Dirichlet Forms, Caccioppoli Sets and the Skorohod Equation

Abstract

Let R ^d be the Euclidean d—space and m be the Lebesgue measure on it. We consider a bounded domain D ⊂ R ^d with m(∂ D) = 0 and and the Sobolev space

$$ {H^1}\left( D \right) = \left\{ {u \in {L^2}\left( D \right):{\partial _i}u \in {L^2}\left( D \right),1 \leqslant i \leqslant d} \right\} $$

(1.1)

with inner product

$$ \varepsilon \left( {u,v} \right) = \frac{1}{2}{\smallint _D}\nabla u \cdot \nabla vdx,u,v \in {H^1}\left( D \right) $$

(2.1)

(H ¹(D), ε) is a specific Dirichlet space on L ² (D)but it is not necessarily regular.