Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L^p -Sobolev spaces
Résumé
We consider the heat equation in a polygonal domain Ω of the plane in weighted L^p-Sobolev spaces ∂_t u − ∆u = h, in Ω × ] − π, π[, u = 0, on ∂Ω × [−π, π], u(•, −π) = u(•, π), in Ω. Here h belongs to L^p (−π, π; L^p_µ (Ω)), where L^p_µ (Ω) = {v ∈ L^p_loc (Ω) : r^µ v ∈ L p (Ω)}, with a real parameter µ and r(x) the distance from x to the set of corners of Ω. We give sufficient conditions on µ, p and Ω that guarantee that the above problem has a unique solution u ∈ L^p (−π, π; L^p_µ (Ω)) that admits a decomposition into a regular part in weighted L^p-Sobolev spaces and an explicit singular part. The classical Fourier transform techniques do not allow to handle such a general case. Hence we use the theory of sums of operators.
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