Dirichlet Regularity and Degenerate Diffusion

Let Ω ⊂ RN be an open and bounded set and let m : Ω → (0,∞) be measurable and locally bounded. We study a natural realization of the operator m in C0(Ω) := { u ∈ C(Ω) : u|∂Ω = 0 } . If Ω is Dirichlet regular, then the operator generates a positive contraction semigroup on C0(Ω) whenever 1 m ∈ Lploc(Ω) for some p > N 2 . If m(x) does not go fast enough to 0 as x → ∂Ω, then Dirichlet regularity is necessary. However, if |m(x)| ≤ c·dist(x, ∂Ω)2, then we show that m 0 generates a semigroup on C0(Ω) without any regularity assumptions on Ω. We show that the condition for degeneration of m near the boundary is optimal.