Quantitative Runge type approximation theorems for zero solutions of certain partial differential operators

TL;DR

This paper establishes quantitative approximation theorems for smooth solutions of various linear PDEs, including elliptic, parabolic, and wave operators, on convex sets, advancing the understanding of solution approximation in PDE theory.

Contribution

It introduces new quantitative Runge approximation results for zero solutions of multiple classes of linear PDEs with constant coefficients, extending previous qualitative results.

Findings

Quantitative Runge approximation for elliptic, parabolic, and wave operators.

New qualitative approximation theorem for subspace elliptic operators.

Methodology based on linear topological invariants for PDE kernels.

Abstract

We prove quantitative Runge type approximation results for spaces of smooth zero solutions of several classes of linear partial differential operators with constant coefficients. Among others, we establish such results for arbitrary operators on convex sets, elliptic operators, parabolic operators, and the wave operator in one spatial variable. Our methods are inspired by the study of linear topological invariants for kernels of partial differential operators. As a part of our work, we also show a qualitative Runge type approximation theorem for subspace elliptic operators, which seems to be new and of independent interest.