In this paper, we consider an elliptic partial differential equation where a small parameter is multiplied with one or both of the second derivatives. Four types of basic spectral regularization methods such as Showalter’s, Tikhonov’s, Lardy’s and Lavrentiev’s are applied to approximate the solution by introducing another large (or small) parameter. Convergence of the regularized solutions to t...

We study Fredholmness of nonlocal boundary value problems for fourth order elliptic differential-operator equations in UMD Banach spaces. The main condition is given in terms of R-boundedness of some families of bounded operators generated by inverse of the characteristic operator pencil of the equation. Then we prove an isomorphism of the problem on the semi-axis, for some special boundary con...

The issue of convexity is fundamental in the theory of partial differential equations. We discuss some recent progress of convexity estimates for solutions of nonlinear elliptic equations arising from some classical problems in differential geometry. We first review some works in the literature on the convexity of solutions of quasilinear elliptic equations in Rn. The study of geometric propert...

Abstract. We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives, we consider the equation in the context of power scale generated by a strongly elliptic differential operator. Application of semigroup ar...

We develop an a priori error analysis for the finite element Galerkin discretization of parameter identification problems. The state equation is given by an elliptic partial differential equation of second order with a finite number of unknown parameters, which are estimated using point-wise measurements of the state variable.