\documentclass[reqno]{amsart}
\begin{document}
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.~2000(2000), No.~38, pp.~1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 2000 Southwest Texas State University and
University of North Texas.}
\vspace{1cm}
\title[\hfilneg EJDE--2000/17\hfil The maximum principle ]
{ The maximum principle for equations with composite coefficients }
\author[Gary M. Lieberman \hfil EJDE--2000/17\hfilneg]{ Gary M. Lieberman }
\address{ Gary M. Lieberman \hfill\break\indent
Department of Mathematics\\ Iowa State University\\
Ames, Iowa 50011}
\email{lieb@iastate.edu}
\date{}
\thanks{Submitted April 24, 2000. Published May 22, 2000.}
\subjclass{35J25, 35B50, 35J65, 35B45, 35K20 }
\keywords{elliptic differential equations, oblique boundary conditions,
maximum principles, \hfill\break\indent
H\"older estimates, Harnack inequality,
parabolic differential equations }
\begin{abstract}
It is well-known that the maximum of the solution of a linear elliptic equation
can be estimated in terms of the boundary data provided the coefficient of the
gradient term is either integrable to an appropriate power or blows up like a
small negative power of distance to the boundary. Apushkinskaya and Nazarov
showed that a similar estimate holds if this term is a sum of such functions
provided the boundary of the domain is sufficiently smooth and a Dirichlet
condition is prescribed. We relax the smoothness of the boundary and also
consider non-Dirichlet boundary conditions using a variant of the method of
Apushkinskaya and Nazarov. In addition, we prove a Holder estimate for
solutions of oblique derivative problems for nonlinear equations satisfying
similar conditions.
\end {abstract}
\newtheorem {theorem}{Theorem}[section]
\newtheorem {lemma}[theorem]{Lemma}
\newtheorem {corollary}[theorem]{Corollary}
\numberwithin {equation}{section}
\newcommand{\diam}{\operatorname{diam}}
\newcommand{\osc}{\operatorname*{osc}}
\newcommand{\tr}{\operatorname{tr}}
\maketitle
\section*{Introduction}
We are concerned here with various estimates for solutions of the linear
elliptic equation
\[
a^{ij}D_{ij}u+b^iD_iu+cu = f
\]
in some domain $\Omega\subset \mathbb R^n$ under weak hypotheses on the coefficients
$a^{ij}$, $b^i$, and $c$. We always assume that $[a^{ij}]$ is a positive-definite
matrix-valued function with minimum eigenvalue $\lambda$ and determinant $\mathcal D$, and
that $c \le 0$. It was shown in \cite {AL1} that, if $u=0$ on $\partial\Omega$, then $u$ satisfies
an estimate of the form
\[
\sup_\Omega u \le C(n,\|b/\lambda\|_n)(\diam\Omega)\left\| \frac f{\mathcal D^{1/n}}\right\|_n.
\]
On the other hand, if $\partial\Omega$ is sufficiently smooth and the vector $b$ grows like a
(sufficiently small, negative) power of the distance to $\partial\Omega$, then a similar estimate
holds via the maximum principle. To state this estimate, let us use $d$ to denote distance to
$\partial\Omega$. If there are positive constants $\alpha<1$, $B_0$, and $\mu$ such that
$\partial\Omega\in C^{1,\alpha}$, $|b| \le B_0\lambda d^{\alpha-1}$, and $|a^{ij}|
\le\mu\lambda$, then
\[
\sup_\Omega u \le C(n,\alpha,B_0,\mu,\Omega)\sup_\Omega |fd^{1-\alpha}/\lambda|.
\]
(Although this precise form of the maximum principle does not seem to be stated anywhere, we
point out that its proof is contained in arguments that go back as least as far as \cite {GH}; see
also \cite {LC1}.) In \cite {AN1}, the authors showed that a composite condition on $b$ leads to
an analogous maximum principle. Specifically, assume that there are positive constants
$\alpha$, $B_0$, and $\mu$ such that $|a^{ij}| \le \mu\lambda$, $b=b_1+b_2$ with
\[
\left\|\frac {b_1}{\lambda}\right\|_n + \sup_{x\in \Omega} \frac
{|b_2(x)|}{\lambda(x)}(x^n)^{1-\alpha} \le B_0
\]
and $|x'| 0$ in $\Omega$ for some $R \in (0,1)$. Then \cite [Theorem 2.1$'$]
{AN1} states that
\[
\sup_\Omega u \le C(B_0, n, \alpha, \mu )R\left\| \frac f\lambda \right\|_n.
\]
(Actually, the proof of \cite [Theorem 2.1$'$] {AN1} seems to need a smallness condition on
$B_0$ for the reasons discussed in Remark 3.3 of \cite {Krylov}.) These estimates were used in
\cite {AN1} to infer the regularity of solutions to the Dirichlet problem for nonlinear elliptic
equations with coefficients satisfying similar composite conditions; parabolic problems are also
considered in \cite {AN1}.
Our goal here is to prove a maximum principle for problems with composite coefficients
under Dirichlet and non-Dirichlet boundary conditions. In addition, we obtain various
consequences of the maximum principle not used in \cite {AN1}. We shall apply these
estimates, such as the H\"older estimates (Corollaries \ref {C:Holder} and \ref {C:Holdermix}
below), to studying the smoothness of solutions to such problems in \cite {Lhigher}. On the
other hand, we shall not discuss analogs of the boundary gradient estimates from \cite {AN1}.
Our approach is based on that in \cite {AN1}, but there are some technical differences which we
indicate below. In principle, our estimates could be proved by modifying the method in \cite
{AN1}, but our techniques are of independent interest.
