Uniqueness of diffusion on domains with rough boundaries

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• 作者：Juha Lehrbä ; ck^a ; ^{juha.lehrback@jyu.fi" class="auth_mail" title="E-mail the corresponding author} ; Derek W. Robinson^b ; ^{derek.robinson@anu.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：47D07 ; 35J70 ; 35K65
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：131
• 期：Complete
• 页码：60-80
• 全文大小：953 K

文摘

Let Ω$\Omega$ be a domain in $R^d$ and $h\left(\phi \right)={\sum }_{k,l=1}^d({\partial }_k\phi ,c_{kl}{\partial }_l\phi )$ a quadratic form on L₂(Ω)$L_2\left(\Omega \right)$ with domain $C_c^{\infty }\left(\Omega \right)$ where the c_kl$c_{kl}$ are real symmetric L_∞(Ω)$L_{\infty }\left(\Omega \right)$-functions with C(x)=(c_kl(x))>0$C\left(x\right)=\left(c_{kl}\left(x\right)\right)>0$ for almost all x∈Ω$x\in \Omega$. Further assume there are a,δ>0$a,\delta >0$ such that $a^{-1}{d}_{\Gamma }^{\delta }I\le C\le a{d}_{\Gamma }^{\delta }I$ for d_Γ≤1$d_{\Gamma }\le 1$ where d_Γ$d_{\Gamma }$ is the Euclidean distance to the boundary 33c8041d74df801ef1cca3" title="Click to view the MathML source">Γ$\Gamma$ of Ω$\Omega$.

We assume that 33c8041d74df801ef1cca3" title="Click to view the MathML source">Γ$\Gamma$ is Ahlfors s$s$-regular and if s$s$, the Hausdorff dimension of 33c8041d74df801ef1cca3" title="Click to view the MathML source">Γ$\Gamma$, is larger or equal to d−1$d-1$ we also assume a mild uniformity property for Ω$\Omega$ in the neighbourhood of one z∈Γ$z\in \Gamma$. Then we establish that h$h$ is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if δ≥1+(s−(d−1))$\delta \ge 1+(s-(d-1))$. The result applies to forms on Lipschitz domains or on a wide class of domains with 33c8041d74df801ef1cca3" title="Click to view the MathML source">Γ$\Gamma$ a self-similar fractal. In particular it applies to the interior or exterior of the von Koch snowflake curve in $R^2$ or the complement of a uniformly disconnected set in $R^d$.