Comparison principle and De Giorgi-Nash-Moser theorem: Difference between pages

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The comparison principle refers to the general concept that a subsolution to an elliptic equation stays below a supersolution of the same equation. It known to hold under a great generality of assumptions.
The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form.


The comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the [[maximum principle]]. The uniqueness of the solution of the equation is an immediate consequence.
The equation is
\[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \]
in the elliptic case, or
\[ u_t = \mathrm{div} A(x,t) \nabla u(x). \]
Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$,
\[ \langle A v,v \rangle \geq \lambda |v|^2,\]
for every $v \in \R^n$, uniformly in space and time.


== General statement ==
The corresponding result in non divergence form is [[Krylov-Safonov theorem]].


The two statements below correspond to the comparison principle for elliptic and parabolic equations with Dirichlet boundary conditions. The main difference with the local case, is that for nonlocal equations the Dirichlet condition has to be taken in the whole complement of the domain $\Omega$ instead of only the boundary.
For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]].


Other boundary conditions require appropriate modifications.
== Elliptic version ==
For the result in the elliptic case, we assume that the equation
\[ \mathrm{div} A(x) \nabla u(x) = 0 \]
is satisfied in the unit ball $B_1$ of $\R^n$.
===Holder estimate===
The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $B_{1/2}$ and
\[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\]
The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.


=== Elliptic case ===
The result can be scaled to balls of arbitrary radius $r>0$ to obtain
We say that an elliptic equation $Iu=0$ satiesfies the comparison principle if the following statement is true.
\[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]


Given two functions $u : \R^n \to \R$ and $v : \R^n \to \R$ such that $u$ and $v$ are upper and lower semicontinuous in $\overline \Omega$ respectively, where $\Omega$ is an open domain,
Moreover, by covering an arbitrary domain $\Omega$ with balls, one can show that a solution to the equation in $\Omega$ is $C^\alpha$ in the interior of $\Omega$.
$Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in $\Omega$, and
$u \leq v$ in $\R^n \setminus \Omega$, then $u \leq v$ in $\Omega$ as well.


=== Parabolic case ===
===Harnack inequality===
We say that a parabolic equation $u_t - Iu=0$ satiesfies the comparison principle if the following statement is true.
The [[Harnack inequality]] says that if $u$ is a non negative solution of the equation in $B_1$, then its minimum controls its maximum in $B_{1/2}$:
\[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\]
The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only.


Given two functions $u : [0,T] \times \R^n \to \R$ and $v : [0,T] \times\R^n \to \R$ such that $u$ and $v$ are upper and lower semicontinuous in $[0,T] \times \overline \Omega$ respectively,
===Minimizers of convex functionals===
$Iu \leq 0$ and $Iv \geq 0$ in the viscosity sense in $(0,T] \times \Omega$, and
The theorem of De Giorgi, Nash and Moser was used originally to solve one of the famous Hilbert problems. The question was whether the minimizers of Dirichlet integrals
$u \leq v$ in $(\{0\} \times \R^n) \cup ([0,T] \times (\R^n \setminus \Omega))$, then $u \leq v$ in $[0,T] \times \Omega$ as well.
\[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\]
are always smooth if $F$ is smooth and strictly convex. The theorem of De Giorgi-Nash-Moser in its elliptic form can be applied to the differential quotients of the minimizer of $J$ to show that the solution is $C^{1,\alpha}$. Once that initial regularity is obtained, further regularity follows by [[bootstrapping]] with the [[Schauder estimates]] and the smoothness of $F$.


== Assumptions for which the comparison principle holds ==
Note that in order to apply the theorem to these nonlinear equations, it is very important that no smoothness assumption on the coefficients $A(x)$ is made.


=== Proper elliptic equations with $x$ dependence ===
== Parabolic version ==
A fairly general comparison principle <ref name="BI"/> <ref name="BIC2"/> is available for [[fully nonlinear integro-differential equations]] of the form
For the result in the parabolic case, we assume that the equation
\[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \]
\[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \]
where $\{I_\alpha u\}_{\alpha \in A}$ is a family of [[linear integro-differential operators]].
is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$.
===Holder estimate===
The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $[1/2,1] \times B_{1/2}$ and
\[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\]
The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.