We begin in Section \ref {S:note} with a review of the relevant notation. Next, we construct
some supersolutions for the linear operator $L_2=a^{ij}D_{ij} + b_2^iD_i$ in Section \ref
{S:prelim}. Similar supersolutions for the full operator $L=a^{ij}D_{ij} + b^iD_i$ were used in
\cite {AN1}. These supersolutions are used to derive our maximum principles for subsolutions
of elliptic equations with composite coefficients in Section \ref {S:ell1}. Harnack and H\"older
inequalities for solutions of such equations are stated in Section \ref {S:locell}; we include them
for completeness, but their proofs from the maximum principles in the previous section contain
no new ingredients. Results for parabolic problems are given in Section \ref {S:para1}. We
close with some remarks, mostly about comparing our results with those in \cite {Krylov}, in
Section \ref {S:rmk}. Our investigation of this problem was heavily influenced by that paper
although the methods used here are quite different.
\section {Notation} \label {S:note}
Although our notation is generally quite standard, we list here some elements that may
not be immediately apparent. The reader is also directed to \cite {GT} for further information in
the elliptic case and to \cite {Lbook} in the parabolic case.
First, we write $\Omega$ for a bounded domain in $\mathbb R^n$ and we write $d$ for
the function defined by $d(x) = \inf \{ |x-y|: y\in \partial\Omega\}$. Points in $\mathbb R^n$ are
written as $x=(x^1,\dots,x^n)$ and we sometimes abbreviate $x'=(x^1,\dots,x^{n-1})$. For
$x_0\in \mathbb R^n$ and $R>0$, we define
\[
\Omega[x_0,R] = B(x_0,R) \cap \Omega,\ \Sigma[x_0,R] = B(x_0,R) \cap \partial\Omega,
\]
and we suppress $x_0$ from the notation when we assume that $x_0$ is the origin. Note that,
even when we refer to $\Omega[R]$ rather than $\Omega$, $d$ denotes the distance to
$\partial\Omega$.
We say that a vector $\beta$ is \emph {inward pointing} at $x_0 \in \partial\Omega$ if
there is a positive constant $\varepsilon$ such that $x_0+ t\beta \in \overline{\Omega}$ for $t\in
[0,\varepsilon]$. For $x_0 \in \partial\Omega$, a vector $\beta$ which is inward pointing at
$x_0$, a constant $k$, and a function $u\in C^0(\overline{\Omega})$, we say that $\beta \cdot
Du(x_0)
\ge k$ if
\[
\liminf _{t\to 0^+} \frac {u(x_0+t\beta)-u(x_0)}t \ge k.
\]
Similar definitions apply to the conditions $\beta \cdot Du(x_0) \le k$ and $\beta \cdot Du(x_0)
= k$. Moreover, if $\beta$ is a vector field defined on (a portion of) $\partial\Omega$, we say
that $\beta\cdot Du\ge g$ for some function $g$ if $\beta(x_0)\cdot Du(x_0) \ge g(x_0)$ for each
$x_0$ in (the portion of) $\partial\Omega$ with similar definitions for $\beta \cdot Du \le g$ and
$\beta \cdot Du=g$.
We recall that a continuous, increasing function $\zeta$, defined on $[0,1]$ is called {\em
Dini} if the function $\bar\zeta$ defined by $\bar\zeta(s) = \zeta(s)/s$ is in $L^1(0,1)$ and it is
{\em $1$-decreasing} if
\[
\frac {\zeta(s)}s \ge \frac {\zeta(\sigma)}\sigma
\]
for all $s\le \sigma$ in $(0,1)$. If $\zeta$ is Dini, we define
\[
I(\zeta)(s) = \int_0^s \frac {\zeta(\sigma)}\sigma\,d\sigma, \ J(\zeta)(s) = \int_0^s \frac
1{\sigma^2} \int_0^\sigma \zeta(t)\,dt\,d\sigma.
\]
It follows from \cite [Section 5]{Sperner} that $J$ is continuous, increasing, and $1$-decreasing
and that $J(\zeta) \le I(\zeta) \le 2J(\zeta)$. We shall say that $\zeta$ is a $D_1$ function if
$\zeta$ is Dini and $1$-decreasing with $\zeta(1) = 1$.
\section {Construction of supersolutions} \label {S:prelim}
The major step is to show that, for any $D_1$ function $\zeta$ and any $\mu \ge 1$, there is a
nonnegative function $w$ such that $a^{ij}D_{ij}w \le- \lambda\zeta(d/R)/d$ for any $[a^{ij}]$
satisfying
\begin {equation} \label {E:unifella}
|a^{ij}| \le \mu\lambda.
\end {equation}
It will be convenient to construct such a function locally first.
\begin {lemma} \label {L:supersoln}
Suppose that there are constants $\omega_0\ge0$ and $R>0$ along with a function $\omega$,
defined for $|x'|\omega(x'),\ |x| < R\},\ |\omega(x')-\omega(y')| \le \omega_0|x'-y'|
\end {equation}
for all $x'$ and $y'$ with $|x'|,|y'| 0$ along with a function $\omega$,
defined for $|x'|\varepsilon_0\diam\Omega$. But
\[
B_2|Dw_0|\frac {\zeta(\varepsilon_0)}{\varepsilon_0R} = \frac
{\zeta_1(\varepsilon_0)}{6C_0\varepsilon_0R}|Dw_0| \le \frac
{\zeta_1(\varepsilon_0)}{2\diam\Omega}
+\frac {\bar B}R |Dw_0|^2,
\]
where $\bar B = \diam\Omega /(72C_0^2\varepsilon_0R)$ since $\zeta_1$ is $1$-decreasing. It
follows
that
\[
a^{ij}D_{ij}w_1+b^iD_iw_1 \le -\lambda g' \frac {\zeta_1(d/\diam\Omega)}{2d} + \lambda
\left[ \frac
{\bar Bg'}R + g''\right]|Dw_0|^2
\]
wherever $d > \varepsilon_0\diam\Omega$. We now note that there is a positive constant $K_0$
such
that $K_0\zeta_1 \ge 6\zeta$ on the whole interval $[0,1]$, and we choose
\[
g(s)= K_0\frac R{\bar B}\exp(\bar B\sup w_0/R) [1-\exp (-\bar Bs/R)].
