Dirichlet form

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

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*} \mathcal{E}(u) = \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) = \iint_{\R^n \times \R^n} (u(y)-u(x))^2 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 .

References

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