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 refers to a quadratic functional defined by an integral of the form
\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \]
for some nonnegative kernel $K$.


== Linear equations ==
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.


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


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]].
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"/>.


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.
== References ==
(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>
}}


* [[De Giorgi-Nash-Moser]]
<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>
{{stub}}
is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).
 
* [[Krylov-Safonov]]
<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]]
<math>a_{ij}(x) u_{ij} = f</math>
 
with <math>a_{ij}</math> close enough to the identity and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.
 
* [[Cordes-Nirenberg]]
<math>a_{ij}(x) u_{ij} = f</math>
 
with <math>a_{ij}</math> close enough to the identity uniformly and f in L^infty,
then $u$ is in $C^{1,\alpha}$
 
* [[Cordes-Nirenberg improved]] (corollary of work of Caffarelli for nonlinear equations)
<math>a_{ij}(x) u_{ij} = f</math>
 
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)
 
*[[Schauder]]
<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]]
 
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]]
 
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]]
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 $u \in C^{2,\alpha}$.

Revision as of 21:36, 5 February 2012

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

A Dirichlet form refers to a quadratic functional defined by an integral of the form \[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \] for some nonnegative kernel $K$.

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.

Dirichlet forms are natural generalizations to fractional order 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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