Seminario EDPs y Mecánica de Fluidos

Quantitative Runge approximation theorems and linear topological invariants for kernels of partial differential operators

Ponente:  Thomas Kalmes (Chemnitz University of Technology, Faculty of Mathematics)
Fecha:  jueves 23 de marzo de 2023 - 11:30
Lugar:  Aula Naranja, ICMAT


We present quantitative Runge type approximation results for spaces of smooth zero solutions of several classes of constant coefficients linear partial differential operators. Among others, we discuss such results for arbitrary operators on convex sets, elliptic operators, parabolic operators, and the wave operator in one spatial variable. The presented method is based on qualitative approximation results on the one hand and recent results on Vogt's and Wagner's topological invariant \((\Omega)\) for kernels of partial differential operators on the other hand which originates from the structure theory of nuclear Fréchet spaces. The talk is based on joint work with Andreas Debrouwere (University of Brussels).

[1] A. Debrouwere, T. Kalmes. Linear topological invariants for kernels of convolution and differential operators. arXiv-preprint 2204.11733, accepted for publication in J. Funct. Anal.

[2] A. Debrouwere, T. Kalmes. Quantitative Runge type approximation theorems for zero solutions of certain partial differential operators. arXiv-preprint 2209.10794, 2022.

[3] T. Kalmes. An approximation theorem of Runge type for certain non-elliptic partial differential operators. Bull. Sci. Math. 170, Paper No. 103012, 26 pp., 2021.
[4] D. Vogt, M.J. Wagner. Charakterisierung der Quotientenräume von \(s\) und eine Vermutung von Martineau. Studia Math. 67(3):225--240, 1980.