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In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e.,where Ω ⊂ R N is a regular bounded domain, 1 2 < s < 1, 2 * s is the critical fractional Sobolev exponent, 0 ≤ λ ∈ R, ν is the outwards normal to ∂Ω, Σ D , Σ N are smooth (N − 1)-dimensional submanifolds of ∂Ω such that Σ D ∪ Σ N = ∂Ω, Σ D ∩ Σ N = ∅, and Σ D ∩ Σ N = Γ is a smooth (N − 2)-dimensional submanifold of ∂Ω.

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.

In this paper we study a model of an interface between two fluids in a porous medium. For this model we prove several local and global well-posedness results and study some of its qualitative properties. We also provide numerics.

This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator,where Ω is a bounded domain in R N , 0 < β < 1, β < α < β + 1 and λ > 0. In particular, we study the fractional elliptic problem,and we prove existence or nonexistence of positive solutions depending on the parameter λ > 0, up to the critical value of the exponent p, i.e., for 1 < p ≤ 2 * µ − 1 where µ := α − β and 2 * µ = 2N N−2µ is the critical exponent of the Sobolev embedding.