Dirichlet regularity and degenerate diffusion

Abstract

Let Ω C ℝ N be an open and bounded set and let m: Ω→ (0, oo) be measurable and locally bounded. We study a natural realization of the operator mΔ in Co(Ω) := {u ∈ C(Ω): u |∂Ω = 0}. If Ω is Dirichlet regular, then the operator generates a positive contraction semigroup on C 0 (Ω) whenever 1/m ∈ L p loc (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 mΔ 0 generates a semigroup on C 0 (Ω) without any regularity assumptions on Ω. We show that the condition for degeneration of m near the boundary is optimal.