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Zbl 1142.35315
Kawohl, Bernhard; Kutev, Nikolay
Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations.
(English)
[J] Funkc. Ekvacioj, Ser. Int. 43, No. 2, 241-253 (2000). ISSN 0532-8721

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.
MSC 2000:
*35B05 General behavior of solutions of PDE
35B50 Maximum principles (PDE)
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations

Citations: Zbl 0907.35008

Cited in: Zbl 1248.60066

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