0$ and $B_R=\{f\in L^n(\Omega)\colon\ \|f\|_{L^n(\Omega)}\le R\}$. Then the set of all functions $u\in C(\overline \Omega)$ such that there exists $\psi\in{\cal C}$ and $f\in B_R$ for which $u$ is an $L^n$-viscosity solution of both $${\cal P}^-(D^2 u)-\gamma|Du|\le f\text{\it and} -f\le {\cal P}^+(D^2 u)+\gamma|Du| \eq$$ in $\Omega$ and $u=\psi$ on $\partial \Omega$ is precompact in $C(\overline \Omega)$. } \proof{According to [4] Corollary 3.10, if $(\varphi,g)\in C(\partial \Omega)\times L^n(\Omega)$, there exist a unique $U=U(\varphi,g) \in C(\overline \Omega)\cap W^{2,n}_{\rm loc}(\Omega)$ such that $$ {\cal P}^-(D^2 U)-\gamma|DU|=g\text{a.e. in}\Omega\text{and} u =\varphi\text{on}\partial \Omega. $$ We require several facts. First, there exist $p\in(n/2,n)$ depending on $\lambda, \Lambda, n, \gamma\,{\rm diam}\,(\Omega)$ (see [13], [11], [4], [28], [1]) and $C$ such that $$U(\varphi,g)\le \sup_{\partial \Omega}\varphi+ C\left(\int_{\Omega}(g^+)^p\right)^{1\over p}\text{for}(\varphi,g)\in C(\partial \Omega)\times L^n(\Omega).\eq$$ Next the mapping $$ \ C(\partial \Omega)\times L^n(\Omega)\ni(\varphi,g)\rightarrow U(\varphi,g)\text{is sublinear and order preserving.}\eq $$ Finally, if $u\in C(\overline \Omega)$ is an $L^n$-viscosity solution of $$ {\cal P}^-(D^2 u)-\gamma|Du|\le f\text{in}\Omega\text{and}u=\psi\text{on}\partial \Omega, \eq$$ where $f\in L^n(\Omega)$, then $$ u\le U(\psi,f).\eq$$ We review the genesis of these results in Remark 4.4\ below. The second inequality of (4.6) may be restated as $w=-u$ is an $L^n$-viscosity solution of ${\cal P}^-(D^2 w)-\gamma|Dw|\le f$, so if (4.6) holds (4.10) implies $-u\le U(-\psi,f)$ or $-U(-\psi,f)\le u$. All told, (4.6) and $u=\psi$ on $\partial \Omega$ yield $$-U(-\psi,f)\le u\le U(\psi,f).\eq$$ From (4.11) and (4.7) it follows that $u$ remains bounded in $C(\overline \Omega)$ if $(\psi,f)$ remains bounded in $C(\partial \Omega)\times L^n(\Omega)$ (or even in $C(\partial \Omega)\times L^p(\Omega)$). We now use (4.11) to show that $u$ assumes the boundary values $\psi$ in an equicontinuous manner. In this regard, let $f_M=\max(\min(f,M),-M)$ be the standard truncation of $f$ for $M>0$. We note that for $f\in B_R$ $$\|f-f_M\|_{L^p(\Omega)}\le \left({\rm measure}(\{|f|>M\})\right)^{{n-p\over np}}R\le R\left({R\over M}\right)^{{n-p\over p }},$$ which tends to 0 as $M\rightarrow \infty$ uniformly in $f\in B_R$. Using the properties (4.8), (4.7) of $U$ we thus have $$\eqalign{U(\psi,f)\le U(\psi,f_M)+U(0,f-f_M)&\le U(\psi,M)+C\|f- f_M\|_{L^p(\Omega)}\cr &\qquad\le U(\psi,M)+CR\left({R\over M}\right)^{{n-p\over p }}.}$$ According to Proposition 3.2, $U(\psi, M)$ assumes the boundary values $\psi$ in a manner controlled by the modulus of continuity of $\psi$ for fixed $M$. The ``error term" on the right above can be made as small as desired by choosing $M$ sufficiently large, and $u\le U(\psi,f)$ thus guarantees an estimate $u(x)-\psi(y)\le \rho(|x-y|)$ for $x\in\overline \Omega, y\in \partial \Omega$, where $\rho(0+)=0$. Similarly, $-U(-\psi,f)\le u$ provides control of $u-\psi$ at the boundary from below. Finally, once $u$ is bounded, (4.6) guarantees equi-H\"older continuity of $u$ on compact subsets of $\Omega$ so long as $f$ remains bounded in $L^n(\Omega)$ (see, for example, [13] for a sufficiently general statement and Remark 4.7 below). The result follows.