A modiied classical penalty method for solving a Dirichlet boundary value problem is presented. This new ctitious domain/penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domain. It is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast, approximate calculation of boundary integral...

This paper presents a simple, self-contained account of G̊arding’s theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. In a second, independent part of the paper, the relationship of G̊arding’s theo...

Here f ∈ L(Ω) , g ∈ H(∂Ω) and Ω is a bounded domain in R with the smooth boundary ∂Ω ( see Figure 1 ). The method of lines for solving Problem I works well if Ω is a rectangular domain since the finite difference solution is expressed explicitly by use of eigenvalues and eigenvectors for the finite difference scheme([BGN70], [Nak65]). But one says that this method seems difficult to be applied ...

The main aim of this paper is a geometrical approach to simultaneous solutions of the abstract weak Dirichlet problem. We answer partially a question from the paper 2] where a similar problem was discussed from a potential-theoretical point of view for the case of function spaces consisting of harmonic functions.

In this paper we solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in R. The main results apply, in particular, to subequations with a Riesz characteristic p ≥ 2. It is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved ...

The Dirichlet divisor problem is used as a model to give a conjecture concerning the conditional convergence of the Dirichlet series of an L-function.

We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The pr...

We prove a Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. We derive lower bounds for Dirichlet eigenvalues using the Harnack inequality. We also consider a randomization problem in connection with combinatorial games using Dirichlet eigenvalues.

A multigrid preconditioning scheme for solving the Ciarlet-Raviart mixed method equations for the biharmonic Dirichlet problem is presented. In particular, a Schur complement formulation for these equations which yields non-inherited quadratic forms is considered. The preconditioning scheme is compared with a standard W-cycle multigrid iteration. It is proved that a Variable V-cycle preconditio...