# Solution theory for parabolic problems with irregular boundary data

Let $$\Omega \subset \mathbb R^n$$, $$n \in \mathbb N$$, be a smooth bounded domain, $$\mathcal A$$ an elliptic operator (for instance $$\mathcal A = - \Delta$$), $$u_0 \in C^0(\overline \Omega)$$ and $$g \in C^\alpha(\partial \Omega)$$ for some $$\alpha \in (0, 1)$$. What is known about the equation

$$\begin{cases} u_t + \mathcal A u = 0 & \text{in (0, \infty) \times \Omega}, \\ u = g & \text{in (0, \infty) \times \partial \Omega}, \\ u(\cdot, t) = u_0 & \text{in \Omega} \end{cases}$$

(or also about the corresponding elliptic equation)?

If $$g \in C^{2+\beta}(\partial \Omega)$$ for some $$\beta \in (0, 1)$$, then one can extend $$g$$ to a function $$\tilde g \in C^{2+\beta}(\overline \Omega)$$ and consider $$u-\tilde g$$ instead of $$u$$. Of course, this is no longer possible for less regular $$g$$.

However, I would still expect that the problem above (or at least the elliptic version) has a (unique?) classical solution – which may even be $$C^\alpha$$ up to the boundary (but of course not more).

I guess this has been treated somewhere and hence I am thankful for any references.

I think Theorem 9, Chapter 3.4, in Avner Friedman's book, Partial Differential Equations of Parabolic Type, might help you. Basically it says that with suitable assumptions on $$\mathcal{A}$$ and the boundary of $$\Omega$$, you get a unique solution that is in $$C^{2+\alpha}$$ for a $$0<\alpha<1$$, but only in the interior of $$\Omega\times(0,T)$$.

For the proof you don't even need that $$g\in C^\alpha$$. It is enough that $$u=\psi$$ on the parabolic boundary $$\Omega\times\{t=0\} \cup \partial\Omega\times(0,T)$$, where $$\psi$$ is a continuous function on that boundary. For a bounded domanin $$\Omega$$ we can for example apply Tietze's extension theorem to get a $$\psi\in C(\bar{\Omega}\times[0,T]$$). Let $$N$$ be the rectangle in $$\mathbb{R}^{d+1}$$ that contains $$\bar{\Omega}\times[0,T]$$. By the Weierstrass approximation theorem there exists a sequence of polynomilas $$\psi_k$$ that approximate $$\psi$$ uniformly in $$N$$. As you already suggested, you can now apply interior parabolic schauder estimates to $$u-\psi_k$$. You get a sequence of solutions $$u_k$$, by interior parabolic Schauder estimates you can show that $$u_k$$ converges uniformly to a $$u$$. $$u$$ is continuous in $$\bar{\Omega}\times[0,T]$$ and $$u\in C^{2+\alpha}$$ locally in $$\Omega\times(0,T)$$.

Tl;dr, you got the right idea. You consider $$u-g$$, but since $$g$$ is not differentiable, you approximate $$g$$ by a sequence of polynomials $$g_k$$. Interior Schauder estimates show that the corresponding sequence $$u_k$$ converges uniformly to your solution $$u$$.

If you want $$u\in C^{2+\alpha}(\bar{\Omega}\times[0,T])$$, i.e. differentiability up to the boundary, then you need that the boundary condition $$\psi$$ is in $$C^{2+\alpha}$$, see for instance Ladyzenskaja-Solonnikov-Ural'ceva's book Linear and Quasi-linear Equations of Parabolic Type, Theorem 5.2,Chapter IV Section 5.

• That makes a lot of sense, thanks! Which of his books do you refer to? “Partial differential equations“ or “Partial differential equations of parabolic type”?
– Keba
Commented Nov 2, 2020 at 15:12
• And, secondly, is there any hope for a uniqueness theorem?
– Keba
Commented Nov 2, 2020 at 15:12
• Ah, you named the title of the book, sorry.
– Keba
Commented Nov 2, 2020 at 15:23
• @Keba It is "Partial differential equations of parabolic type". Yes, the theorem that I am referring to states that $u$ is unique. Commented Nov 2, 2020 at 15:24
• Thanks again. Will try to get the book then.
– Keba
Commented Nov 2, 2020 at 15:33