Interior regularity results (local) and Dirichlet form: Difference between pages

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Let <math>\Omega</math> be an open domain and <math> u </math> a solution of an elliptic equation in <math> \Omega </math>. The following theorems say that <math> u </math> satisfies some regularity estimates in the interior of <math> \Omega </math> (but not necessarily up the the boundary).
$$
\newcommand{\dd}{\mathrm{d}}
\newcommand{\R}{\mathbb{R}}
$$


A Dirichlet form in $\mathbb{R}^n$ is a bilinear function


== Linear equations ==
\begin{equation*}
\mathcal{E}: D\times D \to \mathbb{R}
\end{equation*}


Regularity results for linear equations are applicable to nonlinear equations as well through the linearization of the equation. However, this process requires some initial regularity knowledge on the solution (since the coefficients of the linearization depend on the solution itself). Therefore, the less regularity required for the coefficients, the more useful the theorem is.
with the following properties


All regularity results that require some modulus of continuity or smallness condition for the coefficients rely on the idea that the solution is locally close to a solution to an equation with constant coefficients. The proof is based on an estimate on how far these two solutions are at small scales. These type of arguments are often called [[perturbation methods]].
1) The domain $D$ is a dense subset of $L^2(\mathbb{R}^n)$


From the results below for linear equations, [[De Giorgi-Nash-Moser]] and [[Krylov-Safonov]] are the only non perturbative results. Their assumptions are scale invariant in the sense that a rescaling of the solution ($u_r(x) = u(rx)$) would solve an elliptic equation with the same bounds as the original.
2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(u,v)=\mathcal{E}(v,u)$ for any $u,v \in D$.


* [[De Giorgi-Nash-Moser]]
3) $\mathcal{E}(u,u) \geq 0$ for any $u \in D$.
<math> {\rm div \,} A(x) Du + b(x) \cdot \nabla u = 0 </math>


then <math>u</math> is Holder continuous if <math>A</math> is just uniformly elliptic and <math>b</math>
4) The set $D$ equipped with the inner product defined by $(u,v)_{\mathcal{E}} := (u,v)_{L^2(\mathbb{R}^n)} + \mathcal{E}(u,v)$ is a real Hilbert space.
is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).  


* [[Krylov-Safonov theorem]]
5) For any $u \in D$ we have that $u_* = (u\vee 0) \wedge 1 \in D$ and $\mathcal{E}(u_*,u_*)\leq \mathcal{E}(u,u)$
<math>a_{ij}(x) u_{ij} + b \cdot \nabla u = f </math>


with <math>a_{ij}</math> unif elliptic, <math>b \in L^n</math> and <math>f \in L^n</math>, then the
solution is <math>C^\alpha</math>


* [[Calderon-zygmund estimate]]
An example of a Dirichlet form is given by  any integral of the form
<math>a_{ij}(x) u_{ij} = f</math>
\begin{equation*}
\mathcal{E}(u,v) = \iint_{\R^n \times \R^n} (u(y)-u(x))(v(y)-v(x))k(x,y)\, \dd x \dd y
\end{equation*}
where $K$ is some non-negative symmetric kernel.


with <math>a_{ij}</math> close enough to the identity (or continuous) and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.
If the kernel $K$ satisfies the bound $K(x,y) \leq \Lambda |x-y|^{-n-s}$, then the quadratic form is bounded in $\dot H^{s/2}$ . If moreover, $\lambda |x-y|^{-n-s} \leq K(x,y)$, then the form is comparable to the norm in $\dot H^{s/2}$ squared and in that case the set $D \subset L^2(\mathbb{R}^n)$ defined above is given by  $H^{s/2}(\mathbb{R}^n)$


* [[Cordes-Nirenberg estimate]]
Dirichlet forms are natural generalizations of the Dirichlet integrals
<math>a_{ij}(x) u_{ij} = f</math>
\[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \]
where $a_{ij}$ is elliptic.


with <math>a_{ij}</math> close enough to the identity uniformly and $f \in L^\infty$,
The Euler-Lagrange equation of a Dirichlet form is a fractional order version of elliptic equations in divergence form. They are studied using variational methods and they are expected to satisfy similar properties <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.
then $u$ is in $C^{1,\alpha}$


* [[Cordes-Nirenberg estimate improved]] (corollary of work of Caffarelli for nonlinear equations)
== References ==
<math>a_{ij}(x) u_{ij} = f</math>
(There should be a lot more references here)
{{reflist|refs=
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</ref>
<ref name="BBCK">{{Citation | last1=Barlow | first1=Martin T. | last2=Bass | first2=Richard F. | last3=Chen | first3=Zhen-Qing | last4=Kassmann | first4=Moritz | title=Non-local Dirichlet forms and symmetric jump processes | url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 | doi=10.1090/S0002-9947-08-04544-3 | year=2009 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=361 | issue=4 | pages=1963–1999}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
}}


with $a_{ij}$ close enough to the identity in a scale invariant Morrey
norm in terms of $L^n$ and $f \in L^n$, then $u$ is in $C^{1,\alpha}$.