\]
Then straightforward calculations show that \eqref {E:superglobal} and \eqref {E:w1ests} hold.
\end {proof}
\section {The elliptic composite maximum principle} \label {S:ell1}
In order to prove our maximum principle for elliptic equations with composite
coefficients, we first prove an intermediate result which is a variant of the Aleksandroff
maximum principle. Instead of the usual upper contact set (see \cite [(9.5)]{GT}), for a function
$u \in C^0(\overline{\Omega})$ and a constant $\varepsilon \in (0,1)$, we introduce
$\Gamma_\varepsilon(u)$, the set of all $y\in \Omega$ such that $u(y) \ge 0$ and there is a
vector
$p$ with $|p|\le \varepsilon\sup u/(\diam\Omega +\beta_0)$ and $u(x) \le u(y)+p\cdot (x-y)$ for
all $x \in
\Omega$. We
also have the normal mapping $\chi$ defined by
\[
\chi(y) = \big\{p\in \mathbb R^n: u(x) \le u(y)+p\cdot (x-y)
\text { for all } x \in \Omega\big\}.
\]
\begin {lemma} \label {L:mpell}
Let $\Omega\subset \mathbb R^n$ be a Lipschitz domain, and define the operators $L$ and
$\mathcal M$ by
\begin {subequations} \label {E:defnLM}
\begin {gather} \label {E:Ldef}
Lu=a^{ij}D_{ij}u + b^iD_iu+cu, \\
\mathcal Mu = -u+\beta \cdot Du, \label {E:Mdef1}
\end {gather}
\end {subequations}
with $[a^{ij}]$ a positive-definite matrix-value function and $c \le 0$. Suppose there are
constants $B_1$ and $\beta_0$ such that
\begin {subequations} \label {E:SClower}
\begin {gather}
\left\| \frac {b}{\mathcal D^{1/n}}\right\|_{n,\Omega}\le B_1, \\
|\beta| \le \beta_0. \label {E:SCbeta}
\end {gather}
\end {subequations}
Let $u \in W^{2,n}_{\text{loc}}(\Omega) \cap C^0(\overline{\Omega})$ and suppose there is a
nonpositive function $f$ with $f/\mathcal D^{1/n}\in L^n(\Omega)$ such that $Lu \ge f$ in
$\Omega$ and $\mathcal Mu \ge 0$ on $\partial\Omega$. Then, there is a constant $C(n,B_1)$
such that, for any $\varepsilon \in (0,1)$,
\begin {equation} \label {Mpconcl}
\sup u \le C\frac {\diam\Omega+\beta_0}\varepsilon \left\| \frac {f}{\mathcal
D^{1/n}}\right\|_{n,\Gamma_\varepsilon(u)}.
\end {equation}
\end {lemma}
\begin {proof} As in the proof of \cite [Lemma 9.4] {GT} (see also \cite [Proposition 2.1]
{Lnonsmooth}), there is a constant $R_0$ with
\begin {equation} \label {E:assert}
R_0 \le C(B_1,n) \left \| \frac {f}{\mathcal D^{1/n}}\right\|_{n,\Gamma_\varepsilon(u)}.
\end {equation}
such that, for any $\delta >0$, there is $p_0\in \mathbb R^n\setminus
\chi(\Gamma_\varepsilon(u))$ with $|p_0| \le R_0 +\delta$. If $|p_0| \le \varepsilon \sup
u/(\diam\Omega+\beta_0)$, we proceed as in \cite [Lemma 1.1] {LTAMS} to see that
\[
\sup_\Omega u \le (\diam\Omega +\beta_0)|p_0|,
\]
which implies that $\sup u=0$. On the other hand, if $|p_0| >\varepsilon \sup
u/(\diam\Omega+\beta_0)$,
then
\[
\sup u \le \frac 1{\varepsilon} (R_0+\delta)
\]
for any $\delta > 0$. Combining these two cases and using \eqref {E:assert} yields the desired
estimate.
\end {proof}
We are now ready to state and prove our main maximum estimate.
\begin {theorem} \label {T:mpell}
Let $\Omega \subset \mathbb R^n$ be Lipschitz with Lipschitz constants $N$, $R$, and
$\omega_0$, and define the operators $L$ and $\mathcal M$ by \eqref {E:defnLM} with
$[a^{ij}]$ a positive-definite matrix-value function and $c \le 0$. Suppose there is a constant
$\beta_0$ such that condition \eqref {E:SCbeta} holds. Suppose also that $\partial\Omega \in
C^{0,1}$ and that there is a constant $\mu$ such that condition \eqref {E:unifella} holds.
