A strong maximum principle for parabolic systems in a convex set with arbitrary boundary

Evans, Lawrence

doi:10.1090/S0002-9939-2010-10495-1

A strong maximum principle for parabolic systems in a convex set with arbitrary boundary
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Proc. Amer. Math. Soc. 138 (2010), 3179-3185 Request permission

Abstract:

In this paper we prove a strong maximum principle for certain parabolic systems of equations. In particular, our methods place no restriction on the regularity of the boundary of the convex set in which the system takes its values, and therefore our results hold for any convex set. We achieve this through the use of viscosity solutions and their corresponding strong maximum principle.

References

K. N. Chueh, C. C. Conley, and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), no. 2, 373–392. MR 430536, DOI 10.1512/iumj.1977.26.26029
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
Francesca Da Lio, Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations, Commun. Pure Appl. Anal. 3 (2004), no. 3, 395–415. MR 2098291, DOI 10.3934/cpaa.2004.3.395
Louis Nirenberg, A strong maximum principle for parabolic equations, Comm. Pure Appl. Math. 6 (1953), 167–177. MR 55544, DOI 10.1002/cpa.3160060202
Xuefeng Wang, A remark on strong maximum principle for parabolic and elliptic systems, Proc. Amer. Math. Soc. 109 (1990), no. 2, 343–348. MR 1019284, DOI 10.1090/S0002-9939-1990-1019284-8
Hans F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat. (6) 8 (1975), 295–310 (English, with Italian summary). MR 397126