We prove interior H2s−ε regularity for weak solutions of linear elliptic integrodifferential equations close to the fractional s-Laplacian. The result is obtained via intermediate estimates in Nikol’skii spaces, which are in turn carried out by means of an appropriate modification of the classical translation method by Nirenberg.

This paper aims to investigate the existence of solutions for fractional integral boundary value problems with p(t)-Laplacian operator. By using the fixed point theorem and the coincidence degree theory, some new results are obtained, which enrich existing literatures. Some examples are supplied to verify our main results.

We deal with symmetry properties for solutions of nonlocal equations of the type (−∆) s v = f (v) in R n , where s ∈ (0, 1) and the operator (−∆) s is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation −div (x α ∇u) = 0 on R

For the discretization of integral fractional Laplacian $(-\Delta )^s$, $0 < s 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual posteriori error estimator. In order to compensate for lack $L^2$-regularity in regime $3/4 this estimator includes as an additional weight power distance from mesh skeleton. We prove optimal convergence rates $h$-adaptive al...

Lions (1959, Bull. Soc. Math. France, \textbf{87}, 245--273) introduced the Navier-Stokes equations with a viscous diffusion in form of fractional Laplacian; subsequently, he (1969, Dunod, Gauthiers-Villars, Paris) claimed uniqueness its solution when exponent is not less than five quarters case spatial dimension three. Following work Hofmanov$\acute{\mathrm{a}}$, Zhu and (2019, arXiv:1912.1184...

We consider a class of partial differential equations with the fractional Laplacian and the homogeneous Dirichlet boundary data. Some sufficient condition under which the solutions of the equations considered depend continuously on parameters is stated. The application of the results to some optimal control problem is presented. The methods applied in the paper make use of the variational struc...

We prove an optimal Hardy inequality for the fractional Laplacian on the half-space. 1. Main result and discussion Let 0 < α < 2 and d = 1, 2, . . .. The purpose of this note is to prove the following Hardy-type inequality in the half-space D = {x = (x1, . . . , xd) ∈ R : xd > 0}. Theorem 1. For every u ∈ Cc(D), (1) 1 2 ∫

Scattering theory between the fractional power H0 = κ−1(−∆)κ/2 (κ ≥ 1) of negative Laplacian and the Hamiltonian H = H0+V perturbed by shortand long-range potentials is developed. The existence and asymptotic completeness of wave operators are proved. AMS Subject Classification: Primary 35P25, 81U05 ; Secondary 47A40, 35J10, 35S30.

We prove a Liouville-type theorem for bounded stable solutions v ∈ C(R) of elliptic equations of the type (−∆)v = f(v) in R, where s ∈ (0, 1) and f is any nonnegative function. The operator (−∆) stands for the fractional Laplacian, a pseudo-differential operator of symbol |ξ|.

In this paper, we investigate the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the nonlinearities, various existence results for positive solutions are derived in terms of different values of parameters via t...