Suppose finally that there are constants $B_1$ and $B_2$, vector-valued functions $b_1$ and
$b_2$, and a $D_1$ function $\zeta$ such that $b=b_1+b_2$ and
\begin {subequations} \label {E:SCb}
\begin {gather}
\left\|\frac {b_1}{\mathcal D^{1/n}}\right\| \le B_1, \\
|b_2| \le B_2\lambda \frac {\zeta(d/\diam\Omega)}d
\end {gather}
\end {subequations}
Let $u \in W^{2,n}_{\text{loc}}(\Omega) \cap C^0(\overline{\Omega})$ and suppose there are
nonpositive functions $f_1$ and $f_2$ with $f_1/\mathcal D^{1/n}\in L^n(\Omega)$ and
$f_2d/(\lambda\zeta(d/\diam\Omega)) \in L^\infty(\Omega)$ and a nonpositive constant $g$ such
that $Lu \ge
f_1+f_2$ in $\Omega$ and $\mathcal Mu \ge g$ on $\partial\Omega$. Then, there are
constants $C$ and $B^*$, determined only by $n$, $\mu$, $B_2$, $N$, $R/\diam\Omega$, and
$\omega_0$ such that $B_1 \le B^*$ implies that
\begin {equation} \label {E:mpconclfull}
\sup u \le |g| +C (\diam\Omega+\beta_0)\left[ \left\| \frac {f_1}{\mathcal
D^{1/n}}\right\|_{n,\Gamma^*(u)} + \left\| \frac {f_2d}{\lambda
\zeta(d/\diam\Omega)}\right\|_{\infty} \right],
\end {equation}
where
\begin {equation} \label {E:Gstar}
\Gamma^* = \{ x\in \Omega: u(x) >0,\ |Du| \le \frac {\sup u}{\diam\Omega+\beta_0} + C \left\|
\frac
{f_2d}{\lambda\zeta(d/\diam\Omega)}\right\|_{\infty}\}.
\end {equation}
\end {theorem}
\begin {proof} We define the operators $L_k$ for $k=1,2$ by $L_ku=a^{ij}D_{ij}u
+b_k^iD_iu+cu$, and, for $A\ge 0$ and $\varepsilon \in (0,1/2)$ constants to be determined, we
set $\bar u = u+g$, $\bar w_1 =w_1+\beta_0\sup |Dw_1|$, and $v= \bar u-A\bar w_1$. Next, we
assume that $B^* \le 1$ and we apply Lemma \ref {L:mpell} to $v$ using the operator $L_1$ in
place of $L$. It follows that
\begin {equation} \label {E:mpv}
\sup v \le \frac {CR_0}{\varepsilon}\left\| \frac {(f^*)^-} {\mathcal
D^{1/n}}\right\|_{n,\Gamma},
\end {equation}
where we have used the abbreviations
\[
f^*=f_1 + f_2 -b_2^iD_iv-AL_2w_1-Ab_1^iD_iw_1,
\]
$\Gamma=\Gamma_ \varepsilon(v)$, and $R_0=\diam\Omega+\beta_0$. To proceed, we obtain
a lower
bound for $f^*$ using the abbreviation $z=\zeta(d/\diam\Omega)$.
First, we note that $\sup v \le \sup \bar u$ and hence
\[
-b_2^iD_iv \ge - \lambda\frac {B_2}2\,\sup \bar u\frac z{R_0d}
\]
on $\Gamma$. Then we set
\[
F_1 = \left\|\frac {f}{\mathcal D^{1/n}}\right\|_{n,\Gamma},\ F_2= \left\| \frac
{f_2d}{\lambda\zeta(d/\diam\Omega)}\right\|_{\infty},
\]
and note that
\[
f_2-b_2^iD_iv -AL_2w_1 \ge \lambda(-F_2-\frac{B_2}R_0 \varepsilon \sup \bar u + A)\frac zd.
\]
In addition, $-Ab_1^iD_iw_1 \ge -CA|b_1|$. Taking $A= F_2+\varepsilon B_2 \sup \bar u/R_0$
then yields
\[
f^* \ge f_1 -CF_2|b_1| -C\frac {B_2}{2R_0} |b_1|\sup \bar u
\]
Now we use this inequality and our choice of $A$ along with \eqref {E:mpv} to see that
\[
\sup v \le C\left(1+\frac 1\varepsilon\right) F_2R_0 + \frac{C}\varepsilon F_1R_0 +
CB_1B_0\sup \bar u .
\]
On the other hand,
\[
\sup v \ge \sup \bar u - CF_2\diam\Omega - CB_2\varepsilon \sup \bar u.
\]
By choosing $B^*=\varepsilon =1/(4+4CB_2)$, we find that
\[
\sup \bar u \le C(F_1+ F_2)R_0,
\]
and that
\[
|Du| \le |Dv| + A|Dw_1| \le \frac {(1+B_2)\varepsilon}{R_0} \sup \bar u + CF_2
\]
on $\Gamma$. Combining these two inequalities and recalling our choice of $\varepsilon$ easily
implies \eqref {E:mpconclfull} since $\bar u \le u$.
\end {proof}
Note that the smallness condition on $B_1$ can be modified. By paying more attention to the
values of the constants generically denoted by $C$ in this proof, we see that \eqref
{E:mpconclfull}
holds provided $B_1$ and $B_2$ satisfy the joint condition
\[
K_1(B_1)K_2(B_2)B_1B_2 <1,
\]
where $K_1(B_1)$ is the constant from Lemma \ref {L:mpell} and $K_2(B_2)$ is the constant
from Lemma \ref {L:supersolnglobal}.
