Quantitative Runge type approximation theorems for zero solutions of certain partial differential operators
arXiv:2209.10794 · doi:10.1016/j.aim.2025.110413
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.
arXiv comment27 pages; comments welcome; v2: fixed some typos; reference to implicit approximation results as well as a global approximation result due to Malgrange has been added in the introduction which allows to choose approximating global solutions as linear combinations of exponential solutions; slight refomulation of Theorems 1.1, 1.2 and 1.3; Remark 5.10 and Example 5.11 have been removed
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