Dirichlet forms and degenerate elliptic operators

Abstract

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B)\mapsto d(A;B)\in [0,\infty ]$ over pairs of measurable subsets of $\Ri^d$. Then \[ |(ϕ_A,S_tϕ_B)|\leq e^{-d(A;B)^2(4t)^{-1}}\|ϕ_A\|_2\|ϕ_B\|_2 \] for all $t>0$ and all ${\varphi }_A\in L_2(A)$, ${\varphi }_B\in L_2(B)$. Moreover $S_t{L}_2(A)\subseteq L_2(A)$ for all $t>0$ if and only if $d(A;A^c)=\infty$ where $A^c$ denotes the complement of $A$.

22 pages

Cited by in corpus (3)

• Conservation and invariance properties of submarkovian semigroups
• Hardy inequalities, Rellich inequalities and local Dirichlet forms
• Ellipticity and Ergodicity