Year: 1968
Type: article
Abstract: Let / be a real-valued Lebesque integrable function on a domain Ω in Euclidean space E 2m , and let / be doubly-harmonic on Ω so that it satisfies for In this paper, a new proof of the maximum principle is given for nonconstant functions / satisfying the preceding conditions.The proof depends on the fact that the associated forms φ( -H , f)= Σ h? • h> / d*f where A e Ω 9 are either indefinite or identically 0 for each p^l.The authors previously proved this under weaker hypotheses on /, 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.l Introduction* We say that / is doubly-harmonic on a domain
Source: Pacific Journal of Mathematics
Institutions: Pennsylvania State University, United States Army
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2095249228