It is known that the Poisson kernel function is an analogue of the Cauchy kernel for the analytic functions and the Poisson integral formula solves the Dirichlet problem
for the inhomogeneous Laplace equation.
In this paper we study the following parametric nonlinear Dirichlet problem
Consequently [v.sub.k] is the solution of Dirichlet problem
As a quasioptimal preconditioner, we suggest to use the spectrally adapted matrix of the corresponding Dirichlet problem
with homogeneous isotropic coefficients in the same computational domain.
We recall that we are considering the shape optimization problem (7) where [u.sub.D] solves the pure Dirichlet problem
(4) and [u.sub.N] solves the Neumann problem (6).
Zhang; Nonexistence of positive classical solutions of a singular nonlinear Dirichlet problem
with a convection term, Nonlinear Analysis 8 (1996) 957-961
Grebennikov, The study of the approximation quality of GR-method for solution of the Dirichlet problem
for Laplace equation.
From the uniqueness theorem for the exterior Dirichlet problem
[5,6] we obtain div [u.sub.s] = 0 in [D.sub.e].
No date is cited for the first edition, but to this second has been added a final chapter on the existence of solutions, primarily the Dirichlet problem
for various types of elliptic equations.
Many authors have investigated the Dirichlet problem
in simply connected domains.
Zhang, Existence of solutionas for p(x)-Laplacian Dirichlet problem
, Nonlinear Analysis: TMA, 52(2003), 1843-1852.
They cover the Cauchy problem, the Dirichlet problem
, the Neumann problem, the Neumann problem for a nonlocal nonlinear diffusion equation, nonlocal p-Laplacian evolution problems, the nonlocal total variation flow, and nonlocal models for sandpiles.