Harnack inequality and Dirichlet form: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>Luis
 
imported>Nestor
No edit summary
 
Line 4: Line 4:
$$
$$


The Harnack inequality refers to a control of the maximum of a nonnegative solution of an equation by its minimum. Unlike the local case (either [[De Giorgi-Nash-Moser theorem]] or [[Krylov-Safonov theorem]]), for nonlocal equations one needs to assume that the function is nonnegative in the full space.
A Dirichlet form in $\mathbb{R}^n$ is a bilinear function


The Harnack inequality is tightly related to [[Holder estimates]] for solutions to elliptic/parabolic equations. For a large class of problems both statements are equivalent. But there are simple cases (stable processes with the spectral measure consisting of atoms) where the Harnack inequality fails but [[Hölder estimates]] still hold true.
\begin{equation*}
\mathcal{E}: D\times D \to \mathbb{R}
\end{equation*}


The result can hold either in the parabolic or elliptic setting. The parabolic Harnack inequality trivially implies the elliptic one. The reverse implication is not automatic, and the proof in the parabolic case may have some extra difficulties compared to the elliptic case.
with the following properties


== Elliptic case ==
1) The domain $D$ is a dense subset of $\mathbb{R}^n$


In the elliptic setting, the Harnack inequality refers to the following type of result: if a function $u: \R^n \to \R$ satisfies an elliptic equation $ L_x u (x) = f(x)$ in the unit ball $B_1$ and is nonnegative in the full space $\R^n$, then
2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(u,v)=\mathcal{E}(v,u)$ for any $u,v \in D$.
\[ \sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + \|f\| \right). \]


The norm $\|f\|$ may depend on the type of equation.
3) $\mathcal{E}(u,u) \geq 0$ for any $u \in D$.


== Parabolic case ==
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.


In the parabolic setting, the Harnack inequality refers to the following type of result: if a function $u: [-1,0] \times \R^n \to \R$ satisfies a parabolic equation $ u_t - L_x u (x) = f(x)$ in the unit cylinder $(-1,0) \times B_1$ and is nonnegative in the full space $[-1,0] \times \R^n$, then
5) For any $u \in D$ we have that $u_* = (u\vee 0) \wedge 1 \in D$ and $\mathcal{E}(u_*,u_*)\leq \mathcal{E}(u,u)$
\[ \sup_{[-1/2,-1/4] \times B_{1/2}} u \leq \left(\inf_{[-1/4,0] \times B_{1/2}} u + ||f|| \right). \]


The norm $||f||$ may depend on the type of equation.


== Concrete examples ==
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.


The Harnack inequality as above is known to hold in the following situations.
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  $H^{s/2}(\mathbb{R}^n)$


* '''Generalizad elliptic and parabolic [[Krylov-Safonov]]'''. If $L_x u(x)$ is a symmetric integro-differential operator of the form
Dirichlet forms are natural generalizations of the Dirichlet integrals
\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)) K(x,y) \dd y \]
\[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \]
with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda |y|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda |y|^{-n-s}$.
where $a_{ij}$ is elliptic.


In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $||f||$ refers to $||f||_{L^\infty(B_1)}$ <ref name="CS"/><ref name="lara2011regularity"/>. It is a generalization of [[Krylov-Safonov]] theorem.
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"/>.
 
* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form
\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)- y \cdot \nabla u(x) \chi_{B_1}(y)) K(x,y) \dd y \]
with uniformly elliptic of variable order: $\lambda |y|^{-n-s_1} \leq K(x,y) \leq \Lambda |y|^{-n-s_2}$ and $0<s_1 < s_2 < 2$ and $s_2 - s_1 < 1$, then
the elliptic Harnack inequality holds if $f \equiv 0$<ref name="BK"/>. The constants in this result blow up as $s_2 \to 2$, so it does not generalize [[Krylov-Safonov]] theorem. The proof uses probability and was based on a previous result with fixed order <ref name="BL"/>.
 
It is conceivable that a purely analytic proof could be done using the method of the corresponding [[Holder estimate]] <ref name="S"/>, but such proof has never been done.
 
* '''Gradient flows of symmetric Dirichlet forms with variable order'''. If $u_t - L_x u(x)=0$ is the gradient flow of a [[Dirichlet form]]:
\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 K(x,y)\, \dd x \dd y. \]
for kernels $K$ such that $K(x,y)=K(y,x)$ and $\lambda |x-y|^{-n-s_1} \leq K(x,y) \leq \Lambda |x-y|^{-n-s_2}$ for some $0<s_1<s_2<2$ and $|x-y|$ sufficiently small. Then the parabolic Harnack inequality holds if $f \equiv 0$ for some constant $C$ which a priori '''blows up as $s_2 \to 2$''' <ref name="BBCK"/>.
 
It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.


== References ==
== References ==
(There should be a lot more references here)
{{reflist|refs=
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
<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="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="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Harnack inequalities for non-local operators of variable order | url=http://dx.doi.org/10.1090/S0002-9947-04-03549-4 | doi=10.1090/S0002-9947-04-03549-4 | year=2005 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=357 | issue=2 | pages=837–850}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin | first2=David A. | title=Harnack inequalities for jump processes | url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 | year=2002 | journal=Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis | issn=0926-2601 | volume=17 | issue=4 | pages=375–388}}</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="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>
<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>
<ref name="lara2011regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | journal=Calculus of Variations and Partial Differential Equations | year=2011 | pages=1--34}}</ref>
}}
}}
{{stub}}

Revision as of 16:58, 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 $\mathbb{R}^n$

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 


This article is a stub. You can help this nonlocal wiki by expanding it.