Invariant convex bodies for strongly elliptic systems
- 35 Downloads
We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies are obtained for linear systems without zero order term on bounded domains and quasilinear systems of special form on bounded domains and on 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.
Unable to display preview. Download preview PDF.
- Yu. D. Burago and V. G. Maz’ya, Potential Theory and the Function Theory for Irregular Regions, Zap. Nauchn. Sem. LOMI 3 (1967); English translation: Sem. in Mathematics, V. A. Steklov Math. Inst., Leningrad 3, Consultants Bureau, New York, 1969.Google Scholar
- Ya. B. Lopatinskiĭ, On a method of reducing boundary value problems for systems of differential equations of elliptic type to regular integral equations, Ukrain. Mat. Žurnal 5 (1953), 123–151 (Russian).Google Scholar
- Z. Ya. Shapiro, The first boundary value problem for an elliptic system of differential equations, Mat. Sb. 28(70) (1951), 55–78 (Russian).Google Scholar
- H. F. Weinberger, Some remarks on invariant sets for systems, in Maximum Principles and Eigenvalue Problems in Partial Differential Equations, Longman Scientific & Technical, 1988, pp. 189–207.Google Scholar