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Estimates of eigenvalues and eigenfunctions in periodic homogenization

Carlos E. Kenig, Fanghua Lin, Zhongwei Shen (2013)

Journal of the European Mathematical Society

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an O ( ϵ ) estimate in H 1 for solutions with Dirichlet condition.

Estimates of solutions to linear elliptic systems and equations

Heinrich Begehr (1992)

Banach Center Publications

Whenever nonlinear problems have to be solved through approximation methods by solving related linear problems a priori estimates are very useful. In the following this kind of estimates are presented for a variety of equations related to generalized first order Beltrami systems in the plane and for second order elliptic equations in ℝ m . Different types of boundary value problems are considered. For Beltrami systems these are the Riemann-Hilbert, the Riemann and the Poincaré problem, while for elliptic...

Estimates of the principal eigenvalue of the p -Laplacian and the p -biharmonic operator

Jiří Benedikt (2015)

Mathematica Bohemica

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet p -Laplacian and the Navier p -biharmonic operator on a ball of radius R in ℝ N and its asymptotics for p approaching 1 and ∞ . Let p tend to ∞ . There is a critical radius R C of the ball such that the principal eigenvalue goes to ∞ for 0 < R ≤ R C and to 0 for R > R C . The critical radius is R C = 1 for any N ∈ ℕ for the p -Laplacian and R C = 2 N in the case of the p -biharmonic operator. When p approaches 1 , the principal eigenvalue of the Dirichlet...

Estimates on the solution of an elliptic equation related to Brownian motion with drift (II).

Joseph G. Conlon, Peder A. Olsen (1997)

Revista Matemática Iberoamericana

In this paper we continue the study of the Dirichlet problem for an elliptic equation on a domain in R3 which was begun in [5]. For R &gt; 0 let ΩR be the ball of radius R centered at the origin with boundary ∂Ω R. The Dirichlet problem we are concerned with is the following:(-Δ - b(x).∇) u(x) = f(x),   x ∈ Ω R,with zero boundary conditionsu(x) = 0,   x ∈ ∂Ω R.

Estimates on the solution of an elliptic equation related to Brownian motion with drift.

Joseph G. Conlon, Juan Redondo (1995)

Revista Matemática Iberoamericana

In this paper we are concerned with studying the Dirichlet problem for an elliptic equation on a domain in R3. For simplicity we shall assume that the domain is a ball ΩR of radius R. Thus:ΩR = {x ∈ R3 : |x| &lt; R}.The equation we are concerned with is given by(-Δ - b(x).∇) u(x) = f(x),    x ∈ ΩR,with zero Dirichlet boundary conditions.

Evaluation of the condition number in linear systems arising in finite element approximations

Alexandre Ern, Jean-Luc Guermond (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper derives upper and lower bounds for the ℓ p -condition number of the stiffness matrix resulting from the finite element approximation of a linear, abstract model problem. Sharp estimates in terms of the meshsize h are obtained. The theoretical results are applied to finite element approximations of elliptic PDE's in variational and in mixed form, and to first-order PDE's approximated using the Galerkin–Least Squares technique or by means of a non-standard Galerkin technique in L1(Ω). Numerical...

Exact solutions to some external mixed problems in potential theory

Valery I. Fabrikant (1986)

Aplikace matematiky

A new and elegant procedure is proposed for the solution of mixed potential problems in a half-space with a circular line of division of boundary conditions. The approach is based on a new type of integral operators with special properties. Two general external problems are solved; i) An arbitrary potential is specified at the boundary outside a circle, and its normal derivative is zero inside; ii) An arbitrary normal derivative is given outside the circle, and be potential is zero inside. Several...

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