Invariant convex bodies for strongly elliptic systems

TL;DR

This paper establishes algebraic conditions under which convex bodies remain invariant for strongly elliptic second-order systems, covering both bounded and unbounded domains, with new necessary and sufficient criteria that impose no restrictions on coefficient matrices.

Contribution

It provides new algebraic criteria for the invariance of convex bodies in strongly elliptic systems, including necessary and sufficient conditions without restrictions on coefficients.

Findings

Derived algebraic conditions for invariance in linear systems.

Established criteria for quasilinear systems in various domains.

Identified necessary and sufficient conditions for elliptic homogeneous systems.

Abstract

We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies obtained for linear systems without zero order term in bounded domains and quasilinear systems of special form in bounded and in a class of unbounded domains. These conditions are formulated in algebraic form. They describe relation between the geometry of the invariant convex body and the coefficients of the system. Next, necessary conditions, which are also sufficient, for the invariance of some convex bodies are found for elliptic homogeneous systems with constant coefficients in a half-space. The necessary conditions are derived by using a criterion on the invariance of convex bodies for normalized matrix-valued integral transforms also obtained in the paper. In contrast with the previous studies of invariant sets for elliptic systems no a priori restrictions on the coefficient matrices are imposed.