The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

TL;DR

This paper develops Gevrey class regularity bounds for implicit solution mappings in parametric operator equations, with applications to uncertainty quantification in semilinear elliptic PDEs, enhancing understanding of solution smoothness under randomness.

Contribution

It introduces a Gevrey class implicit mapping theorem and applies it to establish regularity bounds for solutions of semilinear elliptic PDEs with parametric and random inputs.

Findings

Gevrey bounds on derivatives of solution mappings

Regularity results for parametric elliptic PDEs

Framework applicable to uncertainty quantification

Abstract

This article is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under $s$-Gevrey assumptions on on the residual equation, we establish $s$-Gevrey bounds on the Fr\'echet derivatives of the local data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.