Different Domains Induce Different Heat Semigroups on C 0(Ω)

Let Ω₁,Ω₂ ⊂ ℝ ^N be two open, connected sets which are regular in the sense of Wiener. Denote by Δ 0 Ω j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter1_1_1.tif"/> the Laplacian on C ₀(Ω _j ), j = 1, 2. Assume that there exists a non-zero linear mapping U: C ₀(Ω₁) → C ₀(Ω₂) such that

∣Uf∣ = U∣f∣ (f ∈ C ₀(Ω₁)) and

U e t Δ 0 Ω 1 = e t Δ 0 Ω 2 U                 ( t ≥ 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter1_1_2.tif"/> .

Then it is shown that Ω₁ and Ω₂ are congruent. This result complements [2] where the Laplacian on L^p was considered and U was supposed to be bijective.