Abstract:
We introduce a new boundary Harnack principle in Lipschitz domains for equations with a right hand side.
Our approach, which uses comparisons and blow-ups, seems likely to adapt to more general domains, as well as other types of operators.
We consider operators of the form
$$
\mathcal{L} u = (a^{ij}u_i)_j + b^i u_i + cu ,
$$
with the following ellipticity conditions
$$
\Lambda^{-1} |\xi|^2 \leq \langle a^{ij}(x) \xi, \xi \rangle \leq \Lambda |\xi|^2,
$$
for some $\Lambda >0$ and for all nonzero $\xi \in \R^n$. Furthermore, $a^{ij}(x)$ is a real $n \times n$ matrix. For the lower order terms we assume
$$|c(x)|, \sum |b^i(x)| \leq \Lambda-1$$
and that $c(x) \leq 0$.
We say $u \in \mathcal{S_{\mathcal{L}}}(\Omega_{L,R})$ if
$$
\begin{aligned}
\mathcal{L} u(x) &=0 \text{ in } \Omega_{L,R} \cap B_R ,\\
u(x)&=0 \text{ on } \Omega_{L,R}^c\cap B_R,
\end{aligned}
$$
and that
$u \in \mathcal{S_{\mathcal{L}}}(\Omega_{L,R},d^{\gamma})$ if
$$
\begin{aligned}
|\mathcal{L} u(x)| &\leq (\text{dist}(x,\partial \Omega_{L,R} \cap B_R))^{\gamma} \ \text{ in } \Omega_{L,R} , \\
u(x)&=0 \ \text{ on } \Omega_{L,R}^c\cap B_R.
\end{aligned}
$$
To apply the Hölder continuity estimates for elliptic operators we will require that
$\gamma>-2/n $.
Since the boundary is Lipschitz, this will ensure the correct integrability
assumptions for the right hand side.
Theorem:
Let $0 \in \partial \Omega_L$ with $L< M$, and fix $x^0 \in \Omega_L$. Assume further that
$B_1 \cap \{x_n > 1/4\} \subseteq \Omega_L$.
Assume $u,v \geq 0$ and $u,v \in \mathcal{S}(\Omega_{L},d^{\gamma})$ with
$\mathcal L u, \mathcal L v \leq 0$ and $u(x^0)=v(x^0)=1$, and also assume that $2-\alpha+\gamma>0$, with $\gamma>-2/n$ and $\alpha$ be such that
$$
\sup_{B_{r}(x)} u \geq c_1 u(e_n/2) r^{\alpha}.
$$
Then there exists a uniform constant $C>0$
(depending only on dimension $n$,
Lipschitz constant $M$, $M-L$, and dist$(x^0,\partial \Omega_L)$)
such that
\begin{equation} \label{e:ine3}
C^{-1} v(x) \leq u(x) \leq Cv(x)
\end{equation}
for all $x \in B_{1/2}$.
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