Abstract
AbstractThe necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.
Funder
Università degli Studi di Roma La Sapienza
Publisher
Springer Science and Business Media LLC
Reference18 articles.
1. Agmon, S.: On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. Methods Funct. Anal. Theory Ellipt. Equ. 19–52 (1982)
2. Berestycki, H., Caffarelli, L.A., Nirenberg, L.: Inequalities for second-order elliptic equations with applications to unbounded domains I. Duke Math. J. 81(2), 467–494 (1996)
3. Berestycki, H., Caffarelli, L.A., Nirenberg, L.: Monotonicity for elliptic equations in unbounded Lipschitz domains. Commun. Pure Appl. Math. 50(11), 1089–1111 (1997)
4. Berestycki, H., Nirenberg, L., Varadhan, S.R.S.: The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Commun. Pure Appl. Math. 47(1), 47–92 (1994)
5. Berestycki, H., Rossi, L.: Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains. Commun. Pure Appl. Math. 68(6), 1014–1065 (2015)