Regularity of solutions to a fractional elliptic problem with mixed Dirichlet–Neumann boundary data

Abstract

In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.

Concave-Convex critical problems for the spectral fractional Laplacian with mixed boundary conditions

“…Then, by an iteration argument, we get f (•, u) ∈ L r (Ω), r > N 2s after a finite number of steps. Because of [13,Theorem 3.7] we conclude u ∈ L ∞ (Ω).…”

A Strong Maximum Principle for the fractional Laplace equation with mixed boundary condition

“…As we commented before, since f ∈ L ∞ (Ω) and g ∈ L p (Ω), with p > N/s, repeating step by step the proof of [2, Theorem 4.7] we get that U, V ∈ L ∞ (C Ω ). Let us also stress that u, v ∈ L ∞ (Ω) by [8,Theorem 3.7]. Moreover, since g ≥ 0, by comparison with the respective Dirichlet problem, we can assume that v ≥ 0 and…”

“…We will obtain a similar result for the mixed boundary data problem (P s ) by adapting the approach of Dávila to our fractional setting. To that end we will also use the regularity results proved in [8]. Let us remark that in [1] the authors proved a fractional strong maximum principle, but dealing with a different fractional operator which is defined by means of a singular integral.…”

Semilinear fractional elliptic problems with mixed Dirichlet-Neumann boundary conditions

“…To this aim we will make use of the estimates proved in [12, Theorem 1.1] that guarantee the compactness needed in order to accomplish this limit step. Then, with the same ideas, we prove Theorem 1.3 using the uniform estimates proved in [12,Corollary 1.1] for the moving boundary conditions (as in hypotheses (B 1 )-(B 3 )).…”

“…To carry out this step we need some compactness properties for the sequence {v k } in order to guarantee the convergence in some sense. By [12,Theorem 1.1] the sequence {v k } is uniformly bounded in C γ (Ω k ) for some γ ∈ 0, 1 2 . Then, by the Ascoli-Arzelá Theorem, there exists a subsequence {v k } uniformly convergent over compact sets in…”