Ngô Quốc Anh

June 16, 2010

De Giorgi’s class and De Giorgi’s theorem

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 18:55

In this entry, we introduce that De Giorgi’s class DG(\Omega) and prove that functions in DG(\Omega) are Holder continuous. This has as a consequence the celebrated De Giorgi’s theorem saying that solutions of second-order elliptic equations with bounded measurable coefficients are Holder continuous.

Let us study H^1-solutions of

\displaystyle\int_\Omega {{A^{\alpha \beta }}(x){\nabla _\alpha }u{\nabla _\beta }\varphi dx} = 0, \quad \varphi \in H_0^1(\Omega )

where A is assumed to be of class L^\infty(\Omega). Our aim is to show that u is in fact Holder continuous. The key idea is to use the Cacciopoli inequality on level sets of u.

Definition (De Giorgi’s class). We define the De Giorgi’s class (DG(\Omega)) to consist of all u \in H^1(\Omega) which satisfy

\displaystyle \int_{{B_\rho }} {{{\left| {\nabla {{(u - k)}^ + }} \right|}^2}dx} \leqslant \frac{c}{{{{(R - \rho )}^2}}}\int_{{B_R}} {{{\left| {{{(u - k)}^ + }} \right|}^2}dx} , \quad \forall k \in \mathbb{R}.

The following remark is useful.

  1. If u is a sub-solution then u \in DG(\Omega).
  2. If u is a super-solution then -u \in DG(\Omega).
  3. If u \in DG(\Omega) then u+{\rm const} \in DG(\Omega).
  4. In the definition of the De Giorgi class, exponent p=2 can be replaced by any p>1.

If we denote by A the level set of u, that is

A(k,r)=\{x \in B_r : u(x)>k\}

then if u \in DG(\Omega) by choosing a cut-off function

\displaystyle \eta \in C_0^\infty\left(B_\frac{R+\rho}{2}\right), \quad \eta \equiv 1, \quad {\rm on } B_\rho

we get

\displaystyle\int_{{B_{\frac{{R + \rho }}{2}}}} {{{\left| {\nabla {{(\eta (u - k))}^ + }} \right|}^2}dx} \leqslant \frac{c}{{{{(R - \rho )}^2}}}\int_{{B_R}} {{{\left| {{{(u - k)}^ + }} \right|}^2}dx} ,\quad \forall k \in \mathbb{R}

or

\displaystyle\int_{A(k,\rho )} {{{\left| {\nabla (u - k)} \right|}^2}dx} \leqslant \frac{c}{{{{(R - \rho )}^2}}}\int_{A(k,R)} {{{\left| {u - k} \right|}^2}dx} , \quad \forall k \in \mathbb{R}.

By using the Sobolev inequality

\displaystyle {\left( {\int_{{B_\rho }} {{{\left| {{{(u - k)}^ + }} \right|}^{{2^ \star }}}dx} } \right)^{\frac{2}{{{2^ \star }}}}} \leqslant \int_{{B_{\frac{{R + \rho }}{2}}}} {{{\left| {\nabla {{(\eta (u - k))}^ + }} \right|}^2}dx}

and the Holder inequality

\displaystyle \int_{{B_\rho }} {{{\left| {{{(u - k)}^ + }} \right|}^2}dx} \leqslant {\left| {x \in {B_\rho }:u(x) > k} \right|^{1 - \frac{2}{{{2^ \star }}}}}{\left( {\int_{{B_\rho }} {{{\left| {{{(u - k)}^ + }} \right|}^{{2^ \star }}}dx} } \right)^{\frac{2}{{{2^ \star }}}}}

we conclude that

\displaystyle\int_{A(k,\rho )} {{{\left| {u - k} \right|}^2}dx} \leqslant \frac{c}{{{{(R - \rho )}^2}}}{\left| {A(k,R)} \right|^{\frac{2}{n}}}\int_{A(k,R)} {{{\left| {u - k} \right|}^2}dx}.

For h>k

\displaystyle {\left| {h - k} \right|^2}\left| {A(h,\rho )} \right| = \int_{A(h,\rho )} {{{\left| {h - k} \right|}^2}dx} \leqslant \int_{A(k,\rho )} {{{\left| {u - k} \right|}^2}dx} .

Set

\displaystyle a(h,\rho ) = \left| {A(h,\rho )} \right|, \quad u(h,\rho ) = \int_{A(h,\rho )} {{{\left| {u - h} \right|}^2}dx}

and write the above inequalities in the form

\displaystyle\begin{gathered} u(h,\rho ) \leqslant \frac{c}{{{{(R - \rho )}^2}}}u(k,R)a{(k,R)^{\frac{2}{n}}}, \hfill \\ a(h,\rho ) \leqslant \frac{1}{{{{(h - k)}^2}}}u(k,R). \hfill \\ \end{gathered}

For some positive numbers \xi and \eta  we find

\displaystyle u{(h,\rho )^\xi }a{(h,\rho )^\eta } \leqslant \frac{{{c^\xi }}}{{{{(R - \rho )}^{2\xi }}}}\frac{1}{{{{(h - k)}^{2\eta }}}}u{(k,R)^{\xi + \eta }}a{(k,R)^{\frac{{2\xi }}{n}}}.