} \remark{4.3}{ Proposition 4.2\ can be reformulated by saying that if $u$ satisfies (4.6) and $u=\psi$ on $\partial \Omega$ then $u$ has a modulus of continuity on $\overline\Omega$ that only depends on the parameters of the cone condition, $\lambda,\Lambda,n,\gamma$, ${\rm diam}\,(\Omega)$, $R$ and the modulus of continuity of $\psi$.} \remark{4.4}{The inequality (4.7) generalizes the original work of Fabes and Stroock [12] and is proved in [13] in the spirit of this work, but it could also be deduced from Cabr\'e [1]; its relevance in this arena was first shown by Escauriaza [11]. In fact, the existence of $U(\varphi,g)$ for $\gamma=0$ was proved in [11] relying on (4.7) with $\gamma=0$. The properties (4.8) are a consequence of the positive homogeneity and superlinearity of $(p,X)\rightarrow {\cal P}^-(X)-\gamma|p|$ and (4.7) ($p=n$ suffices). For example, the superadditivity implies that $W=U(\varphi,g)-U(\hat\varphi,\hat g)$ solves $${\cal P}^-(D^2 W )-\gamma|DW|\le g-\hat g\le 0$$ if $g\le\hat g$ and an application of (4.7) ($p=n$ suffices) then proves the order preserving property. The relation (4.10) given (4.9) follows upon observing that $v=u-U(\psi,f)$ is an $L^n$-viscosity solution of ${\cal P}^-(D^2 v)-\gamma|Dv|\le 0$ and the Alexandrov-Bakelman-Pucci maximum principle for viscosity solutions proved in [2] ($\gamma=0$), [30], [4]. Finally, Proposition 4.2\ itself appears in [3], Theorem 4.14, in the situation where $\gamma=0$, $\Omega$ is a ball, and all functions $f$ appearing in (4.6) are continuous. This proof could be adapted, with effort, to the current case. The current proof uses the work already done in Section 3.} \proofof{Theorem 4.1}{First we assume that $F(x,r,p,X)$ is defined for all $(r,p,X)$ for all $x\in{\Bbb R}^n$ and satisfies the structure conditions (1.2), (1.3) and (4.1) for all $x\in{\Bbb R}^n$. To achieve this, if necessary extend $F(x,r,p,X)$ to be ${\cal P}^-(X)-\gamma|p|$ (or ${\cal P}^+(X)+\gamma|p|$) for those $x$'s where it was not originally defined. Now mollify $F$ in $x$: $$F_\epsilon(x,r,p,X)={1\over\epsilon^n} \int_{{\Bbb R}^n}\eta\left({x-y\over\epsilon}\right)F(y,r,p,X)\,dy, $$ where $\eta\in C_0^\infty({\Bbb R}^n)$ satisfies $\eta\ge 0$ and $\int_{{\Bbb R}^n}\eta(x)\,dx=1$. The structure conditions are preserved under this sort of averaging, so $F_\epsilon\in{\cal SC}$. Clearly $F_\epsilon$ satisfies (4.1), (4.4) and (4.5) as well as $F$. Moreover, the bound (4.5) on $|F|$ gives us $$|F_\epsilon(x,r,p,X) -F_\epsilon(y,r,p,X)| \le \frac C\epsilon|x-y|\left(\Lambda\|X\|+\gamma|p|+\beta(r)\right) $$ for some $C$. Fix $f\in L^n(\Omega)$ and let $f_j\in C(\overline \Omega)$ satisfy $$\|f_j-f\|_{L^n(\Omega)}\rightarrow 0\text{as}j\rightarrow \infty.$$ Since $F_\epsilon\in{\cal SC}$ is continuous, according to Theorem 1.1\ the problem $$F_\epsilon(x,u,Du,D^2u)=f_j\text{in}\Omega\text{and} u=\psi\text{on}\partial \Omega\eq$$ has a $C$-viscosity solution (and hence $L^n$-viscosity solution) $u=u_{\epsilon,j}$. Clearly $u=u_{\epsilon,j}$ also solves $${\cal P}^-(D^2u)-\gamma|Du|+F_\epsilon(x,u,0,0)\le f_j\text{and}f_j\le {\cal P}^+(D^2u)+\gamma|Du|+F_\epsilon(x,u,0,0). $$ Since $F_\epsilon(x,u,0,0)\ge F_\epsilon(x,0,0,0)=0$ if $u\ge 0$, the first relation above and the maximum principle for viscosity solutions implies $$u\le \sup_{\partial \Omega}u^++C\|f_j\|_{L^n(\Omega)}\le\sup_{\partial \Omega}\psi^+ +C\sup_j\|f_j\|_{L^n(\Omega )} $$ and we conclude that the $u_{\epsilon,j}$ are bounded above independently of $\epsilon, j$. Likewise, the $u_{\epsilon,j}$ are bounded below independently of $\epsilon, j$, and