Regularity analysis for semilinear elliptic PDEs with random data

In this thesis we consider a class of semilinear elliptic partial differential equations with uncertainty regarding the diffusion coefficient. We derive the analytic and Gevrey smooth dependence of the solution of such equations on the diffusion coefficient, both as an element of a suitable function space and when given by some (possibly nonlinear) parametric expansion. Such smoothness results then enables one to quantify the uncertainty in the solution using state-of-the-art numerical methods.
We approach this task in a modular and abstract fashion. We first introduce the necessary combinatorical objects and results. We then review the definitions and behaviour of linear and multilinear maps as well as powers, polynomials and formal power series between (possibly infinite-dimensional) vector spaces. Following this, their corresponding continuous versions between Banach spaces are considered. With this we also review the definitions and behaviour of Peano and (higher-order) Fréchet differentiability, as well as analyticity and Gevrey smoothness. Especially, we prove various qualitative and quantitative results concerning both the composition of maps and implicit map theorems for polynomials and power series, as well as Peano differentiable, Fréchet differentiable, analytic or Gevrey smooth maps. These mathematical tools enable our analysis of the class of semilinear elliptic partial differential equations.