\section {Local estimates for elliptic problems} \label {S:locell}
Next, we discuss various local estimates for elliptic oblique derivative problems. Our
main concern is with a H\"older estimate for $u$, which will be useful in applications, so we just
sketch the major ideas. First, for a positive-definite matrix-valued function $\mathcal
A=[a^{ij}]$ and a continuous increasing function $\zeta$, we say that a measurable function $f$
is an $(n,\zeta, \mathcal A)$-composite function if there is a decomposition $f=f_1+f_2$ along
with constants $F_1$ and $F_2$ such that
\[
\|\frac {f_1}{\mathcal D^{1/n}}\|_{n,\Omega[R]} \le F_1,\ |f_2/\lambda| \le F_2\frac
{\zeta(d/R)}d
\]
in $\Omega[R]$. We call $(F_1,F_2)$ the composite norm of $f$. In general, there will be more
than one such decomposition and hence this norm is not unique, so we shall choose any
convenient choice. In particular, if $f$ is nonnegative or nonpositive, then we shall assume that
$f_1$ and $f_2$ are both nonnegative or nonpositive, respectively. We also write
\[
F_1(\rho) = \|f_1/\mathcal D^{1/n}\|_{n,\Omega[\rho]}
\]
for $\rho \in (0,R)$. We use similar notation for the coefficients $b$ and $c$.
Now suppose that there are positive constants $\varepsilon<1$, $R$, and $\omega_0$ such
that
\begin {subequations} \label {E:intcone}
\begin {gather} \label {E:intconea}
\{x\in \mathbb R^n: x^n > \omega_0|x'|, |x|n$ and $|b_2| \le
B_2d^{\alpha-1}$ for some $\alpha \in (0,1)$.
We conclude this section with a maximum principle for mixed boundary value problems
in $\Omega[\rho]$. This maximum principle will be useful in studying gradient estimates for
oblique derivative problems; see \cite {Lhigher}.
\begin {lemma} \label {L:Aleks}
Let $R>0$, and suppose conditions \eqref {E:unifella} and \eqref {E:oblique} are satisfied. Let
$\rho \in (0,R)$, let $\zeta$ be a $D_1$ function, let $b$ and $c$ be
$(n,\zeta,\mathcal A)$-composite function with $c \le 0$, and define the operator $L$ by \eqref
{E:Ldef}. Suppose $\beta$ is an inward pointing direction field defined on $\Sigma[\rho]$ with
$|\beta| \le \mu_1\beta^n$ for some constant $\mu_1$. Let $u\in
W^{2,n}_{\text{loc}}(\Omega[\rho]) \cap C(\overline{\Omega[\rho]})$ and suppose there are
nonnegative constants $\mu_0$ and $g$ along with a nonpositive $(n,\zeta,\mathcal
A)$-composite function $f$ such that $Lu \ge f$ in $\Omega[\rho]$ and $\beta \cdot Du \ge
-g\beta^n$ on $\Sigma[\rho]$. Then there are positive constants $\varepsilon_1$ and $\rho_0$,
determined only by $n$, $\mu$, and $\mu_1$ such that $B_1 \le \varepsilon_1$ and $\rho \le
\rho_0R$ imply that
\begin {equation} \label {E:Aleks}
\sup_{\Omega[\rho]} u \le \sup_{E^+(\rho)} u^+ + C(n,B_2,\mu,\mu_1)\rho[ g + F_1(\rho) +
F_2I(\zeta_1)(\rho/R) ],
\end {equation}
where $E^+(\rho)= \{x\in \Omega: |x| =\rho\}$.
\end {lemma}
\begin {proof} We define $v$ by
\[
v(x)= \exp(\varepsilon x^n/\rho)(u(x)- \sup_{E^+(\rho)} u^+),
\]
and we define $\bar L$ by
\[
\bar Lu = a^{ij}D_{ij}u+ \left( b^i - \frac {2\varepsilon}\rho a^{in}\right) D_iu + cu.
\]
It's easy to see that
\[
\bar Lv \ge \exp(\varepsilon x^n/\rho)f -\left( \varepsilon \frac {|b|}\rho + \frac
{\varepsilon^2a^{nn}}{\rho^2}\right)v
\]
in $\Omega[\rho]$. Now we set $\bar \beta = (\rho\beta)/(\varepsilon\beta^n(0))$ and
extend $\bar\beta$ to be zero on $E^+(\rho)$. Then $\bar\beta \cdot Dv -v \ge - (\rho
g/\varepsilon)$ on $\partial (\Omega[\rho])$.
We now assume that $\varepsilon_1 \le B^*$, the constant from Theorem \ref {T:mpell}, and
we apply that theorem to $v$ with $\bar L$ and $\bar \beta$ replacing $L$ and $\beta$,
respectively. In this way, we obtain
\begin {align*}
\sup_{\Omega[\rho]} v &\le C(1+ \frac {\mu_1}\varepsilon) \rho[ g + F_1(\rho)] +
F_2I(\zeta_1)(\rho/R)
\\
&+ C(\varepsilon+\mu_1)(\varepsilon_1+\varepsilon+B_2I(\rho/R)) \sup_{\Omega[\rho]} v.
\end {align*}
The proof is completed by choosing $\varepsilon$, $\varepsilon_1$, and $\rho_0$ sufficiently
small and rewriting the resulting inequality in terms of $u$.
\end {proof}
\section {The parabolic composite maximum principle} \label {S:para1}
For parabolic problems, we modify our notation slightly. Let $\Omega$ be a bounded domain
in $\mathbb R^{n+1}$ with parabolic boundary $\mathcal P\Omega$ and suppose $R_0$ is so
large that $|x| \le R_0$ for all $X=(x,t) \in \Omega$. Let $u$ be a continuous function defined on
$\overline{\Omega}\setminus \mathcal P\Omega$. For constants $\beta_0 \ge 0$ and
$\varepsilon >0$,
we define $E_\varepsilon(u,\beta_0)$ to be the set of all $X\in \overline{\Omega}\setminus
\mathcal
P\Omega$ such that there is $\xi\in \mathbb R^n$ with $u(Y) \le u(X) + \xi\cdot (x-y)$ for all
$Y\in \Omega$ with $s\le t$, $u(X) >0$, and
\[
\frac {R_0+\beta_0}\varepsilon |\xi| \le u(X)-\xi\cdot x < \frac 12 \sup_{\Omega} u.
