Smoothness improvement for temperatures in terms of the Besov regularity of initial and Dirichlet data

Hugo A. Aimar, Ivana Gómez


Jerison and Kenig in J. Funct. Anal. 130 (1995), no.1, 161-219, gave a precise region $\mathcal{R}$ in the square $[0,1]^2$ for the pairs $(s,\tfrac{1}{p})$ for which every harmonic function in the Lipschitz domain $D$, with Dirichlet data in $B^s_p(\partial D)$, belongs to $B^{s+\tfrac{1}{p}}_p(D)$. We prove that every temperature $u$ in $\Omega=D\times (0,T)$ belongs to $\mathbb{B}^{\alpha}_{\tau}(\Omega)$ with $\tfrac{1}{\tau}=\tfrac{1}{p}+\tfrac{\alpha}{d}$, $0<\alpha<\min\bigl\{d\tfrac{p1}{p},(s+\tfrac{1}{p})\tfrac{d}{d-1}\bigr\}$ provided that the Dirichlet data $f$ belongs to $B^s_p(\partial D)$ and that the initial condition $g$ belongs to $B^{s+\tfrac{1}{p}}_p(D)$, whenever $(s,\tfrac{1}{p})\in \mathcal{R}$. The result follows from those by T. Jakab and M. Mitrea in Math. Res. Lett. 13 (2006), no.5-6, 825-831 and from Parabolic Besov regularity for the heat equation by the authors accepted in Constructive Approximation.

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