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Invariant convex bodies for strongly elliptic systems

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Ariel University, Ariel, 40700, Israel

    Gershon Kresin

  2. Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, UK

    Vladimir Maz′ya

  3. Department of Mathematics, Linköping University, SE-58183, Linköping, Sweden

    Vladimir Maz′ya

  4. RUDN University, 6, Miklukho-Maklay St., Moscow, 117198, Russia

    Vladimir Maz′ya

Authors
  1. Gershon Kresin
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  2. Vladimir Maz′ya
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Correspondence to Gershon Kresin.

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Kresin, G., Maz′ya, V. Invariant convex bodies for strongly elliptic systems. JAMA 135, 203–224 (2018). https://doi.org/10.1007/s11854-018-0033-z

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  • DOI: https://doi.org/10.1007/s11854-018-0033-z

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