# Dirichlet form

(Difference between revisions)
 Revision as of 21:59, 18 November 2012 (view source)Nestor (Talk | contribs)← Older edit Revision as of 22:41, 18 November 2012 (view source)Nestor (Talk | contribs) Newer edit → Line 23: Line 23: - An example of a Dirichlet form is given by  any integral of the form + In other words, a Dirichlet form is nothing but a positive symmetric bilinear form defined in a dense subset of $L^2(\mathbb{R}^n)$ such that 4) and 5) hold. Alternatively, the quadratic form $u \to \mathcal{E}(u,u)$ itself is known as the Dirichlet form and it is still denoted by $\mathcal{E}$, so $\mathcal{E}(u):=\mathcal{E}(u,u)$. + + The best known Dirichlet form is the Dirichlet energy + \begin{equation*} + \int_{\mathbb{R}^n} |\nabla u|^2\;dx + \end{equation*} + + which gives rise to the space $H^1(\mathbb{R}^n)$. Another example of a Dirichlet form is given by \begin{equation*} \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 \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*} \end{equation*} - where $K$ is some non-negative symmetric kernel. + 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  $H^{s/2}(\mathbb{R}^n)$ 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 + Then we see that Dirichlet forms are natural generalizations of the Dirichlet integrals $\int a_{ij}(x) \partial_i u \partial_j u \dd x,$ $\int a_{ij}(x) \partial_i u \partial_j u \dd x,$ - where $a_{ij}$ is elliptic. + where $a_{ij}(x)$ is a positive matrix. - 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 . + The Euler-Lagrange equation of a Dirichlet form is a non-local analogue  of an elliptic equations in divergence form. Equations of this type are studied using variational methods and they are expected to satisfy similar properties . == References == == References ==

## Revision as of 22:41, 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 $\mathcal{E}(u_*,u_*)\leq \mathcal{E}(u,u)$

In other words, a Dirichlet form is nothing but a positive symmetric bilinear form defined in a dense subset of $L^2(\mathbb{R}^n)$ such that 4) and 5) hold. Alternatively, the quadratic form $u \to \mathcal{E}(u,u)$ itself is known as the Dirichlet form and it is still denoted by $\mathcal{E}$, so $\mathcal{E}(u):=\mathcal{E}(u,u)$.

The best known Dirichlet form is the Dirichlet energy \begin{equation*} \int_{\mathbb{R}^n} |\nabla u|^2\;dx \end{equation*}

which gives rise to the space $H^1(\mathbb{R}^n)$. Another example of a Dirichlet form is given by \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 $H^{s/2}(\mathbb{R}^n)$

Then we see that Dirichlet forms are natural generalizations of the Dirichlet integrals $\int a_{ij}(x) \partial_i u \partial_j u \dd x,$ where $a_{ij}(x)$ is a positive matrix.

The Euler-Lagrange equation of a Dirichlet form is a non-local analogue of an elliptic equations in divergence form. Equations of this type 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
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
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