Related papers: Large solutions for subordinate spectral Laplacian

We give a complete characterization, as "stadium-like domains", of convex subsets $\Omega$ of $\mathbb{R}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity…

A kind of nonlocal reaction-diffusion equations on an unbounded domain containing fractional Laplacian operator is analyzed. To be precise, we prove the convergence of solutions of the equation governed by the fractional Laplacian to the…

Consider the fractional powers $(A_{\operatorname{Dir}})^a$ and $(A_{\operatorname{Neu}})^a$ of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator $A$ on a smooth bounded subset $\Omega $ of…

We describe all self-adjoint realizations of the restricted fractional Laplacian $(-\Delta)^a$ with power $a \in (\frac{1}{2}, 1)$ on a bounded interval by imposing boundary conditions on the functions in the domain of a maximal…

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate…

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem $$ \begin{cases} -\Delta\upsilon= e^{\upsilon}-s\phi_1-4\pi\alpha\delta_p-h(x)\,\,\,\,…

We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -\Delta_1 u = g(u)|D u|+h(u)f & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega,…

We show a convergence result of the fractional Laplacian for sequences of nonnegative functions without uniform boundedness near infinity. As an application, we construct a sequence of solutions to the fractional Nirenberg problem that…

We consider the linear stationary equation defined by the fractional Laplacian with drift. In the supercritical case, that is the case when the dominant term is given by the drift instead of the diffusion component, we prove local…

We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second…

For the Laplacian $\Delta$ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann boundary conditions. That is, we construct…

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nu…

We consider the fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain $D$ of $\mathbb{R}^d$, with $d \geq 2$. In this paper, we perform a stochastic gradient descent algorithm that approximates the…

This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-\Delta)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $\Omega$ in $R^n$. The…

We study the structure of solutions of the interior Bernoulli free boundary problem for $(-\Delta)^{\alpha/2}$ on an interval $D$ with parameter $\lambda > 0$. In particular, we show that there exists a constant $\lambda_{\alpha,D} > 0$…

In this paper, we prove the global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems set in a bounded open subset of $\mathbb{R}^N$. The diffusion operators are of the form $u_i \mapsto d_i…

We establish sharp interior and boundary regularity estimates for solutions to $\partial_t u - L u = f(t, x)$ in $I\times \Omega$, with $I \subset \mathbb{R}$ and $\Omega \subset\mathbb{R}^n$. The operators $L$ we consider are…

We consider the supercritical elliptic problem -\Delta u = \lambda e^u, \lambda > 0, in an exterior domain $\Omega = \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal…

The initial-value problem for the drift-diffusion equation arising from the model of semiconductor device simulations is studied. The dissipation on this equation is given by the fractional Laplacian. When the exponent of the fractional…

In this paper we discuss the topic of correct setting for the equation $(-\Delta )^s u=f$, with $0<s <1$. The definition of the fractional Laplacian on the whole space $\mathbb R^n$, $n=1,2,3$ is understood through the Fourier transform,…