We present a quasi maximum principle stating roughly that holomorphic solutions of a given partial differential equation with constant coefficents in C n ,
achieve essentially their maximal growth on a certain algebraic hypersurface 1 related to the operator. We prove it in the case where P is homogeneous and 1 is the conjugate dual cone, and also in the case where
and 1 is the complexified real sphere. We obtain a weak (semi-local) variant of the quasi maximum principle for certain non-homogeneous operators P(D), in which case 1 is the conjugate dual cone related to the principal part of the operator. This weaker variant is closely intertwined with several other notions. One of them is a quasi balayage principle for solutions of (-), involving the ``sweeping'' of measures in C n onto 1. 1997 Academic Press Contents 1. Introduction. 1.1 Partial differential equations in the holomorphic category. 1.2. Scope of the present paper. 1.3. Notations. 2. The weak quasi maximum principle. 2.1. The main result. 2.2. Operators with non-closed ranges. 3. Fischer majorants and the quasi maximum principle. 3.1. The main result. 3.2. An example where the WQMP holds, but the QMP fails. 4. A quasi maximum principle for the sphere.
- INTRODUCTION 1.1. Partial Differential Equations in the Holomorphic Category Partial differential equations in the holomorphic category i.e., where the derivatives in the equation are holomorphic derivatives, where the coefficients and data of the problem are holomorphic, and where we search for article no.