Analysis and Applications

A class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and n known, separated microscopic length scales ε_i, i = 1, …, n in a bounded domain D ⊂ ℝ^d is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge ℙ-a.s, as ε_i → 0, to a stochastic, elliptic one-scale limit problem in a tensorized domain of dimension (n + 1)d. It is shown that this stochastic limit problem admits best N-term "polynomial chaos" type approximations which converge at a rate σ > 0 that is determined by the summability of the random inputs' Karhúnen–Loève expansion. The convergence of the polynomial chaos expansion is shown to hold ℙ-a.s. and uniformly with respect to the scale parameters ε_i. Regularity results for the stochastic, one-scale limiting problem are established. An error bound for the approximation of the random solution at finite, positive values of the scale parameters ε_i is established in the case of two scales, and in the case of n > 2, scales convergence is shown, albeit without giving a convergence rate in this case.

Uniqueness and non-existence results of entire solutions to the maximal surface equation and to the constant mean curvature spacelike surface equation on certain complete non-compact Riemannian surfaces are obtained.

We study the large-time behavior of the solution of an initial-boundary value problem for the equations of 1D motions of a compressible viscous heat-conducting gas coupled with radiation through a radiative transfer equation.

Assuming only scattering processes between matter and photons (neglecting absorption and emission) and suitable hypotheses on the transport coefficients, we prove that the unique weak solution of the problem converges toward the static state.

In this paper, we investigate a hemivariational inequality involving Leray–Lions type operator with critical growth. Some existence and multiple results are obtained through using the concentration compactness principle of P. L. Lions and some nonsmooth critical point theorems.