Related papers: Petviashvilli's Method for the Dirichlet Problem

We study the Dirichlet problem $-\div(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $\Omega$. The main issue is to give a proper sense to what…

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate…

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain…

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-\Delta)^s u = g$ in $\Omega$, $u \equiv 0$ in $\R^n\setminus\Omega$, for some…

We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus \Omega)^2} = f(x,u) & \quad \mbox{in } \Omega \\ u = 0 &…

This study investigates Dirichlet boundary condition related to a class of nonlinear parabolic problem with nonnegative $L^1$-data, which has a variable-order fractional $p$-Laplacian operator. The existence and uniqueness of renormalized…

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $\lambda$ and $\beta$: [{{array} [c]{rcll}% -\Delta_{p}u & = & \lambda h(x,u)+\beta f(x,u,\nabla u) & \text{in}\Omega u & = & 0 &…

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $\Omega$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f…

We consider the quasi-linear eigenvalue problem $-\Delta_p u = \lambda g(u)$ subject to Dirichlet boundary conditions on a bounded open set $\Omega$, where $g$ is a locally Lipschitz continuous functions. Imposing no further conditions on…

In the paper we study the 2D div-curl problem in the exterior domain which models the flow with given vorticity, divergency, boundary condition at infinity, and Dirichlet condition on the solid surface. We will find the relations on…

We prove the uniform boundedness of all solutions for a general class of Dirichlet anisotropic elliptic problems of the form $$-\Delta_{\overrightarrow{p}}u+\Phi_0(u,\nabla u)=\Psi(u,\nabla u) +f $$ on a bounded open subset $\Omega\subset…

In this work, first we prove that for any compact set $K\subset\mathbb{R}^{n}$ and any continuous function $\phi$ defined on $\partial K$, there exists a bounded weak solution in $C(\bar{\mathbb{R}^{n}\backslash K}) \cap…

We prove that the set of solutions to the parabolic singular $p$-Laplace equation with Dirichlet boundary conditions on a bounded Lipschitz domain $\Omega$ for all space dimensions is continuous in the parameter $p\in [1,+\infty)$ and the…

A standard Hilbert-space proof of Dirichlet's principle is simplified, using an observation that a certain form of min-problem has unique solution, at a specified point. This solves Dirichlet's problem, after it is recast in the required…

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…

We present an algorithm to solve $- \lap u - f(x,u) = g$ with Dirichlet boundary conditions in a bounded domain $\Omega$. The nonlinearities are non-resonant and have finite spectral interaction: no eigenvalue of $-\lap_D$ is an endpoint of…

We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular…

We derive a priori bounds for positive supersolutions of $ - \Delta_{p} u = \rho(x) f(u) $, where $p>1$ and $\Delta_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply…

In this paper we study the existence of solutions of a one-dimensional eigenvalue problem $-\left(|\phi_x|^{p-2}\phi_x\right)_x=\lambda \left(|\phi|^{q-2}\phi-f(\phi)\right)$ such that $\phi(0)=\phi(1)=0$, where $p,q>1$, $\lambda$ is a…

We have considered the following semi linear elliptic problem on the unit disk $B$ $-\Delta u = \lambda_1 u+e^u+f $ in $B$ with the Dirichlet boundary condition and $f$ satisfying the following condition : $f\in L^r(B)$, for some $r>2$ and…