Dirichlet form - Revision history

imported>Nestor at 22:49, 18 November 2012

2012-11-18T22:49:12Z

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

imported>Nestor at 22:49, 18 November 2012

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


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imported>Nestor at 22:44, 18 November 2012

2012-11-18T22:44:52Z

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	Then we see that Dirichlet forms are natural generalizations of the Dirichlet integrals		Then we see that Dirichlet forms are natural generalizations of the Dirichlet integrals
	\[ \mathcal{E}(u) = \int (A\nabla u,\nabla u)\; \dd x, \]		\[ \mathcal{E}(u) = \int (A\nabla u,\nabla u)\; \dd x, \]
	where $A(x)$ is a positive symmetric matrix.		where $A(x)$ is a positive symmetric 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 <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.

	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 <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.

	== References ==		== References ==

imported>Nestor at 22:44, 18 November 2012

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	Then we see that Dirichlet forms are natural generalizations of the Dirichlet integrals		Then we see that Dirichlet forms are natural generalizations of the Dirichlet integrals
	\[ \~~int a_~~{ij}(x) \~~partial_i~~ u \~~partial_j~~ u \dd x, \]		\[ \mathcal{E}(u) = \int (A\nabla u,\nabla u)\; \dd x, \]
	where $~~a_{ij}~~(x)$ is a positive matrix.		where $A(x)$ is a positive symmetric 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 <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.		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 <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.

imported>Nestor at 22:43, 18 November 2012

2012-11-18T22:43:41Z

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

	which gives rise to the space $H^1(\mathbb{R}^n)$. Another example of a Dirichlet form is given by		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) = \iint_{\R^n \times \R^n} (u(y)-u(x))^2 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.

imported>Nestor at 22:41, 18 November 2012

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	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 <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.		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 <ref name="BBCK"/><ref name="K"/><ref name="CCV"/>.

	== References ==		== References ==

imported>Nestor at 21:59, 18 November 2012

2012-11-18T21:59:09Z

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	with the following properties		with the following properties

	1) The domain $D$ is a dense subset of $\mathbb{R}^n$		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 $u,v \in D$.		2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(u,v)=\mathcal{E}(v,u)$ for any $u,v \in D$.

imported>Nestor at 21:58, 18 November 2012

2012-11-18T21:58:29Z

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

imported>Nestor at 21:57, 18 November 2012

2012-11-18T21:57:43Z

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

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

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

	4) The set $D$ equipped with the inner product defined by $(f,g)_{\mathcal{E}} := (f,g)_{L^2(\mathbb{R}^n)} + \mathcal{E}(f,g)$ is a real Hilbert space.		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 $f \in D$ we have that $f_* = (f\vee 0) \wedge 1 \in D$ and $\mathcal{E}(f_,f_)\leq \mathcal{E}(f,f)$		5) For any $u \in D$ we have that $u_* = (u\vee 0) \wedge 1 \in D$ and $\mathcal{E}(u_,u_)\leq \mathcal{E}(u,u)$


	~~A particular case~~ of a Dirichlet form ~~are defined~~ by ~~integrals~~ of the form		An example of a Dirichlet form is given by any integral of the form
	\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \]		\mathcal{E}(u,v) = \[ \iint_{\R^n \times \R^n} (u(y)-u(x))(v(y)-v(x))k(x,y)\, \dd x \dd y, \]
	~~for some nonnegative kernel~~ $K$.		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.		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		Dirichlet forms are natural generalizations of the Dirichlet integrals

imported>Nestor at 21:53, 18 November 2012

2012-11-18T21:53:59Z

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

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

	with the following properties		with the following properties

	* The domain $D$ is a dense subset of $\mathbb{R}^n$ ~~and~~		1) The domain $D$ is a dense subset of $\mathbb{R}^n$
	* $\mathcal{E}$ is symmetric, that is $\mathcal{E}(f,g)=\mathcal{E}(g,f)$ for any $f,g \in D$.
	* $\mathcal{E}(f,f)\geq 0$ for any $f \in D$.
	* The set $D$ equipped with the inner product defined by

	\~~begin~~{equation*}		2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(f,g)=\mathcal{E}(g,f)$ for any $f,g \in D$.
	(f,~~g)_{~~\mathcal{E}~~} :=~~ (f,g)~~_{L^2(\mathbb{R}^n)} +~~ \mathcal{E}(f,g)
	\~~end{equation*}~~

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

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

	A Dirichlet form ~~refers to a quadratic functional~~ defined by ~~an integral~~ of the form		5) For any $f \in D$ we have that $f_* = (f\vee 0) \wedge 1 \in D$ and $\mathcal{E}(f_,f_)\leq \mathcal{E}(f,f)$


			A particular case of a Dirichlet form are defined by integrals of the form
	\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \]		\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \]
	for some nonnegative kernel $K$.		for some nonnegative kernel $K$.
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	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.		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		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 $a_{ij}$ is elliptic.