From zero solutions with partially bounded supports to solvability and Runge approximation for partial differential equations

Abstract

The existence of non-trivial solutions of homogeneous partial differential equations with prescribed support properties plays a fundamental role in the theory of linear partial differential operators. In this article, we establish the existence of smooth zero solutions with partially bounded supports for a broad class of constant coefficient partial differential operators. As applications, we obtain new geometric results concerning solvability and approximation for partial differential equations. In particular, we derive geometric characterizations of $P$-convexity for supports for a large class of non-elliptic operators, thereby extending classical geometric ideas of H\"ormander. By a theorem of Malgrange, $P$-convexity for supports is equivalent to the surjectivity of the differential operator on spaces of smooth functions and, more generally, on local subspaces of distributions of finite order. As a further consequence, we prove that the kernel of every surjective semi-elliptic operator satisfies condition ($\Omega$) implying parameter dependence results. We also investigate Runge-type approximation phenomena. We obtain geometric characterizations of Runge pairs for smooth functions, distributions, and spaces of smooth Whitney jets. Finally, we develop a general framework for Runge approximation for square systems of constant coefficient partial differential equations. As applications, we recover by alternative methods known Runge approximation results for Beltrami fields and for the three-dimensional unsteady Stokes system, and we complement them by corresponding approximation theorems in spaces of smooth Whitney jets. In particular, this yields approximation up to the boundary for solutions on suitable domains with H\"older continuous boundary.