Parabolic layer potentials and initial boundary value problems in Lipschitz cylinders with data in Besov spaces
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We adapt the method of boundary layer potentials to the Poisson problem for the heat operator [partial differential]t [delta] in a bounded Lipschitz cylinder, with Dirichlet and Neumann boundary conditions. When the lateral datum has a fractional amount of smoothness measured at the Besov scale with parabolic anisotropy, the well-posedness of these problems is obtained in a constructive fashion. More specifically, the solution can be represented as a double layer potential in the Dirichlet case, and as a single layer potential in the Neumann case. The main theorems we prove extend, generalize and bring together many earlier results, such as the work of E. Fabes and N. Rivière (1978) for C¹ domains; R. Brown (1990) for integer amount of smoothness; D. Jerison and C. Kenig (1995),and E. Fabes, O. Mendez and M. Mitrea (1998) who dealt with the case of the Laplacian.