ε > 0} C^{2 s - ε} (\bar{normalΩ})

. Anyway, we do not discuss this point here but we refer the reader to in which a deeper analysis on the regularity of some spectral fractional problems is given.…”

Section: Introductionmentioning

confidence: 99%

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An Ambrosetti–Prodi type result for fractional spectral problems

Ambrosio

2019

Mathematische Nachrichten

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“…Remark 2.1 A Moser iteration argument combined with the use of the Caffarelli-Silvestre extension [15] shows that any finite energy solution u of (1.1) is bounded (see, for example, [25]), and elliptic regularity results guarantee that u is continuous up to the boundary (refer to [16,37] for the spectral case and [52] for the restricted case). Hence, it makes sense to say that a finite energy solution u to (1.1) has zero boundary values.…”

Section: Definition Of Sobolev Spaces and Fractional Laplaciansmentioning

confidence: 98%

Classification of finite energy solutions to the fractional Lane–Emden–Fowler equations with slightly subcritical exponents

Choi

Kim

2016

Annali di Matematica

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We study qualitative properties of solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents where the associated fractional Laplacian is defined in terms of either the spectra of the Dirichlet Laplacian or the integral representation. As a consequence, we classify the asymptotic behavior of all finite energy solutions. Our method also provides a simple and unified approach to deal with the classical (local) LaneEmden-Fowler equation for any dimension >2.

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“…Proof. The proof follows applying the discrete version of J-Method for interpolation, see [4] and also [14].…”

Section: Dirichlet Spectral Fractional Ellipticmentioning

confidence: 99%

Initial-boundary value problem for a fractional type degenerate heat equation

Huaroto

Neves

2018

Math. Models Methods Appl. Sci.

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The aim of the paper is to generalize the author's previous work [15]. We extend the argument [15] for any uniformly elliptic operator in divergence form Lu = −div(A(x)∇u), more precisely, we study a fractional type degenerate elliptic equation posed in bounded domains with homogeneous boundary conditionswhere L −s is the inverse s-fractional elliptic operator for any s ∈ (0, 1). This work consists of two part. The first part is devoted to state how the boundary condition will be consider (in the spirit of F. Otto [25]), and to give a formulation for the IBVP. In the second part, It is shown the existence of mass-preserving, non-negative weak solutions satisfying energy estimates for measurable and bounded non-negative initial data.

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Regularity of spectral fractional Dirichlet and Neumann problems

Cited by 69 publications

References 57 publications

An Ambrosetti–Prodi type result for fractional spectral problems

An Ambrosetti–Prodi type result for fractional spectral problems

Classification of finite energy solutions to the fractional Lane–Emden–Fowler equations with slightly subcritical exponents

Initial-boundary value problem for a fractional type degenerate heat equation

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