The comparison principle holds provided that the following assumptions are satisfied:
===Harnack inequality===
* All integro-differential operators $I_\alpha$ are in the [[Levy-Ito form]]
The [[Harnack inequality]] says that if $u$ is a non negative solution of the equation in $[0,1] \times B_1$, then its minimum controls its maximum in a previous time:
\[ I_\alpha = \int_{\R^n} (u(x+j_\alpha(x,z)) - u(x) - \nabla u(x) \cdot j_\alpha(x,z) \mathbb{1}_B(z)) \mu_\alpha(dz). \]
\[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \]
* The measures $\mu_\alpha$ and the functions $j_\alpha$ satisfy the following assumptions for some large constant $C$ independent of $\alpha$.
\begin{align*}
\int_B |j(x,z)|^2 \mu(dz) &\leq C, \qquad \int_{\R^n \setminus B} \mu(dz) \leq C,\\
\int_{\R^n} |j(x,z)-j(y,z)|^2 \mu(dz) &\leq C|x-y|^2, \text{ and } \int_{\R^n \setminus B} |j(x,z)-j(y,z)|^2 \mu(dz) \leq C|x-y|.
\end{align*}
* There exist a $\gamma>0$ such that for all $x \in \R^n$, $u,v \in \R$, $p \in \R^n$, $X$ a symmetric matrix in $\R^{n \times n}$ and $\{i_\alpha\} \in \R^A$,
\[ F(x,u,p,X,\{i_\alpha\}) - F(x,v,p,X,\{i_\alpha\}) \geq \gamma (v-u).\]
* One of the following two holds
*# For any $R >0$, there exist moduli of continuity $\omega$ and $\omega_R$ such that, for any $|x|,|y|<R$, $|v|<R$, $\{i_\alpha\} \in \R^A$, and for any two symmetric matrices $X$ and $Y$ satisfying \begin{equation} \label{e1} \left[ \begin{matrix} X & 0 \\ 0 & -Y \end{matrix} \right] \leq \frac 1 \varepsilon \left[ \begin{matrix} I & -I \\ -I & I \end{matrix} \right] +r(\beta) \left[ \begin{matrix} I & 0 \\ 0 & I \end{matrix} \right] \end{equation} for some $\varepsilon>0$ and $r(\beta)\to 0$ as $\beta\to 0$, then, if $s(\beta)\to 0$ as $\beta \to 0$, we have \[ F(y,v,\varepsilon^{-1}(x-y),Y,\{i_\alpha\})-F(x,u,\varepsilon^{-1}(x-y)+s(\beta),X,\{i_\alpha\})\geq\omega(\beta) +\omega_R(|x-y|+\varepsilon^{-1}|x-y|^2)\]
*# For any $R>0$, $F$ is uniformly continuous on $R^n\times [−R,R] \times B_R \times D_R \times R^A$ where $D_R := \{X \in \R^{n \times n}, \text{symmetric}, |X|\leq R\}$ and there exist a modulus of continuity $\omega_R$ such that, for any $x,y\in \R^n$ , $|v|\leq R$, $\{i_\alpha\}\in \R^A$ and for any $X,Y \in \R^{n\times n}$ symmetric satisfying \eqref{e1} and $\varepsilon >0$, we have \[ F(y,v,\varepsilon^{-1}(x-y),Y,\{i_\alpha\})-F(x,u,\varepsilon^{-1}(x-y),X,\{i_\alpha\})\geq \omega_R(\varepsilon^{-1}|x-y|^2+|x-y|+r(\beta))\]
* $F(x,u,p,X,\{i_\alpha\})$ is Lipschitz continuous in $\{i_\alpha\}$ uniformly respect to all other variables.


Note that the assumptions are designed so that Ishii's lemma can be reproduced in this setting.
===Gradient flows===
 
The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth.
<div style="background:#DDEEFF;">
\[ u_t + \partial_u J[u] = u_t - \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\]
<blockquote>
The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients.
'''Note'''. Any linear integro-differential operator which is written in Levy-Ito form can be written in the usual form (with $j(x,z)=z$ and an $x$ dependent jump measure $\mu(x,dz)$) by a change of variables. Conversely, any integro-differential operator with an $x$ dependent $\mu$ can be written in Levy-Ito form for several choices of $j$, as long as $\int_{\R^n} \mu(x,dz) = +\infty$ for all $x$. The reason for assuming that all operators $I_\alpha$ are in the Levy-Ito form is to state the right continuity condition in the second assumption. A comparison principle for a family of integro-differential operators $I_\alpha$ with measures $\mu(x,dz)$ (or kernels $K(x,y)$) which are continuous with respect to $x$, but without any continuity assumption with respect to their Levy-Ito form, is currently an open problem.
</blockquote>
</div>
 
