Related papers: Fractional Laplacians on ellipsoids

This survey hinges on the interplay between regularity and approximation for linear and quasi-linear fractional elliptic problems on Lipschitz domains. For the linear Dirichlet integral Laplacian, after briefly recalling H\"older regularity…

This paper studies the properties of solutions to a class of elliptic and parabolic problems involving the fractional Laplacian. By applying the mountain pass theorem, we prove the existence of bounded classical positive solutions in the…

In this work we consider higher dimensional thin domains with the property that both boundaries, bottom and top, present oscillations of weak type. We consider the Laplace operator with Neumann boundary conditions and analyze the behavior…

We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps, such as the well-known infinite Bernoulli convolution…

In dimension two or three, the weak maximum principal for biharmonic equation is valid in any bounded Lipschitz domains. In higher dimensions (greater than three), it was only known that the weak maximum principle holds in convex domains or…

In this paper we study the obstacle problems for the fractional Lapalcian of order $s\in(0,1)$ in a bounded domain $\Omega\subset\mathbb R^n$, under mild assumptions on the data.

The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order…

In this paper we discuss the existence, uniqueness and regularity of solutions of the following system of coupled semilinear Poisson equations on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$: \[ \left\{{llll} \mathcal{A}^s u= v^p &…

This note is devoted to several results about frequency localized functions and associated Bernstein inequalities for higher order operators. In particular, we construct some counterexamples for the frequency-localized Bernstein…

This paper studies the growth of local extrema of Laplacian eigenfunctions on post-critically finite (p.c.f.) fractals. We establish the sharp two-sided estimate $\#\mathrm{Extr}(u_\lambda)\asymp\lambda^{d_S/2}$ for the Sierpinski gasket,…

We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $\Omega^c$, where…

This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass…

The fractional Laplacian $(-\Delta)^{\alpha/2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open problems in the numerical analysis of the…

We consider an elliptic equation with the fractional Laplacian operator $(-\Delta)^{\frac{\alpha}{2}}$ in the dissipative term, a singular integral operator ${\bf A}(\cdot)$ in the nonlinear term, and an external source $f$. The key example…

In the following we show the strong comparison principle for the fractional $p$-Laplacian, i.e. we analyze functions $v,w$ which satisfy $v\geq w$ in $\mathbb{R}^N$ and \[ (-\Delta)^s_pv+q(x)|v|^{p-2}v\geq (-\Delta)^s_pw+q(x)|w|^{p-2}w…

In this note we introduce some nonlinear extremal nonlocal operators that approximate the, so called, truncated Laplacians. For these operators we construct representation formulas that lead to the construction of what, with an abuse of…

We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains $\Omega \subset \mathbb R^n$ of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along…

Let $\Omega \subset \mathbb{R}^{n}$ be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to $\Omega$ maps $L^{p}(\Omega) \to W^{1,p}(\Omega)$ for all $p > 1$, when the smoothness…

In this paper we extend two nowadays classical results to a nonlinear Dirichlet problem to equations involving the fractional $p-$Laplacian. The first result is a existence in a non-resonant range more specific between the first and second…

We study the symmetry properties for solutions of elliptic systems of the type (-\Delta)^{s_1} u = F_1(u, v), (-\Delta)^{s_2} v= F_2(u, v), where $F\in C^{1,1}_{loc}(\R^2)$, $s_1,s_2\in (0,1)$ and the operator $(-\Delta)^s$ is the so-called…