Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm

Abstract

We determine the exact regularity of the trace of a function $u \in L_{q} (0,T; W_{p}^{2}(\Omega ))$ $\cap W^{1}_{q} (0,T; {L_{p} (\Omega ))}$ and of the trace of its spatial gradient on $\partial \Omega \times ( 0,T )$ in the regime $p \le q$. While for $p=q$ both the spatial and temporal regularity of the traces can be completely characterized by fractional order Sobolev-Slobodetskii spaces, for $p \neq q$ the Lizorkin-Triebel spaces turn out to be necessary for characterizing the sharp temporal regularity.