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Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations. (English) Zbl 1142.35315
In [Arch. Math. (Basel) 70, No. 6, 470–478 (1998; Zbl 0907.35008)], we investigated the class of fully nonlinear elliptic equations (1) $F\left(x,u,Du,{D}^{2}u\right)=0$ in ${\Omega }\subset {ℝ}^{n}$ which are proper and strictly elliptic but not uniformly elliptic and not uniformly proper. Under this main assumption we proved a strong maximum principle for semicontinuous viscosity sub- and supersolutions of (1). However, the comparison principle given in [loc. cit.] is valid only when one of the semicontinuous viscosity sub- or supersolutions is a piecewise ${C}^{2}$ smooth function. In the present paper we investigate only the class of autonomous elliptic equations (2) $F\left(u,Du,{D}^{2}u\right)=0$ in ${\Omega }$ or degenerate parabolic equations (3) $G\left(Du\right){u}_{t}+F\left(u,Du,{D}^{2}u\right)=0$ in $Q={\Omega }×\left(0,T\right)$, with $G\left(p\right)\ge 0$, and show that under the same assumptions as in [loc. cit.] an unrestricted comparison principle for semicontinuous viscosity sub- and supersolutions of (2) and (3) holds. In other words, the ${C}^{2}$ assumption can be dropped. By means of the Perron procedure this comparison principle guarantees existence of a unique continuous viscosity solution to the Dirichlet problems (4) $F\left(u,Du,{D}^{2}u\right)=0$ in ${\Omega },u=g$ on $\partial {\Omega }$, or (5) $G\left(Du\right){u}_{t}+F\left(u,Du,{D}^{2}u\right)=0$ in $Q,u=g$ on ${\Gamma }:=\left({\Omega }×0\right)\cup \left(\partial {\Omega }×\left[0,T\right]\right)$ provided (2) and (3) have a sub- and supersolution satisfying the boundary data. If additionally a continuous viscosity solution is Lipschitz continuous on the boundary $\partial {\Omega }$ or on ${\Gamma }$, then the solution is Lipschitz continuous in the whole domain, i.e. regularity is inherited from the boundary to the interior of the domain.

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35B50 Maximum principles (PDE) 35J60 Nonlinear elliptic equations 35K55 Nonlinear parabolic equations