4. Juni 2024
Thomas Kalmes (TU Chemnitz)
Linear topological invariants for kernels of differential operators
Let P(D) be a constant coefficient linear partial differential operator which is surjective on a given space of (generalized) functions F(X) on an open subset X of Rd. Given data (fλ)λ in F(X) that depend "regularly" on a parameter λ, it is a natural problem whether there exist solutions uλ in F(X) to the equations P(D)uλ=fλ such that (uλ)λ also depend regularly on λ. By abstract functional analytic methods, it turns out that in many relevant situations, this problem has an affirmative solution when the kernel of P(D) in F(X) satisfies a certain linear topological invariant.
In the talk, we discuss these linear topological invariants for important classes of partial differential operators on the space of smooth functions and the space of distributions, respectively. Among others, we consider (subspace) elliptic operators and parabolic operators, and we give a complete characterization for arbitrary operators in case of d=2.
Parts of the talk are based on joint work with Andreas Debrouwere (Vrije Universiteit Brussel, Belgium).