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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 2 u)=0 in Ω 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 2 u)=0 in Ω or degenerate parabolic equations (3) G(Du)u t +F(u,Du,D 2 u)=0 in Q=Ω×(0,T), with G(p)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(u,Du,D 2 u)=0 in Ω,u=g on Ω, or (5) G(Du)u t +F(u,Du,D 2 u)=0 in Q,u=g on Γ:=(Ω×0)(Ω×[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 Ω or on Γ, 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:
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B50Maximum principles (PDE)
35J60Nonlinear elliptic equations
35K55Nonlinear parabolic equations