Some probabilistic problems and methods in singular perturbations

Singular Perturbations of Some Nonlinear Problems

In this paper we deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behaviour of these solutions when ε → 0. Then these abstract results are applied to some boundary value p...

Spectral Problems in Elasticity. Singular Boundary Perturbations

A. The three-dimensional spectral elasticity problem is studied in an anisotropic and inhomogeneous solid with small defects, i.e., inclusions, voids, and microcracks. Asymptotics of eigenfrequencies and the corresponding elastic eigenmodes are constructed and justified. New technicalities of the asymptotic analysis are related to variable coefficients of differential operators, vectoria...

Some Singular Mixed Problems.

Nonconvex Variational Problems with General Singular Perturbations

We study the effect of a general singular perturbation on a nonconvex variational problem with infinitely many solutions. Using a scaling argument and the theory of T-convergence of nonlinear functionals, we show that if the solutions of the perturbed problem converge in L1 as the perturbation parameter goes to zero, then the limit function satisfies a classical minimal surface problem. Introdu...

Topics in Singular Perturbations and Hybrid Asymptotic-Numerical Methods

Hybrid asymptotic-numerical methods are used to study various singular perturbation problems where innnite logarithmic asymptotic expansions occur. In particular, they are used to sum the innnite logarithmic expansions that arise for a low Reynolds number ow problem and for certain two-dimensional eigenvalue problems in perforated domains. Asymptotic and numerical methods are also used to analy...