Different Domains Induce Different Heat Semigroups on C 0(Ω)
Wolfgang Arendt · 2019
Let Ω 1 ,Ω 2 ⊂ 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 (Ω 1 ) → C 0 (Ω 2 ) such that ∣ Uf ∣ = U ∣ f ∣ ( f ∈ C 0 (Ω 1 )) 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 Ω 1 and Ω 2 are congruent. This result complements [ 2 ] where the Laplacian on L p was considered and U was supposed to be bijective.