Normal approximation for a random elliptic equation

Random approximation of a general symmetric equation

In this paper, we prove the Hyers-Ulam stability of the symmetric functionalequation $f(ph_1(x,y))=ph_2(f(x), f(y))$ in random normed spaces. As a consequence, weobtain some random stability results in the sense of Hyers-Ulam-Rassias.

Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L and tensorized H 0 simultaneously on a standard k-dimensional cube. In the special case k = 2 the suggested ...

A two-phase free boundary problem for a semilinear elliptic equation

In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary‎. ‎We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...

Shape-Measure Method for Solving Elliptic Optimal Shape Problems (Fixed Control Case)

Extreme Analysis of a Random Ordinary Differential Equation

In this paper, we consider a one dimensional stochastic system described by an ellipticequation. A spatially varying random coefficient is introduced to account for uncertainty orimprecise measurements. We model the logarithm of this coefficient by a Gaussian process andprovide asymptotic approximations of the tail probabilities of the derivative of the solution.