Necessity of parameter-ellipticity for multi-order systems of differential equations

R. Denk, M. Faierman


Abstract. In this paper we investigate parameter-ellipticity conditions for multi-order systems of differential equations on a bounded domain. Under suitable assumptions on smoothness and on the order structure of the system, it is shown that parameter-dependent a priori-estimates imply the conditions of parameter-ellipticity, i.e., interior ellipticity, conditions of Shapiro-Lopatinskii type, and conditions of Vishik-Lyusternik type. The mixed-order systems considered here are of general form; in particular, it is not assumed that the diagonal operators are of the same order. This paper is a continuation of an article by the same authors where the sufficiency was shown, i.e., a priori-estimates for the solutions of parameter-elliptic multi-order systems were established.

Oper. Theory Adv. Appl. 221  (2012), 255-268.

The paper is available here (Preprint version).

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