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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) $F(x,u,Du,D^2u)=0$ in $\Omega\subset\bbfR^n$ 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) $F(u,Du,D^2u)=0$ in $\Omega$ or degenerate parabolic equations (3) $G(Du)u_t+F(u,Du,D^2u)=0$ in $Q=\Omega\times(0,T)$, with $G(p)\geq0$, 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) $F(u,Du,D^2u)=0$ in $\Omega, u=g$ on $\partial\Omega$, or (5) $G(Du)u_t+F(u,Du,D^2u)=0$ in $Q, u=g$ on $\Gamma := (\Omega\times{0})\cup(\partial\Omega\times[0,T])$ 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.

35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B50Maximum principles (PDE)
35J60Nonlinear elliptic equations
35K55Nonlinear parabolic equations