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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.
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<div style="font-size:162%; border:none; margin:0; padding:.1em; color:#000;">'''Welcome!'''</div>
<div style="top:+0.2em; font-size:95%;">This is the Nonlocal Equations Wiki</div>
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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.


In this wiki we collect several results about nonlocal elliptic and parabolic equations. If you want to know what a nonlocal equation refers to, a good starting point would be the [[Intro to nonlocal equations]]. If you want to find information on a specific topic, you may want to check the [[list of equations]] or use the search option on the left.
The corresponding result in non divergence form is [[Krylov-Safonov theorem]].


We also keep a list of [[open problems]] and of [[upcoming events]].
For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]].


The wiki has an assumed bias towards regularity results and consequently to equations for which some regularization occurs. But we also include some topics which are tangentially related, or even completely unrelated, to regularity.  
== 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 $||A||_{L^\infty}$.


Some answers, including how to participate, can be found in the section about [[frequently asked questions]].
The result can be scaled to balls of arbitrary radius $r>0$ to obtain
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\[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]


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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 $\Omega$.
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===Harnack inequality===
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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.


<div style="font-size:150%; border:none; margin:0; padding:.1em; color:#000;">Why nonlocal equations?</div>
===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 $F$.


All partial differential equations are a limit case of nonlocal equations. A good understanding of nonlocal equations can ultimately provide a better understanding of their limit case: the PDEs. However, there are some cases in which a nonlocal equation gives a significantly better model than a PDE. Some of the most clear examples in which it is necessary to resort to nonlocal equations are
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.
* Optimal control problems with [[Levy processes]] give rise to the [[Bellman equation]], or in general any equation derived from jump processes will be some [[fully nonlinear integro-differential equation]].
* In [[financial mathematics]] it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the [[obstacle problem]].
* [[Nonlocal electrostatics]] is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.
* The denoising algorithms in [[nonlocal image processing]] are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the [[nonlocal mean curvature flow]].
* The [[Boltzmann equation]] models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified [[kinetic models]] can be used to derive the [[fractional heat equation]] without resorting to stochastic processes.
* In conformal geometry, the Paneitz operators encode information about the manifold, they include fractional powers of the Laplacian, which are nonlocal operators.
* In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the [[surface quasi-geostrophic equation]].
* Models for [[dislocation dynamics]] in crystals.
* Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the [[Nonlocal porous medium equation]], the [[Hamilton-Jacobi equation with fractional diffusion]], [[conservation laws with fractional diffusion]], etc...


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== 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 $||A||_{L^\infty}$.


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===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:
\[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \]


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===Gradient flows===
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The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth.
 
\[ u_t + \partial_u J[u] = u_t - \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\]
<div style="font-size:150%; border:none; margin:0; padding:.1em; color:#000;"> Suggested first reads </div>
The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients.
 
* [[Intro to nonlocal equations]]
 
* [[Fractional Laplacian]]
 
* [[Linear integro-differential operator]]
 
* [[Fully nonlinear integro-differential equations]]
 
* [[Myths about nonlocal equations]]
 
* [[Surface quasi-geostrophic equation]]
 
* [[Levy processes]]
 
* [[Obstacle problem]]
 
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Revision as of 15:14, 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.