A class of integral equations and approximation of p-Laplace equations

Abstract

Let ⊂ R n be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equationwith the boundary condition u = g 0 on ∂ , where f 0 ∈ C( ) and g 0 ∈ C(∂ ) are given functions and M is the singular integral operator given by

Perron’s Method and Wiener’s Theorem for a Nonlocal Equation

“…The so defined L s p is a non-linear and non-local "fractional differential operator" often called the fractional p-Laplacian. Operators of this type were first studied in [19], but in a slightly different form and in the viscosity sense. The notation suggested, for instance, in [5], [20], [15] and [11] stems from the linear case p = 2, when one can use the Fourier transform to define…”

Local behavior of fractional p-minimizers

“…Nonetheless, some partial results are known. Firstly, we would like to cite the higher regularity contributions for viscosity solutions in the case when s is close to 1 proven in the recent interesting paper [1]; see, also, [16]. Secondly, the analysis in the papers [3] and [18] considers the special case when p is suitably large -thus falling in the Morrey embedding case when concerning regularity.…”

Large solutions for a class of semilinear integro-differential equations with censored jumps

“…The proof of this result follows the classical lines of the viscosity theory, for instance see [20] and references therein.…”

“…On the other hand, since β ∈ (0, 1) we have Note that by Corollary 7.2, we can take C large enough in order to have w ,h ,w ,h are respective viscosity sub and supersolution to (7.3). Then, by Proposition 4.2, we can state a version for Perron's method in the same way as in [20] and conclude the existence of a (discontinuous) solution u ,h for the problem (7.3) in the sense that u * ,h is a viscosity subsolution and (u ,h ) * is a viscosity supersolution. In fact, u ,h satisfies, for all , h > 0…”

“…It is also possible to show a well-posedness result for problem (1.1) with u = g on ∂Ω in the case the boundary condition g is merely continuous. We refer to [20] for a proof of this kind of result.…”