Maximum Theorems for Solutions of Higher Order Elliptic Equations by Shmuel Agmon

The classical maximum modulus theorem for solutions of second order elliptic equations was recently extended by C. Miranda [4] to the case of real higher order elliptic equations in two variables. Previously Miranda [3] has derived a maximum theorem for solutions of the biharmonic equation in two variables. In the case of more variables it was observed by Agmon-Douglis-Nirenberg [2 ] that a maximum theorem holds in the special case of elliptic operators with constant coefficients with no lower order terms when the domain of definition is a half-space. In this note we describe a very general maximum theorem for solutions of (complex) higher order elliptic equations in any number of variables. We shall obtain various estimates in the maximum norm which will contain as a special case the extension of Miranda's results to any number of variables. We denote by G a bounded domain in En with boundary dG and closure G. For a function u(E.C(G) we introduce the usual maximum norm :