A new proof of the maximum principle for doubly-harmonic functions

Henry B. Mann, Josephine Mitchell and Lowell Schoenfeld

Vol. 27 (1968), No. 3, 567–571

DOI: 10.2140/pjm.1968.27.567

Abstract

Let f be a real-valued Lebesque integrable function on a domain Ω in Euclidean space E_2m, and let f be doubly-harmonic on Ω so that it satisfies

In this paper, a new proof of the maximum principle is given for nonconstant functions f satisfying the preceding conditions.

The proof depends on the fact that the associated forms

where A ∈ Ω, are either indefinite or identically 0 for each p ≧ 1. The authors previously proved this under weaker hypotheses on f, but the proof used the strong form of the maximum principle for solutions of linear elliptic partial differential equations of the second order with constant coefficients. By means of the theory of distributions, the authors now prove that the φ_p(H;f) have the stated property without using the maximum principle. Consequently, they obtain a new proof of this principle.

Mathematical Subject Classification

Primary: 31.30

Milestones

Received: 12 September 1967

Published: 1 December 1968