Strong barrier function and generalized linear elliptic equation

Suppose \Omega is a domain in \mathbb{R}^n. x_0\in \partial \Omega satisfies the exterior sphere condition. That is there exists a ball B=B_R(y) such that \bar{B}\cap \overline{\Omega}=x_0. Then the function
\displaystyle w(x)=\begin{cases}R^{2-n}-|x-y|^{2-n} \quad n\geq 3\\log\frac{|x-y|}{R} \qquad n=2\end{cases}
is a barrier function for \Delta at x_0.
For a strong barrier function w at x_0, we mean \Delta u\leq -1 or Lu\leq -1, at such x_0 Define


Then \displaystyle Lw=K\alpha|x-y|^{-\alpha-4}\left[-(\alpha+2)a^{ij}(x_i-y_i)(x_j-y_j)+|x-y|^2( a^{ii}+b^i(x_i-y_i))\right]+cw
Assume c\leq 0 and let |x-y|=r
Lw\leq K\alpha r^{-\alpha-4}\left[-(\alpha+2)a^{ij}(x_i-y_i)(x_j-y_j)+r^2(a^{ii}+b^i(x_i-y_i))\right]\leq -1
when \alpha=\alpha(\lambda, \Lambda,diam(\Omega),R) and K large enough.
\mathbf{Problem: } Let u\in C^2(\Omega)\cap C^0(\overline{\Omega}) be a solution of Lu=f in a bounded C^1 domain \Omega n\geq 3. Suppose x_0\in\partial \Omega satisfy the exterior ball condition with B_R(y). Moreover, assume

a^{ij}\xi_i\xi_j\geq \lambda |\xi|^2
|a^{ij}|, |b^i|, |c|\leq \Lambda

Suppose u|_{\partial \Omega}=\phi, \phi\in C^2(\overline{\Omega}). Show that u satisfies a Lipschitz condition at x_0

|u(x)-u(x_0)|\leq K(x-x_0), x\in \Omega.

where K=K(\lambda,\Lambda,R,diam(\Omega), \sup|f|). If the sign c is unrestricted, show that the same result holds provided K also depends on \sup|u|.

\mathbf{Proof:} When c\leq 0 suppose w_0 is the strong barrier function of L at x_0, then there exists M=M(\sup\limits_{\Omega}|f|, |\phi|_{2;\Omega}) such that \displaystyle w=Mw_0 satisfies
w+u(x_0)-u\geq 0 on \partial \Omega and L(w+u(x_0)- u)\leq 0

-w+u(x_0)-u\leq 0 on \partial \Omega and L(-w+u(x_0)-u)\geq 0
By the maximum principle, we know that

-w(x)\leq u(x)-u(x_0)\leq w(x) |u(x)-u(x_0)|\leq w(x)=w(x)-w(x_0)\leq K|x-x_0|.

When the sign of c is undefined, let \tilde{L}u=Lu-c^+u=f-c^+u, apply the above procedure to \tilde{L}, only need to adjust that M=M(\sup\limits_{\Omega}|f|, |\phi|_{2;\Omega},\Lambda,\sup|u|).

\text{Q.E.D}\hfill \square

\mathbf{Remark:}Gilbarg Trudinger’s book. Chapter, exercise 3.6.

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