Now we choose \xi and \eta in such a way that for some \theta>1

\displaystyle \xi + \eta = \theta \xi ,\frac{{2\xi }}{n} = \theta \eta .

Solving these equations gives \eta=1 and \xi=\frac{n}{2}\theta.

Thus we arrive at

\displaystyle\phi (h,\rho ) \leqslant \frac{{{c^\xi }}}{{{{(R - \rho )}^{2\xi }}{{(h - k)}^{2\eta }}}}\phi {(k,R)^\theta }, \quad \theta > 1

where \phi=u^\xi a^\eta.

Proposition. For any k_0 and any \sigma \in (0,1) we have

\phi(k_0+d,R_0-\sigma R_0)=0

for some d.

Consequently, if we choose \sigma=\frac{1}{2} we get

Theorem. If u \in DG(\Omega) then

\displaystyle\mathop {\sup }\limits_{x \in {B_{\frac{R}{2}}}} |u(x)| \leqslant {k_0} + c{\left( {\frac{1}{{{R^n}}}\int_{A({k_0},R)} {{{\left| {u - {k_0}} \right|}^2}dx} } \right)^{\frac{1}{2}}}{\left( {\frac{{A({k_0},R)}}{{{R^n}}}} \right)^{\frac{{\theta - 1}}{2}}}.

In particular when k_0=0 we have

Theorem. If u \in DG(\Omega) then

\displaystyle\mathop {\sup }\limits_{{B_{\frac{R}{2}}}} {u^ + } \leqslant c{\left( {\overline\int_{{B_R}} {{{\left| {{u^ + }} \right|}^2}dx} } \right)^{\frac{1}{2}}}

and

Theorem. If -u \in DG(\Omega) then

\displaystyle\mathop {\sup }\limits_{{B_{\frac{R}{2}}}} {u^ - }  \leqslant c{\left( {\overline\int_{{B_R}} {{{\left| {{u^-}}  \right|}^2}dx} } \right)^{\frac{1}{2}}}

From these two estimates we infer

Theorem. Solution u of PDE

\displaystyle\int_\Omega {{A^{\alpha \beta }}(x){\nabla _\alpha  }u{\nabla _\beta }\varphi dx} = 0, \quad \varphi \in H_0^1(\Omega )

is bounded and

\displaystyle\mathop {\sup }\limits_{{B_{\frac{R}{2}}}} |u|   \leqslant c{\left( {\overline\int_{{B_R}} {{{\left| u   \right|}^2}dx} } \right)^{\frac{1}{2}}}.

We now come to the main result of this entry

Theorem (De Giorgi). If u and -u belong to DG(\Omega) then u is \alpha-Holder continuous. Moreover, for \rho<R<R_0 we have

\displaystyle\int_{{B_\rho }} {{{\left| {\nabla u} \right|}^2}dx} \leqslant c{\left( {\frac{\rho }{R}} \right)^{n + 2\alpha - 2}}\int_{{B_R}} {{{\left| {\nabla u} \right|}^2}dx} .

The proof of the Holder continuity is based an estimate concerning the oscillation

\displaystyle\mathop {\rm osc}\limits_{{B_R}} u = \mathop {\sup }\limits_{{B_R}} u - \mathop {\inf }\limits_{{B_R}} u.

Precisely, there exists some constant \alpha such that

\displaystyle\mathop {\rm osc}\limits_{{B_\rho }} u \leqslant c{\left( {\frac{\rho }{R}} \right)^\alpha }\mathop {\rm osc}\limits_{{B_R}} u.

Thus by scaling and iterating this estimate, one obtains the Holder continuity of u since the oscillation of u is easily seen to decay as a power of radius. To see this, let pick x and y arbitrary in \Omega. Denote \rho=|x-y|, then

\displaystyle\left| {u(x) - u(y)} \right| \leqslant \mathop {\sup }\limits_{{B_\rho } \cap \Omega } u - \mathop {\inf }\limits_{{B_\rho } \cap \Omega } u \leqslant \mathop {\rm osc}\limits_{{B_\rho }} u \leqslant c\frac{{\mathop {\rm osc}\limits_{{B_R}} u}}{{{R^\alpha }}}{\rho ^\alpha } = c\frac{{\mathop {\rm osc}\limits_{{B_R}} u}}{{{R^\alpha }}}{\left| {x - y} \right|^\alpha }.

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