Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition

Laplace Equation in the Half-space with a Nonhomogeneous Dirichlet Boundary Condition

We deal with the Laplace equation in the half space. The use of a special family of weigted Sobolev spaces as a framework is at the heart of our approach. A complete class of existence, uniqueness and regularity results is obtained for inhomogeneous Dirichlet problem.

Regularization for a Laplace Equation with Nonhomogeneous Neumann Boundary Condition

We consider the following problem  uxx + uyy = 0, (x, y) ∈ (0, π)× (0, 1) u(0, y) = u(π, y) = 0, y ∈ (0, 1) uy(x, 0) = g(x), 0 < x < π u(x, 0) = φ(x), 0 < x < π The problem is shown to be ill-posed, as the solution exhibits unstable dependence on the given data functions. Using the new method, we regularize of problem and obtain some new results. Some numerical examples are given to illu...

Spectral concentration phenomena for the Laplace opera- tor with the Dirichlet boundary condition on a cavity

A Nonlocal p-Laplacian Evolution Equation with Nonhomogeneous Dirichlet Boundary Conditions

Abstract. In this paper we study the nonlocal p-Laplacian-type diffusion equation ut(t, x) = ∫ RN J(x−y)|u(t, y)−u(t, x)|p−2(u(t, y)−u(t, x)) dy, (t, x) ∈]0, T [×Ω, with u(t, x) = ψ(x) for (t, x) ∈ ]0, T [×(RN \Ω). If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div(|∇u|p−2∇u) with Dirichlet boundary condition u(t, x) = ψ(x) on (t, x)...

Self-similar Solutions of the Porous Medium Equation in a Half-space with a Nonlinear Boundary Condition. Existence and Symmetry

We find existence of a nonnegative compactly supported solution of the problem ∆u = uα in R+ , ∂u ∂ν = u on ∂R+ . Moreover, we prove that every nonnegative solution with finite energy is compactly supported and radially symmetric in the tangential variables.