# Schauder regularity heat equation

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 Laplacian 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?

For a partial answer, see Theorem 6.48 in Second order parabolic differential equations, 1996, by Gary Lieberman.

There is an additional assumption that $$f$$ belongs to the Morrey space $$M^{1,n+1+\alpha}$$ defined page 130 of the book. In particular, $$L^{\infty}\subset M^{1,n+1+\alpha}\subset L^1$$.

I do not know if you can remove this additional assumption or not.

• Could you include the statement of Theorem 6.48 for those who don't have immediate access to the book? Aug 26 at 12:54
• @MichaelAlbanese: That book is the prime example about how not to write a book if you want it to be read. It contains no enlighting ideas, just boring and heavy technique, and theorems (and notations) always depend on previous chapters and results. I could edit this post to include the stament of theorem 6.48, but you wouldn't understand a thing of it. So much so, that it is impossible to me to decide whether that theorem does answer the question or not... Aug 26 at 13:14
• @AlexM. the theorem does answer the question, you just have to check what means every notation used in the statement (which is not straight forward at all). However, I agree with the fact that there is no way to check the details of the proofs or to rewrite it by yourself.
– ZZZ
Aug 26 at 13:28
• @MichaelAlbanese, it would take me pages to introduce all the notations used in the statements. Moreover, the functional spaces are not the most usual, so I think that it is better to check directly in the book.
– ZZZ
Aug 26 at 14:02
• @ZZZ: Fair enough. Aug 26 at 14:21

There are results for parabolic Schauder theory in $C^{1,\alpha}(\bar \Omega)$. Namely, for the rhs $f\equiv0$, $u_0\in C^{1,\alpha}$, $g\in C^{1,\alpha}$ see Baderko E.A. "Parabolic problems and boundary integral equations", Math. Methods Appl. Sci. 1997, V.20, P. 449-459. The results are for unbounded cylinder $\Omega\times[0,\infty)$ and for parabolic equations with Holder coefficients. There are references here for her earlier works in bounded wrt $t$ cylinders.
As for the rhs $f$, it is shown in M. F. Cherepova, “On some properties of the parabolic potential of bulk masses. I”, Differ. Uravn., 35:12 (1999), 1701–1706; Differ. Equ., 35:12 (1999), 1726–1732, if it satisfies the condition $|f|\le C d^{\alpha-1}$, where $d$ is the parabolic distance to the parabolic boundary, then the volume potential $Vf$ belongs to $C^{1,\alpha}$ in the closure of the domain and therefore solutions of the first BVP with such $f$ (and $u_0$, $g$) belong to the same class.