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 problemWe 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.