## Tag Archives: barrier function

### General approach for fully nonlinear elliptic equation

Consider the Dirichlet problem in a bounded domian $\Omega\subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$

$\displaystyle \begin{cases} F(D^2u)=\psi(x) \text{ in } \Omega\\\quad \quad u=\phi\quad\text{ on }\Omega\end{cases}\quad(1)$

The function $F$ are represented by a smooth symmetric function

$\displaystyle F(D^2u)=f(\lambda_1,\lambda_2,\cdots,\lambda_n)$

here $\lambda_1,\lambda_2,\cdots,\lambda_n$ are the eigenvalues of $D^2u$. In order to be elliptic, we require

$\displaystyle \frac{\partial f}{\partial \lambda_i}>0$, $\forall\, i>0\quad (2)$

$f$ is defined in an open convex cone $\Gamma\subset \mathbb{R}^n$ with vertex at origin, and

$\displaystyle \bigcap\limits_{i=1}^n \{\lambda_i>0\}\subset \Gamma\subset \left\{\sum\lambda_i>0\right\}$

Since $f$ is symmetric, we also require $\Gamma$ to be symmetric.

The following are the assumptions for  (1) to be solvable,

$\psi\in C^\infty(\overline{\Omega})$, $\phi\in C^\infty(\partial \Omega)$

$\displaystyle \psi_0= \min_{\overline{\Omega}}\psi\leq \max_{\overline{\Omega}}\psi=\psi_1\quad (4)$

$f$ is a concave function  $(5)$

$\displaystyle \overline{\lim\limits_{\lambda\to\partial \Gamma}}f(\lambda)\leq \tilde{\psi_0}<\psi_0\quad (6)$

For every compact set $K$ in $\Gamma$ and every constant $C>0$, there is a number $R=R(K,C)$ such that

$\displaystyle f(\lambda_1,\lambda_2,\cdots,\lambda_n+R)\geq C$ for all $\lambda\in K\quad(7)$

$\displaystyle f(R\lambda)\geq C$ for all $\lambda\in K\quad (8)$

Also we need restrict $\partial \Omega$. There exists sufficiently large constant $R$ such that for every point on $\partial \Omega$, if $\kappa_1,\kappa_2,\cdots,\kappa_{n-1}$ are the principle curvatures of $\partial \Omega$

$\displaystyle (\kappa_1,\kappa_2,\cdots,\kappa_{n-1},R)\in\Gamma\quad (9)$

$\mathbf{Thm(CNS):}$ If $(2-9)$ are satisfied, then $(1)$ has a unique solution $u\in C^\infty(\overline{\Omega})$ with $\lambda(D^2u)\in \Gamma$.

$\mathbf{Proof:}$ The existence get from continuity method.

Krylov has shown how from a priori estimates

$|u|_{C^2(\overline{\Omega})}\leq C\quad (10)$

and uniform ellipticity of the linearized opeartor $L=\sum{F_{ij}}\partial_{ij}$ to derive

$\displaystyle |u|_{C^{2,\nu}({\overline{\Omega}})}\leq C$

So we only need to derive $(10)$ for any solution of $(1)$.

Using a maximum principle of fully nonlinear equation and $\psi\leq \psi_1$ and $(9)$, it is possible to construct a subsolution $\underline{u}$of $(1)$. So $u\geq \underline{u}$

If $\lambda(D^2u)\in \Gamma$ then $u$ can be bounded above by a harmonic function $v$ in $\Omega$.

$\underline{u}\leq u\leq v$

Additionally,                                      $|\nabla u|\leq C$ on $\partial \Omega$

Then differentiate $F(D^2u)=\psi$ to get a linear elliptic function. Using maximum principle to get $|u|_{C^1}\leq C$.

For the second derivative estimate, it is also estimate $u_{ij}$ on the boundary first and then differentiate $F(D^2u)=\psi$ another time to get a linear elliptic function of $u_{ij}$. And then apply maximum principle to get interior gradient estimate.

The bound of $u_{\alpha n}$, $\alpha is achieved from  constructing a barrier function.

The estimate of $u_{nn}$ is complicate. Still constructing a barrier function.

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

$\mathbf{Remark:}$Cafferelli, Nirenberg, Spruck: The Dirichlet problem for nonlinear second order elliptic equations III.

