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

Abstract

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.