 
=== Uniformly elliptic equations without $x$ dependence ===
A simple comparison principle can be shown for [[fully nonlinear integro-differential equations]] which are translation invariant and uniformly elliptic respect to some family $\mathcal L$ of linear operators satisfying some assumptions <ref name="CS"/>. In fact, if $Iu \geq 0$ and $Iv \leq 0$ in some domain $\Omega$, then $M_{\mathcal L}^+ (u-v) \geq 0$ where $M_{\mathcal L}$ is the [[extremal operator]] respect to the class $\mathcal L$.
 
The family $\mathcal L$ is assumed to be a family of purely integro-differential [[linear integro-differential operators]] which are translation invariant and satisfy the following assumption:
 
<blockquote> There is a constant $R_0\geq 1$ so that for every $R>R_0$, there exists
a $\delta>0$ (which could depend on $R$) such that for any operator $L$ in $\mathcal L$, we have that
$L \varphi > \delta$ in $B_R$, where $\varphi$ is given by $\varphi(x)=\min(1,|x|^2/R^3)$.
</blockquote>
 
This assumption on the family $\mathcal L$ is a very mild form of uniform ellipticity. The strong assumption is translation invariance. The proof of this comparison principle uses inf and sup-convolutions.
 
 
== References ==
{{reflist|refs=
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="BI">{{Citation | last1=Barles | first1=Guy | last2=Imbert | first2=Cyril | title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited | url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 | doi=10.1016/j.anihpc.2007.02.007 | year=2008 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=25 | issue=3 | pages=567–585}}</ref>
<ref name="BIC2">{{Citation | last1=Barles | first1=G. | last2=Chasseigne | first2=Emmanuel | last3=Imbert | first3=Cyril | title=On the Dirichlet problem for second-order elliptic integro-differential equations | url=http://dx.doi.org/10.1512/iumj.2008.57.3315 | doi=10.1512/iumj.2008.57.3315 | year=2008 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=57 | issue=1 | pages=213–246}}</ref>
}}

Revision as of 14:14, 14 March 2012

The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form.

The equation is \[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \] in the elliptic case, or \[ u_t = \mathrm{div} A(x,t) \nabla u(x). \] Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$, \[ \langle A v,v \rangle \geq \lambda |v|^2,\] for every $v \in \R^n$, uniformly in space and time.

The corresponding result in non divergence form is Krylov-Safonov theorem.

For nonlocal equations, there are analogous results both for Holder estimates and the Harnack inequality.

Elliptic version

For the result in the elliptic case, we assume that the equation \[ \mathrm{div} A(x) \nabla u(x) = 0 \] is satisfied in the unit ball $B_1$ of $\R^n$.

Holder estimate

The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $B_{1/2}$ and \[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\] The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.

The result can be scaled to balls of arbitrary radius $r>0$ to obtain \[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]

Moreover, by covering an arbitrary domain $\Omega$ with balls, one can show that a solution to the equation in $\Omega$ is $C^\alpha$ in the interior of $\Omega$.

Harnack inequality

The Harnack inequality says that if $u$ is a non negative solution of the equation in $B_1$, then its minimum controls its maximum in $B_{1/2}$: \[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\] The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only.

Minimizers of convex functionals

The theorem of De Giorgi, Nash and Moser was used originally to solve one of the famous Hilbert problems. The question was whether the minimizers of Dirichlet integrals \[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\] are always smooth if $F$ is smooth and strictly convex. The theorem of De Giorgi-Nash-Moser in its elliptic form can be applied to the differential quotients of the minimizer of $J$ to show that the solution is $C^{1,\alpha}$. Once that initial regularity is obtained, further regularity follows by bootstrapping with the Schauder estimates and the smoothness of $F$.

Note that in order to apply the theorem to these nonlinear equations, it is very important that no smoothness assumption on the coefficients $A(x)$ is made.

Parabolic version

For the result in the parabolic case, we assume that the equation \[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \] is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$.

Holder estimate

The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $[1/2,1] \times B_{1/2}$ and \[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\] The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.

Harnack inequality

The Harnack inequality says that if $u$ is a non negative solution of the equation in $[0,1] \times B_1$, then its minimum controls its maximum in a previous time: \[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \]

Gradient flows

The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth. \[ u_t + \partial_u J[u] = u_t - \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\] The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients.

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