\]
The rest of the notation from Section \ref {S:note} is then modified in the obvious way.
Before presenting our main maximum principle, we begin with a simple variant of an
intermediate result.
\begin {lemma} \label {L:72}
Suppose $\beta$ is an inward pointing direction field on $\mathcal P\Omega$ with $\beta^{n+1}
\equiv0$ on $\mathcal P\Omega$ and $\beta\equiv0$ on $B\Omega$. Suppose also that there is
a constant $\beta_0$ such that $|\beta| \le \beta_0$ on $\mathcal P\Omega$. If $u\in
W^{2,1}_{n+1;\text{loc}}(\Omega) \cap C^0(\overline{\Omega})$ and $\beta\cdot Du \ge u$ on
$\mathcal P\Omega$, then
\begin {equation} \label {E:73}
\sup_\Omega u \le C(n) \left(\frac {R_0+\beta_0}\varepsilon \right)^{n/(n+1)} \left(
\int_{E_\varepsilon(u,\beta_0)} |u_t \det D^2u|\,dX \right)^{1/(n+1)}.
\end {equation}
\end {lemma}
\begin {proof} The proof is virtually identical to that of \cite [Lemma 7.2]{Lbook} (which is
modeled, in turn, on that in \cite {Tso}), so we only give a sketch. First, we assume that $u \in
C^2(\Omega) \cap C^0(\overline{\Omega})$ and we define the function $\Phi\colon\Omega\to
\mathbb
R_0^{n+1}$ by $\Phi(X)= (Du(X), u(X)-x\cdot Du(X))$. Since the Jacobian determinant of this
function is just $u_t\det D^2u$, it follows that
\[
\int_{E} |u_t \det D^2u|\,dX \ge |\Phi(E)|,
\]
where we use $E$ to abbreviate $E_\varepsilon(u,\beta_0)$. Next, we set $M=\sup_\Omega u$
and we define
\[
\Sigma=\{\Xi=(\xi,h)\in \mathbb R_0^{n+1}: \frac {R_0+\beta_0}\varepsilon |\xi| < h < \frac
M2\}.
\]
The discussion on p. 107 of \cite {Lbook} shows that $\Sigma \subset \Phi(E)$, so
\[
\int_{E} |u_t \det D^2u|\,dX \ge |\Sigma| = C(n) \left( \frac {\varepsilon}{R_0+\beta_0}\right)^n
M^{n+1},
\]
and the desired result (for smooth $u$) follows from this one by simple algebra. The hypothesis
$u \in C^2$ is relaxed to $u \in W^{2,1}_{n+1;\text{loc}}$ as in \cite [Proposition 7.3]
{Lbook}.
\end {proof}
In analogy to the elliptic definition for composite functions, for a positive-definite
matrix-valued function $\mathcal A=[a^{ij}]$ and a continuous increasing function $\zeta$, we
say that a measurable function $f$ is an $(n+1,\zeta, \mathcal A)$-composite function if there is
a decomposition $f=f_1+f_2$ along with constants $F_1$ and $F_2$ such that $\|f_1/\mathcal
D^{1/(n+1)}\|_{n+1,\Omega[R]} \le F_1$
and
\[
|f_2/\lambda| \le F_2\frac {\zeta(d/R)}d
\]
in $\Omega[R]$. We call $(F_1,F_2)$ the composite norm of $f$.
Several different measures of regularity for $\mathcal P\Omega$ will be used to quantify
the dependence of the estimates on the domain. First, we refer to p. 76 of \cite {Lbook} for the
definition of $\mathcal P\Omega\in H_1$ although we shall rewrite the definition to emphasize
the connection to $\beta$. If $\mathcal P\Omega \in H_1$, then there are positive constants $N$,
$R$, $T_0$, and $\omega_0$ along with points $X_1,\dots, X_N$ in $S\Omega$ such that, after
a translation and rotation (in the $x$-variables only) which takes $X_i$ to the origin, we have
\begin {equation} \label {E:obliquep1}
\Omega[R] = \{X \in \mathbb R^n: |X| < R,\ x^n > \omega(X'),\ t > -T_0\}
\end {equation}
for some function $\omega$ (which will generally be different for each $X_i$) satisfying
\begin {equation} \label {E:obliquep2}
|\omega(x',t)-\omega(y',s)| \le \omega_0|X'-Y'|.
\end {equation}
In addition, $S\Omega$ is covered by the cylinders $Q(X_i,R/(3\kappa))$ with $\kappa =
(1+2\omega_0^2)^{1/2}$. Next, a {\em tusk} is a set of the form
\[
\{X:-T < t<0,\ |x-(-t)^{1/2}x_0| < R(-t)^{1/2}\}
\]
for some point $x_0\in \mathbb R^n$ and positive constants $R$ and $T$. We then say
(compare with \cite [p. 26]{LIST3}) that $\Omega$ satisfies an exterior $\theta_0$-tusk
condition at $X_1\in S\Omega$ (for $\omega_0\in (0,\pi/2)$) if $T=\infty$, and
\[
(t_1-t)^{1/2} <\tan \theta_0\left|x-x_1-\frac {|X-X_1|}{2^{1/2}|x_0|}\,x_0\right|
\]
for $X \in \Omega$. Note that $\theta_0$ can be determined explicitly in terms of $R$ and
$|x_0|$.
We then have the following maximum principle for parabolic operators with composite
coefficients.
\begin {theorem} \label {T:mppar}
Let $\mathcal P\Omega \in H_1$ with $H_1$ constants $N$, $R$, $T_0$, and $\omega_0$.
