Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$. Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.
It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\alpha}([0,T]\times \mathrm{cl}\,\Omega)$, $g \in C^{\frac{m+\alpha}{2};m+\alpha}([0,T]\times\partial\Omega)$, $u_0 \in C^{m,\alpha}(\mathrm{cl}\, \Omega)$ (satisfying some compatibility conditions at $t=0$), then there exists a unique solution $u$ in $C^{\frac{m+\alpha}{2};m+\alpha}([0,T]\times \mathrm{cl}\,\Omega)$ of \begin{cases} \partial_t u -\Delta u = f &\mbox{ in }[0,T]\times \mathrm{cl}\,\Omega,\\ u=g & \mbox{ on } [0,T] \times \partial\Omega,\\ u(0,\cdot) = u_0 & \mbox{ in }\mathrm{cl} \, \Omega. \end{cases}
For the elliptic case holds a very similar result, but in this case we allow $m$ to be also $1$, that is $m \in \mathbb{N}\setminus \{0\}$. If $f \in C^{m-2,\alpha}(\mathrm{cl}\,\Omega)$, $g \in C^{m,\alpha}(\partial\Omega)$, then there exists a unique solution $u$ in $C^{m,\alpha}(\mathrm{cl}\,\Omega)$ of \begin{cases} \Delta u = f &\mbox{ in }\mathrm{cl}\,\Omega,\\ u=g & \mbox{ on } \partial\Omega. \end{cases} In this case, for $m=1$, the space $C^{-1,\alpha}(\mathrm{cl}\,\Omega)$ is the space of distributions which euqals the divergence of an element in $C^{0,\alpha}(\mathrm{cl}\,\Omega,\mathbb{C}^n)$, and the laplacianLaplacian is to be intended in the weak sense.
Then my question is the following:
There exists an analog of the case $m=1$ for the heat (parabolic) equation?