### Possion equation for unbounded function

$\mathbf{Thm:}$ Let $\Omega\subset \mathbb{R}^n$ is a bounded $C^2$ domain. Fix $\beta\in(0,1)$, $f$ is a function in $\Omega$ such that $\displaystyle \sup\limits_{x\in B}d^{2-\beta}(x)|f(x)|\leq N<\infty$, where $d(x)=dist(x,\partial \Omega)$ is the distance function. Then

$\displaystyle \begin{cases} \Delta u=f\quad \text{ in } \Omega\\u=0\quad \text{ on }\partial \Omega\end{cases}$

has a unique solution $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$ satisfying

$\displaystyle \sup\limits_{x\in B}d^{-\beta}(x)|u(x)|\leq CN$

where the constant $C$ depends only on $\beta$ and $\Omega$.

Firstly, recall some basic properties of distance function. By lemma 14.16 in GT’s book, if $\Omega$ is $C^2$ and bounded, there is a neighborhood of $\partial \Omega$ in $\Omega$, say $\Gamma$, such that $d(x)\in C^2(\Gamma)$. And $\Delta d$ is bounded in $\Gamma$.

$\mathbf{Proof:}$ $\Omega$ is $C^2$ means there exists $\delta$ such that $d(x)$ is $C^2$ in $B_\delta(x_\alpha)$ for any $x_\alpha\in\partial \Omega$.

Suppose $\eta\in C^\infty_0(B_\delta(0))$ with $\eta(0)=\frac {1}{\beta(1-\beta)}$. Denote $\eta_\alpha(y)=\eta(y-x_\alpha)$.

Fix $x_\alpha\in \partial \Omega$,

$\Delta(d^\beta\eta_\alpha)=\eta_\alpha\Delta (d^\beta)+2\nabla (d^\beta)\cdot\nabla\eta_\alpha+d^\beta\Delta\eta_\alpha$

$\displaystyle =\left[\beta(\beta-1)d^{\beta-2}|\nabla d|^2+\beta d^{\beta-1}\Delta d\right]\eta_{\alpha}+2\beta d^{\beta-1}\nabla d\cdot\nabla\eta_\alpha+d^\beta\Delta\eta_\alpha$

$=\displaystyle -d^{\beta-2}\left[\beta(1-\beta)\eta_\alpha-\beta d\Delta d-2\beta d\nabla d\cdot\nabla\eta_\alpha-d^2\Delta\eta_\alpha\right]$
$\leq -\frac 12d^{\beta-2}$ $\quad$ if $0 small enough.

$\Delta(d^\beta\eta_\alpha)\leq -\frac 12d^{\beta-2}\quad when \quad x\in B_r(x_\alpha)$

$\Delta(d^\beta\eta_\alpha)\leq C=C(\beta,\Omega)\quad when \quad x\in \Omega\backslash B_r(x_\alpha)$

Since $\partial \Omega$ is compact, there exists finitely many $x_1,x_2,\cdots,x_m$ such that $\displaystyle \bigcup\limits_{i=1}^m B_{r}(x_i)$ covers $\partial \Omega$. Let $w$ is solution of $\Delta v=-mC$ in $\Omega$ and $v=0$ on $\partial \Omega$.

Define $w=\sum\limits_{i=1}^md^\beta\eta_i +v$, then $w=0$ on $\partial \Omega$ and $\Delta w\leq -\frac 12d^{\beta-2}$ in $\Omega$. So

$\displaystyle \Delta(2Nw\pm u)\leq 0$ in $\Omega$ and $2Nw\pm u=0$ on $\partial \Omega$

Consequently, by the maximum principle,

$|u(x)|\leq 2Nw=d^\beta 2N( \sum\limits_{i=1}^m\eta_i )+2Nv$

Since $v$ has an upper bound only depends on the geometry of $\Omega$ and $m, C$, we only need to prove $v(x)\leq C'd^\beta(x)$ when $x$ is near the boundary, where $C'=C'(\beta,\Omega)$. Note that

$\displaystyle \Delta(d^\beta)=d^{\beta-2}[\beta(\beta-1)+\beta d\Delta d]\rightarrow -\infty$ uniformly as $x\to \partial \Omega$

So there exists a neighborhood $\Gamma'$ of $\partial \Omega$ such that $\Gamma'\subset \Gamma$ and $\exists C'=C'(C,\beta,\Omega)$

$\Delta(C'd^\beta-v)\leq 0$ in $\Gamma'$ and $C'd^\beta-v\geq 0$ on $\partial \Gamma'$

By the maximum principle, $v(x)\leq C'd^\beta(x)$ in $\Gamma'$.

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

$\mathbf{Remark:}$ Gilbarg, Trudinger. Chapter 4, exercise 4.6. p71

Also see, J.H. Michael. A general theory for linear elliptic partial differential equations. 1977

Gary M. Lieberman. Elliptic equations with strongly singular lower order terms. 2008

### 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

$w=K(R^{-\alpha}-|x-y|^{-\alpha})$

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.