Define the operators $L$ by
\begin {equation} \label {E:LMdefp}
Lu= -u_t +a^{ij}D_{ij}u+b^iD_iu+cu
\end{equation}
and $\mathcal M$ by \eqref {E:Mdef1} with $[a^{ij}]$ positive definite, $c \le 0$, and $\beta$
satisfying \eqref {E:SCbeta}. Suppose there are positive constants $\lambda_0$ and
$\Lambda_0$ so that $[a^{ij}]$ satisfies
\begin {equation} \label {E:upa}
\lambda_0|\xi|^2 \le a^{ij}\xi_i\xi_j \le \Lambda_0|\xi|^2
\end {equation}
in $\Omega$, and suppose $\beta \equiv 0$
on $B\Omega$ and $\beta^{n+1} \equiv 0$ on $S\Omega$. Let $\zeta$ be a
$D_1$ function and suppose that $b$ and $c$ are $(n+1,\zeta,\mathcal A)$-composite
functions. Let $u \in W^{2,1}_{n+1,\text{loc}} \cap C^0(\overline{\Omega})$ and suppose
there are a
nonpositive, $(n+1,\zeta,\mathcal A)$-composite function $f$ and a nonpositive constant $g$
such that $Lu \ge f$ in $\Omega$, $\mathcal Mu \ge g$ on $\mathcal P\Omega$. If $|x| < R_0$
in $\Omega$, then there is a constant $C$, determined only by $B_2$, $n$, $N$, $R/R_0$,
$T_0$, $\lambda_0$, $\Lambda_0$, and $\omega_0$ such that
\begin {equation} \label {E:mppar}
\sup_\Omega u \le |g| + C(B_1^{n+1}+R_0+\beta_0)^{n/(n+1)}\left[ \left\| \frac {f_1}{\mathcal
D^{1/(n+1)}}\right\|_{n+1,\Gamma^*(u)} + F_2\right],
\end {equation}
with $\Gamma^*$ given by
\begin {equation} \label {E:Gstarp}
\Gamma^* = \{ x\in \Omega: u(x) \ge0,\ |Du| \le \frac {\sup u}{R_0+\beta_0} + C F_2\}.
\end {equation}
\end {theorem}
\begin {proof} We first note that the proof of Lemma \ref {L:supersolnglobal} can be modified
to the parabolic case. The only significant differences are that we use the remarks following
Lemma 13.1 of \cite {LIST3} in place of \cite [Theorem 3.7] {MI} and we replace
$\diam\Omega$ by $R_0$. We denote the resulting function also by $w_1$.
Next, we use the matrix inequality $(\det A\det B)^{n+1} \le (\tr AB)/(n+1)$, true for any
$(n+1)\times(n+1)$, positive semidefinite matrices $A$ and $B$, and we set $v=u+g-A(w_1-
\beta_0\sup |Dw_1|)$ to se that
\[
\sup_\Omega v \le C(n) \left(\frac {R_0+\beta_0}\varepsilon \right)^{n/(n+1)} \left\| \frac
{f^*}{\mathcal D^{1/(n+1)}} \right\|_{n+1,E},
\]
where $f^*= -v_t +a^{ij}D_{ij}v$ and $E=E_\varepsilon(v,\beta_0)$. Some straightforward
calculation shows that
\[
f^* \ge f_1-C\left[F_2 +\frac {\varepsilon \sup_\Omega v}{R_0+\beta_0}\right]|b_1|
\]
on $E$ if $A = F_2+(\varepsilon B_2\sup_\Omega v)/(R_0+\beta_0)$, so
\[
\sup_\Omega v \le C \left(\frac {R_0+\beta_0}\varepsilon \right)^{n/(n+1)} \left (F_1 +F_2 +
\frac
{B_1\varepsilon} {R_0+\beta_0} \sup_\Omega v\right),
\]
and the proof is completed by taking $\varepsilon$ sufficiently small.
\end {proof}
Note that the form of the estimate \eqref {E:mppar} agrees with that in \cite {Tso} and it
improves the form stated in \cite [Theorem 7.1] {Lbook} (although the choice
\[
\mu=(R+\|b/\mathcal D^*\|^{n+1}+\beta_0)^{-1/(n+1)}\|f^-\mathcal/ D^*\|
\]
in the proof of that theorem, on p. 159 of \cite {Lbook}, does give this form). Of course, if we
replace the assumption $c \le 0$ by $c \le K$ for some nonpositive constant $K$, then we can
apply this theorem to $u\exp(-Kt)$ to obtain an analogous estimate for $u$.
We leave the statements of the local estimates for parabolic equations to the reader,
mentioning \cite [Section 7]{Lnonsmooth} as a source for the descriptions. In particular, we
point out that the appropriate hypothesis for $b_1$ is that $b_1/\mathcal D^{1/(n+1)}$ should be
in the Morrey space $M^{n+1,1}$ and that $\beta$ is assumed to satisfy condition \eqref
{E:betaest} under the assumption that \eqref {E:obliquep2} is modified to
\[
|\omega(x',t)-\omega(y',s)| \le \omega_0|x'-y'|+\omega_1|t-s||^{1/2}
\]
for some $\omega_1$.
\section {Additional remarks} \label {S:rmk}
Our method gives an alternative approach for some of the results in \cite {Krylov} which were
used in \cite {AN1}. To illustrate this point, we consider the following result, which is
approximately the elliptic analog of \cite [Lemma 1.2] {Krylov}. (See also Lemma 3.3 from that
paper.)
