Linear Dirichlet Problems Research Articles

Analyse algébrique d'un probléme de Dirichlet non linéaire et singulier

The algebraic structure of a factor ring and the topological structure of a semi-normed algebra permit the construction of many algebras of generalized functions. Some of them are well suitable to solve a non linear Dirichlet problem with irregular data.

On nontrivial solutions of an asymptotically linear Dirichlet problem

Let Ω be a bounded domain in R N ( N ⩾ 2) with smooth boundary ∂Ω. Let ƒ(·): R → R be a continuous function such that ƒ(0) = 0 and lim ζ → ± ∞ ƒ(ζ) ζ exist. We discuss the existence of nontrivial solutions of the Dirichlet problem −Δu = ƒ(u) in Ω, u = 0 on ∂Ω . Our hypotheses seem to allow significantly greater freedom for the numbers ƒ ± = lim ζ → ± ∞ ƒ(ζ) ζ than is usually found in the literature in that the interval with end points ƒ ± is here permitted to contain an eigenvalue of −Δ subject to zero Dirichlet data. Furthermore we do not require any differentiability of ƒ and we weaken other technical conditions on ƒ that are generally assumed in other papers dealing with this problem.

A posteriori error bounds for inhomogeneous linear dirichlet and neumann problems in n dimensions

Second-order inhomogeneous linear Dirichlet and Neumann problems of divergent form in n dimensions (n-3) are considered. For a given Dirichlet or Neumann problem, a conjugate problem of elliptic type in n-1 dependent variables with Neumann-like boundary conditions or Dirichler boundary conditions, respectively, is introduced in such a way that the sum of the minima of the energies of a given problem and its conjugate problem is a calculable constant There result a posteriori error bounds for the energy-space norm of the difference of an approximate solution and the unknown exact solution of the Dirichlet or Neumann problem and of its conjugate problem. The data needed for the conjugate problem can be calculated with knowledge of only the given data of the oriyinal Dirichlet or Neumann problem. When those data are sufficiently smooth, the boundary functions of the original problem and the conjugate problem are related by simple formulas involving are-length differentiation or are length integration along t...

A posteriors error bounds for inhomogeneous linear dirichlet and neumann problems

Second-order inhomogeneous linear Dirichlet and Neumann problems in divergent form on a simply-connected to-dimensional domain with Lipschitz-continious boundary of finite length are considered. Conjugate problems, that is, a pair of one Dirichlet and one Neumann problem the minima of energies of which add to a known constant, are introduced. From the conceppt of conjugate problems, two-sided bounds for the energy of the exact solution of any given Dirichlet or Neumann problem are constructed. These two-sided bounds for the energy at the exact solution are in turn used to obtain easily calculable a posteriori error bounds for the norm of the difference of the approximate and exact solutions of the given problem