hence the family is uniformly bounded. Using this information and (4.5) for $F_\epsilon$, there exists a constant $K$ such that $|F_\epsilon(x,u_{\epsilon,j},0,0)|\le K$ and the $u_{\epsilon,j}$ satisfy $${\cal P}^-(D^2u_{\epsilon,j})-\gamma|Du_{\epsilon,j}|\le g_j\text{and}-g_j\le {\cal P}^+(D^2u_{\epsilon,j})+\gamma|Du_{\epsilon,j}|,$$ where $g_j=|f_j|+K$. Therefore, using Proposition 4.2, there exists $\epsilon_m\downarrow 0$, $j_m\rightarrow\infty$ such that $u_m=u_{\epsilon_m,j_m}$ converges uniformly on $\overline \Omega$ to a limit $u$. By Theorem 3.8 of [4] this $u$ is an $L^n$-viscosity solution of (4.3); indeed, what we need to check to use this result is only that for $\varphi\in W^{2,n}_{\rm loc}(\Omega)$ we have $$F_{\epsilon_m}(x,u_m(x),D\varphi(x),D^2\varphi(x)) \rightarrow F(x,u(x),D\varphi(x),D^2\varphi(x))\eq$$ in $L^n_{\rm loc}(\Omega)$. However, $F_\epsilon(x,r,p,X)\rightarrow F(x,r,p,X)$ whenever $x$ is a Lebesgue point of $F(\cdot,r,p,X)$, and almost every $x$ has this property for all $r,p,X$ by $F\in{\cal SC}$ (see [4], page 382), which together with (4.5) shows that (4.13) holds pointwise a.e. and (locally) dominated, hence in $L^n_{\rm loc}(\Omega)$.} We now turn to the parabolic analogue of Theorem 4.1. In this case the initial boundary value problem can be rewritten as before as $$u_t+F(x,t,u,Du,D^2u)=f(x,t)\text{in}Q=\Omega\times (0,T],\quad u=\psi\text{on} \partial_pQ,\eq $$ where $$F(x,t,0,0,0)\equiv 0.\eq$$ The proof of the theorem below is similar to the one in the elliptic case and is therefore omitted, save for the remarks to follow. \theorem{4.5}{Let $F\in{\cal SC}$ satisfy (4.1) and (4.15), let $f\in L^{n+1}(Q)$, $\psi\in C(\partial_pQ)$ and let $\Omega$ satisfy a uniform exterior cone condition. Then (4.14) has an $L^{n+1}$-viscosity solution.} The parabolic version of the compactness result Proposition 4.2\ is \proposition{4.6}{Let $\Omega$ satisfy a uniform exterior cone condition and ${\cal C}\subset C(\partial_pQ)$ be compact, $R>0$ and $B_R=\{f\in L^{n+1}(Q)\colon\ \|f\|_{L^{n+1}(Q)}\le R\}$. Then the set of all functions $u\in C(\overline Q)$ such that there exists $ \psi \in{\cal C}$ and $f\in B_R$ for which $u$ is an $L^{n+1}$-viscosity solution of both $$u_t+{\cal P}^-(D^2u)-\gamma|Du|\le f\text{\it and} -f\le u_t+{\cal P}^+(D^2u)+\gamma|Du| \eq$$ in $Q$ and $u=\psi$ on $\partial_pQ$ is precompact in $C(\overline Q)$. } A version of the maximum principle and an existence result sufficient for the proof of this proposition is given in [7]. The interior H\"older continuity is established in [10], Section 5. The limit theorem needed to complete the proof of Theorem 4.5\ is proved in [10], Section 6. \remark{4.7}{ The proofs of the existence results above do not require full Propositions 4.2\ and 4.6\ but rather their versions with $B_R$ replaced by $B_R\cap C(\Omega)$ (or $B_R\cap C(Q)$). In this case, the proofs of the versions of the maximum principles and equi-H\"older continuity results found in Caffarelli [2], Trudinger [30] (elliptic case), and Wang [32] (parabolic case) could be used. This leaves aside (4.7), upon which we have commented. The parabolic analogue is proved in [7]. The proofs of the various maximum principles sketched in [7] might interest the reader in any case.} \remark{4.8}{ We note again, for emphasis, that Theorems 4.1\ and 4.5\ are also true if $n$ and $n+1$ are replaced by $p$ in (parameter dependent) appropriate ranges of the form $n-\delta < p $ and $n+1-\delta