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<n 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<d<r 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.