Related papers: On the moving plane method for nonlocal problems i…

For a bounded weak Lipschitz domain we show the so called `Maxwell compactness property', that is, the space of square integrable vector fields having square integrable weak rotation and divergence and satisfying mixed tangential and normal…

We consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal minimal surfaces may stick at the boundary of the domain, even when the domain is smooth and…

In the paper, we derive an existence result for a nonlinear nonautonomous partial elliptic system on an open bounded domain with Dirichlet boundary conditions, containg fractional powers of the weak Dirichlet-Laplace operator that are meant…

We consider the Navier-Stokes-Fourier system with general inhomogeneous Dirichlet-Neumann boundary conditions. We propose a new approach to the local well-posedness problem based on conditional regularity estimates. By conditional…

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated to fully nonlinear elliptic equations of order $2s$, with $s\in (1/2,1)$, and a coercive gradient term with subcritical…

We consider the problem of optimal distribution of a limited amount of conductive material in systems governed by local and non-local scalar diffusion laws. Of particular interest for these problems is the study of the limiting case, which…

We study boundary regularity for the inhomogeneous Dirichlet problem for $2s$-stable operators in generalized H\"older spaces. Moreover, we provide explicit counterexamples that showcase the sharpness of our results. Our approach directly…

We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-\Delta u -\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<\mu<1/4$. The admissible boundary behavior…

In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on…

In this paper, we investigate a nonlocal equation involving the logarithmic Laplacian with indefinite nonlinearities: \begin{equation*} \left\{ \begin{array}{ll} L_\Delta u(x)=a(x_n)f(u), & x\in\Omega, \\ u(x)=0,& x\in…

In this paper we consider the system involving fully nonlinear nonlocal operators: $$ \left\{ \begin{array}{ll} F_{\alpha}(u(x)) = C_{n,\alpha} PV \int_{{R}^n} \frac{G(u(x)-u(y))}{|x-y|^{n+\alpha}} dy=f(v(x)), F_{\beta}(v(x)) = C_{n,\beta}…

We consider the Calder\`on problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic conductivity coefficient from partial…

A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…

In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained…

The present paper studies the fractional $p$-Laplacian boundary value problems with jumping nonlinearities at zero or infinity and obtain the existence of multiple solutions and sign-changing solutions by constructing the suitable…

Here we study the Dirichlet problem for first order linear and quasi-linear hyperbolic PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. By means of a Lyapunov…

We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on…

In this paper we study nonlocal problems that are analogous to the local ones given by the Laplacian or the p-Laplacian with dynamical boundary conditions. We deal both with smooth and with singular kernels and show existence and uniqueness…

In recent works, the authors of this chapter have shown with co-authors how a basis consisting of dilated and shifted $\text{sinc}$-functions can be used to solve fractional partial differential equations. As a model problem, the fractional…

We characterize the set of positive harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of several different chambers. We analyze the asymptotic behavior of the solutions in connection with the changes…