We use a direct variational method in order to investigate the dependence on parameter for the solution for a Duffing type equation with Dirichlet boundary value conditions. Mathematics Subject Classification. 49J02

Let L be an infinitely degenerate second-order linear operator defined on a bounded smooth Euclidean domain. Under weaker conditions than those of Hörmander, we show that the Dirichlet problem associated with L has a unique smooth classical solution. The proof uses the Malliavin calculus. At present, there appears to be no proof of this result using classical analytic techniques. Le Probléme De...

(1.2) PIH : C(Sn−1) −→ C(B) ∩ C∞(Bn), such that u = PIH f solves (1.2A) ∆Hu = 0 on B, u ∣∣ Sn−1 = f. Here, ∆H is the Laplace-Beltrami operator on B, with metric tensor (1.1). We will establish further regularity on u = PIH f when f has some further smoothness on Sn−1, and estimate du(x), in the hyperbolic metric, as x → ∂B. If n = 2, then ∆Hu = 0 if and only if ∆u = 0, where ∆ = ∂ 1 +∂ 2 2 is t...

In this paper we study domain embedding preconditioners for discrete linear systems approximating the Dirichlet problem associated a second order elliptic equation. We observe that if a mixed finite element discretization is used, then such a preconditioner can be constructed in a straightforward manner from the H(div)–inner product. We also use the H(div)–inner product to construct a new preco...

We study the long-time behavior of the unique viscosity solution u of the viscous Hamilton-Jacobi Equation ut−∆u+|Du| m = f in Ω×(0,+∞) with inhomogeneous Dirichlet boundary conditions, where Ω is a bounded domain of RN . We mainly focus on the superquadratic case (m > 2) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on f, the initial...

Let Ω ⊂ Rd, d ≥ 3, be a bounded Lipschitz domain. For Laplace’s equation ∆u = 0 in Ω, we study the Dirichlet and Neumann problems with boundary data in the weighted space L2(∂Ω, ωαdσ), where ωα(Q) = |Q−Q0|α, Q0 is a fixed point on ∂Ω, and dσ denotes the surface measure on ∂Ω. We prove that there exists ε = ε(Ω) ∈ (0, 2] such that the Dirichlet problem is uniquely solvable if 1 − d < α < d − 3 +...

Keywords: Dirichlet boundary value problem p-Laplacian Topological degree theory Critical point theory Weak solution a b s t r a c t In this work, by virtue of topological degree theory and critical point theory, we are mainly concerned with the existence of weak solutions for a Dirichlet boundary value problem with the p-Laplacian operator.

Dirichlet processes have been very popular in recent years, particularly as prior distributions over unknown probability distributions. The purpose of this review is to provide both an intuition into and a mathematical understanding of Dirichlet processes. We will begin by discussing mixture models as a motivating problem. We will find that our model forces us to make some strong assumptions ab...