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

${\Omega}\subset {\mathbb{R}}^{n}$ 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)

$F(u,Du,{D}^{2}u)=0$ in

${\Omega}$ or degenerate parabolic equations (3)

$G\left(Du\right){u}_{t}+F(u,Du,{D}^{2}u)=0$ in

$Q={\Omega}\times (0,T)$, 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)

$F(u,Du,{D}^{2}u)=0$ in

${\Omega},u=g$ on

$\partial {\Omega}$, or (5)

$G\left(Du\right){u}_{t}+F(u,Du,{D}^{2}u)=0$ in

$Q,u=g$ on

${\Gamma}:=({\Omega}\times 0)\cup (\partial {\Omega}\times [0,T\left]\right)$ 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.