TY - JOUR

T1 - A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation

AU - Ohnuma, Masaki

AU - Sakaguchi, Shigeru

N1 - Funding Information:
This research was partially supported by the Grants-in-Aid for Challenging Exploratory Research (♯25610024) and for Scientific Research (B) (♯26287020) of Japan Society for the Promotion of Science.

PY - 2019/4

Y1 - 2019/4

N2 - A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.

AB - A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.

KW - Prescribed mean curvature equation

KW - Semicontinuous viscosity solution

KW - Strong comparison principle

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U2 - 10.1016/j.na.2018.11.010

DO - 10.1016/j.na.2018.11.010

M3 - Article

AN - SCOPUS:85058496363

VL - 181

SP - 180

EP - 188

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -