In this paper we will consider elliptic boundary value problems with oscillatory diffusion coefficient, say A. We will derive regularity estimates in Sobolev norms which are weighted by certain derivatives of A. The constants in the regularity estimates then turn out to be independent of the variations in A. These regularity results will be employed for the derivation of error estimates for hp-...

— In this paper we consider supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold (M, g), where the metric g is analytic. Using an analytic WKB method, we are able to construct an Ansatz for the semiclassical equation for times independent of the small parameter. These approximate solutions will help to show two different types of instabilities. The first is in the en...

In this article we derive local error estimates for some Petrov-Galerkin methods applied to strongly elliptic equations on smooth curves of the plane. The results, e.g., cover the basic first-kind and second-kind integral equations appearing in the boundary element solution of the potential problem. The discretization model includes the Galerkin method and the collocation method using smoothest...

An adaptive nite element–boundary element algorithm is proposed to compute an approximate solution of a given boundary value problem. The convergence in H ( ) is controlled by a boundary element based a-posteriori error estimator from which an adaptive re nement strategy is derived. Corresponding error estimates are given based on appropriate boundary element error estimates in negative Sobolev...

Using measure-capacity inequalities we study new functional inequalities, namely LPoincaré inequalities Varμ(f) 1/q ≤ CP ∫ |∇f | dν and L-logarithmic Sobolev inequalities Entμ ( f )1/q ≤ CLS ∫ |∇f | dν for any q ∈ (0, 1]. As a consequence, we establish the asymptotic behavior of the solutions to the so-called weighted porous media equation ∂u ∂t = ∆u −∇ψ · ∇u for m ≥ 1, in terms of L-norms and ...

We prove almost sharp upper bounds for the Lp norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines techniques from semiclassical analysis with harmonic theory on reductive groups, and makes use of new asymptotic bounds for spherical functions that are of inde...

Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except a) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient a is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative pr...

Let u(x; t) be the solution to the free time-dependent Schrr odinger equation at the point (x; t) in space-time R n+1 with initial data f. We characterize the size of u in terms L p (L q)-estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces _ H s (R n). Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interpo...

We consider the generalized airfoil equation in the situation where the index of the problem is −1. We periodize the problem, then discretize it by a fully discrete version of the trigonometric collocation method and apply the conjugate gradient method to solve the discretized problem. The approximate solution appears to be of optimal accuracy in a scale of Sobolev norms, and the N parameters o...