Dirichlet Forms and Degenerate Elliptic Operators
A. F. M. ter Elst, Derek W. Robinson, Adam Sikora, Yueping Zhu · Birkhäuser Basel eBooks · 2006
It is shown that the theory of real symmetric second-order elliptic operators in divergence form on ℝd can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behavior of the corresponding evolution semigroup S t can be described in terms of a function (A, B) ↦ d(A; B) ∈ [0, ∞] over pairs of measurable subsets of ℝd. Then $$ \left| {\left( {\varphi A,S_t \varphi B} \right)} \right| \leqslant e^{ - d(A;B)^2 (4t)^{ - 1} } ||\varphi A||2||\varphi B||2 $$ for all t > 0 and all ϕ A ↦ L 2(A), ϕ B ∈ L 2(B). Moreover S t L 2(A) ∈ L 2(A) for all t > 0 if and only if d(A;A c) = ∞ where A c denotes the complement of A.