A note on the strong maximum principle for elliptic differential inequalities

AbstractWe consider the strong maximum principle and the compact support principle for quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. This allows us to give necessary and sufficient conditions for the validity of both principles

The strong maximum principle revisited

AbstractIn this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g. (i)two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4);(ii)the exterior Dirichlet boundary value problem (Section 5);(iii)the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7);(iv)Euler–Lagrange inequalities on a Riemannian manifold (Section 9);(v)comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10). The case of p-regular elliptic inequalities is briefly considered in Section 11

Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation

The final publication is available at Springer via https://doi.org/10.1007/s002050050032. Levine, Howard A., and James Serrin. "Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation." Archive for Rational Mechanics and Analysis 137, no. 4 (1997): 341-361. Posted with permission.</p

Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation

The final publication is available at Springer via https://doi.org/10.1007/s002050050032. Levine, Howard A., and James Serrin. "Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation." Archive for Rational Mechanics and Analysis 137, no. 4 (1997): 341-361. Posted with permission.</p