($a_{ij} \in VMO$ is a particular case of this)
{{stub}}
 
*[[Schauder estimate]]
<math>a_{ij}(x) u_{ij} = f</math>
 
with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$
 
== Non linear equations ==
 
* [[De Giorgi-Nash-Moser]]
For any smooth strictly convex Lagrangian $L$, minimizers of functionals
 
$ \int_D L(\nabla u) \ dx $
 
are smooth (analytic if $L$ is analytic).
 
* [[Krylov-Safonov|Krylov-Safonov-Caffarelli]]
 
Any continuous function $u$ such that
 
$M^+(D^2 u) \geq 0 \geq M^-(D^2 u)$
 
in the viscosity sense (where $M^+$ and $M^-$ are the Pucci operators), is Holder continuous.
 
(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)
 
* [[Ishii-Lions Lipschitz estimate]]
 
If $u$ solves a fully nonlinear equation
 
$F(D^2 u, Du, u, x) = 0$
 
which is degenerate elliptic but satisfies some structure conditions and some smoothness assumptions respect to $x$, then $u$ is Lipschitz.
 
(The proof of this is based on the uniqueness technique for viscosity solutions)
 
* [[Lin theorem]]
 
Any continuous function $u$ such that
 
$0 \geq M^-(D^2 u)$
 
in the viscosity sense, is twice differentiable almost everywhere and $D^2 u \in L^\varepsilon$.
 
(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)
 
* [[Krylov-Safonov|Krylov-Safonov-Caffarelli]]
If $u$ solves a fully nonlinear equation
 
$F(D^2 u, x) = 0$
 
which is uniformly elliptic and continuous respect to $x$ ($VMO$ is actually enough), then $u \in C^{1,\alpha}$.
 
* [[Evans-Krylov theorem]]
If $u$ solves a convex (or concave) fully nonlinear equation
 
$F(D^2 u, x) = 0$
 
which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.
 
[[Category:Mini second order elliptic wiki]]

Revision as of 16:59, 18 November 2012

$$ \newcommand{\dd}{\mathrm{d}} \newcommand{\R}{\mathbb{R}} $$

A Dirichlet form in $\mathbb{R}^n$ is a bilinear function

\begin{equation*} \mathcal{E}: D\times D \to \mathbb{R} \end{equation*}

with the following properties

1) The domain $D$ is a dense subset of $L^2(\mathbb{R}^n)$

2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(u,v)=\mathcal{E}(v,u)$ for any $u,v \in D$.

3) $\mathcal{E}(u,u) \geq 0$ for any $u \in D$.

4) The set $D$ equipped with the inner product defined by $(u,v)_{\mathcal{E}} := (u,v)_{L^2(\mathbb{R}^n)} + \mathcal{E}(u,v)$ is a real Hilbert space.

5) For any $u \in D$ we have that $u_* = (u\vee 0) \wedge 1 \in D$ and $\mathcal{E}(u_*,u_*)\leq \mathcal{E}(u,u)$


An example of a Dirichlet form is given by any integral of the form \begin{equation*} \mathcal{E}(u,v) = \iint_{\R^n \times \R^n} (u(y)-u(x))(v(y)-v(x))k(x,y)\, \dd x \dd y \end{equation*} where $K$ is some non-negative symmetric kernel.

If the kernel $K$ satisfies the bound $K(x,y) \leq \Lambda |x-y|^{-n-s}$, then the quadratic form is bounded in $\dot H^{s/2}$ . If moreover, $\lambda |x-y|^{-n-s} \leq K(x,y)$, then the form is comparable to the norm in $\dot H^{s/2}$ squared and in that case the set $D \subset L^2(\mathbb{R}^n)$ defined above is given by $H^{s/2}(\mathbb{R}^n)$

Dirichlet forms are natural generalizations of the Dirichlet integrals \[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \] where $a_{ij}$ is elliptic.

The Euler-Lagrange equation of a Dirichlet form is a fractional order version of elliptic equations in divergence form. They are studied using variational methods and they are expected to satisfy similar properties [1][2][3].

References

(There should be a lot more references here)

  1. ↑ Barlow, Martin T.; Bass, Richard F.; Chen, Zhen-Qing; Kassmann, Moritz (2009), "Non-local Dirichlet forms and symmetric jump processes", Transactions of the American Mathematical Society 361 (4): 1963–1999, doi:10.1090/S0002-9947-08-04544-3, ISSN 0002-9947, http://dx.doi.org/10.1090/S0002-9947-08-04544-3 
  2. ↑ Kassmann, Moritz (2009), "A priori estimates for integro-differential operators with measurable kernels", Calculus of Variations and Partial Differential Equations 34 (1): 1–21, doi:10.1007/s00526-008-0173-6, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-008-0173-6 
  3. ↑ Caffarelli, Luis; Chan, Chi Hin; Vasseur, Alexis (2011), Journal of the American Mathematical Society (24): 849–869, doi:10.1090/S0894-0347-2011-00698-X, ISSN 0894-0347 


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