\begin {lemma} \label {L:Krylov12}
Let $\Omega \subset \mathbb R^n$ and define the operator $L$ by \eqref {E:Ldef} with $c \le
0$. If there is a nonnegative function $w$ such that $Lw \le-|b|$ and if $u \in
W^{2,n}_{\text{loc}} \cap C^0(\overline{\Omega})$ with $u \le 0$ on $\partial\Omega$, then
\begin {equation} \label {E:mpK}
\sup_\Omega u \le C(n)(\sup w + \diam\Omega)\left\| \frac {(Lu)^-}{\mathcal
D^{1/n}}\right\|_{n,\Omega^+},
\end {equation}
where $\Omega^+$ is the subset of $\Omega$ on which $u \ge 0$.
\end {lemma}
\begin {proof} Set $M= \sup_\Omega u$ and, with $\varepsilon \in (0,1)$ to be determined, set
$v= u-\varepsilon M/R$ and $f=Lu^-$. Then $a^{ij}D_{ij}v \ge f$ in $\Gamma_\varepsilon(v)$
and $v \le 0$ on $\partial\Omega$, so Lemma \ref {L:mpell} with $B_0=\beta_0=0$ implies that
\[
\sup_\Omega v \le C(n) \frac {\diam\Omega}\varepsilon \left\| \frac {f}{\mathcal
D^{1/n}}\right\|_{n,\Omega^+},
\]
and hence
\[
M(1- \frac \varepsilon{\diam\Omega}\sup w) \le C(n) \frac {\diam\Omega}\varepsilon \left\| \frac
{f}{\mathcal
D^{1/n}}\right\|_{n,\Omega^+}.
\]
The proof is completed by taking $\varepsilon=\diam\Omega/(2(\sup w+\diam\Omega))$.
\end {proof}
This result is weaker than Krylov's in that he proves the pointwise inequality
\[
u \le C(n)( w + \diam\Omega)\left\| \frac {(Lu)^-}{\mathcal D^{1/n}}\right\|_{n,\Omega^+}.
\]
On the other hand, our method considers situations in which we only have a supersolution to part
of the operator; that is, we only need a function $w$ (like $w_1$ in Section \ref {S:ell1}) such
that $a^{ij}D_{ij}w + b_2^iD_iw \le -|b_2|$ with $b=b_1+b_2$.
Via similar considerations, we can prove essentially all the results in \cite {Krylov} for
solutions of elliptic and parabolic equations. The main differences are that we only obtain global
estimates for $u$ and we always assume that $p=n$. (Here, Krylov's $d$ is the same as our
$n$.) In a future work, we shall examine the case $p>n$.
\begin {thebibliography} {99}
\bibitem {AL1} A.~ D. Aleksandrov, \emph {Majorization of solutions of second-order linear
equations}, Vestnik Leningrad Univ. \textbf {21} (1966), no. 1, 5--25 [Russian]; English transl.
in Amer. Math. Soc. Transl. (2) \textbf {68} (1968), 120--143.
\bibitem {AN1} D.~E. Apushkinskaya and A.~I. Nazarov, \emph {Boundary estimates for the
first-order derivatives of a solution to a nondivergent parabolic equation with composite
right-hand side and coefficients of lower-order derivatives}, Prob. Mat. Anal. \textbf {14}
(1995), 3--27 [Russian]; English transl. in J. Math. Sci. \textbf {77} (1995), 3257--3276.
\bibitem {GH} D. Gilbarg and L. H\"ormander, \emph {Intermediate Schauder estimates}, Arch.
Rational Mech. Anal. \textbf {74} (1980), 297--318.
\bibitem {GT}
D. Gilbarg and N.~S. Trudinger, \emph {Elliptic Partial Differential Equations of Second Order},
Springer-Verlag, Berlin-New York-Heidelberg, 1977. Second Ed., 1983.
\bibitem {Krylov}
N. V. Krylov, \emph {On estimates of the maximum of a solution of a parabolic equation and
estimates of the distribution of a semimartingale}, Mat. Sbornik \textbf {130 (172)},
207--221 [Russian]; English transl. in Math. USSR-Sbornik \textbf {58} (1987), 207--221.
\bibitem {LC1}
G.~M. Lieberman, \emph {The Dirichlet problem for quasilinear elliptic equations with
continuously differentiable boundary data}, Comm. Partial Differential Equations \textbf {11}
(1986), 167--229.
\bibitem {LTAMS}
G.~M. Lieberman, \emph {Local estimates for subsolutions and supersolutions of oblique
derivative problems for general second order elliptic equations}, Trans. Amer. Math. Soc. \textbf
{304} (1987), 343--353.
\bibitem {LIST3}
\bysame, \emph {Intermediate Schauder theory for second order parabolic equations III. The tusk
conditions}, Appl. Anal. \textbf {33} (1989), 25--43.
\bibitem {Lbook}
\bysame, \emph {Second Order Parabolic Differential Equations}, World Scientific,
Singapore, 1996.
\bibitem {Lnonsmooth}
\bysame, \emph {Pointwise estimates for oblique derivative problems in nonsmooth
domains}, preprint, 1999.
\bibitem {Lhigher}
\bysame, \emph {Higher regularity for nonlinear oblique derivative problems in Lipschitz
domains}, preprint, 2000.
\bibitem {MI}
J.~H. Michael, \emph {Barriers for uniformly elliptic equations and the exterior cone condition},
J. Math. Anal. Appl. \textbf {79} (1981), 203--217.
\bibitem {Sperner}
E. Sperner, Jr., \emph {Schauder's existence theorem for $\alpha$-Dini continuous data}, Ark.
Mat. \textbf {19} (1981), 193--216.
\bibitem {Tso}
K. Tso, \emph {On an Aleksandrov-Bakel$'$man type maximum principle for second-order
parabolic equations}, Comm. Partial Differential Equations \textbf {10} (1985), 543--553.
\end {thebibliography}
\end {document}