Gradient estimates for divergence form parabolic systems from composite materials

Abstract

We consider divergence form, second-order strongly parabolic systems in a cylindrical domain with a finite number of subdomains under the assumption that the interfacial boundaries are $C^{1,\text {Dini}}$ and $C^{\gamma _{0}}$ in the spatial variables and the time variable, respectively. Gradient estimates and piecewise $C^{1/2,1}$-regularity are established when the leading coefficients and data are assumed to be of piecewise Dini mean oscillation or piecewise Hölder continuous. Our results improve the previous results in Fan et al. (Electron J Differ Equ 2013:1–24, 2013) and Li and Li (Sci China Math 60(11):2011–2052, 2017) to a large extent, and appear to be the first of its kind for time-dependent subdomains. As a byproduct, we obtain optimal regularity of weak solutions to parabolic transmission problems with $C^{1,\mu }$ or $C^{1,\text {Dini}}$ interfaces. This gives an extension of a recent result in Caffarelli et al. (Regularity for $C^{1,\alpha }$ interface transmission problems. arXiv:2004.07322 [math.AP]) to parabolic systems.

Appendix

In the appendix, we prove a generalization of Lemma 3.5. We say that $w: \mathbb R^{d+1}\rightarrow [0,\infty )$ belongs to $A_{p}$ for $p\in (1,\infty )$ if

$$\begin{aligned} \sup _{Q}\frac{w(Q)}{|Q|}\left( \frac{w^{\frac{-1}{p-1}}(Q)}{|Q|}\right) ^{p-1}<\infty , \end{aligned}$$

where the supremum is taken over all parabolic cylinders Q in ${\mathbb {R}}^{d+1}$. The value of the supremum is the $A_{p}$ constant of w, and will be denoted by $[w]_{A_p}$.

We consider the parabolic systems on ${\mathcal {Q}}:=(-T,0)\times {\mathcal {D}}$, where ${\mathcal {D}}$ is a Reifenberg flat domain and the boundary $\partial {\mathcal {D}}$ satisfies the following assumption with a parameter $\gamma _0\in (0,1/4)$ to be specified later.

Assumption 8.1

($\gamma _0$) There exists a constant $r_{0}\in (0,1]$ such that the following conditions hold.

• (1) In the interior of ${\mathcal {D}}$, $A^{\alpha \beta }$ satisfy (3.7) in some coordinate system depending on $(t_{0},x_0)$ and r.

• (2) For any $x_{0}\in \partial {\mathcal {D}}$, $t\in {\mathbb {R}}$, and $r\in (0,r_{0}]$, there is a coordinate system depending on $(t_{0},x_0)$ and r such that in this new coordinate system, we have

  $$\begin{aligned} \{(y',y^{d}): x_0^{d}+\gamma _{0}r<y^{d}\}\cap B_{R}(x_0)\subset {\mathcal {D}}\cap B_{R}(x_0)\subset \{(y',y^{d}): x_0^{d}-\gamma _{0}r<y^{d}\}\cap B_{R}(x_0) \end{aligned}$$

  and

  $$\begin{aligned} \fint _{Q_{r}^{-}(z_{0})}|A(t,x)-(A)_{Q^{-'}_{r}(z'_{0})}|\,dx\ dt\le \gamma _{0}, \end{aligned}$$

  where $(A)_{Q^{-'}_{r}(z'_{0})}=\fint _{Q^{-'}_{r}(z'_{0})}A(z',x^{d})\,dz'$.

Lemma 8.2

Let $p\in (1,\infty )$ and w be an $A_p$ weight. There exists a constant $\gamma _0\in (0,1/4)$ depending on d, p, $\nu$, $\Lambda$, and $[w]_{A_p}$ such that, under Assumption 8.1, for any $u\in {\mathcal {H}}_{p,w}^{1}((-T,0)\times {\mathcal {D}})$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {P}}u-\lambda u={\text {div}}f&{}\text{ in }~{\mathcal {Q}},\\ u=0&{}\text{ on }~\partial _p {\mathcal {Q}}, \end{array}\right. } \end{aligned}$$

(8.1)

where $\lambda \ge 0$ and $f\in L_{p,w}({\mathcal {Q}})$, we have

$$\begin{aligned} \Vert u\Vert _{{\mathcal {H}}_{p,w}^{1}({\mathcal {Q}})}\le N\Vert f\Vert _{L_{p,w}({\mathcal {Q}})}, \end{aligned}$$

where $N=N(n,d,p,\nu ,\Lambda ,[w]_{A_{p}},r_{0},T)$. Moreover, for any $f\in L_{p,w}({\mathcal {Q}})$, (8.1) admits a unique solution $u\in {\mathcal {H}}_{p,w}^{1}({\mathcal {Q}})$.

Proof

The case when $\lambda >\lambda _{0}$ is proved in [13, Theorem 7.2 and Section 8], where $\lambda _{0}>0$ is a sufficiently large constant depending on $n,d,p,\nu ,\Lambda ,[w]_{A_{p}}$ and $r_{0}$. For $0\le \lambda \le \lambda _{0}$, we set

$$\begin{aligned} v:=ue^{-\lambda _{0}t}. \end{aligned}$$

Then we have

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {P}}v-(\lambda +\lambda _{0})v=e^{-\lambda _{0}t}{\text {div}}f&{} \text{ in }~{\mathcal {Q}},\\ v=0&{} \text{ on }~\partial _p {\mathcal {Q}}. \end{array}\right. } \end{aligned}$$

By using [13, Theorem 7.2], we have

$$\begin{aligned} \Vert v\Vert _{{\mathcal {H}}_{p,w}^{1}({\mathcal {Q}})}\le N\Vert e^{-\lambda _{0}t}f\Vert _{L_{p,w}({\mathcal {Q}})}\le N\Vert f\Vert _{L_{p,w}({\mathcal {Q}})}, \end{aligned}$$

where $N=N(n,d,p,\nu ,\Lambda ,[w]_{A_{p}},r_{0},T)$. Hence, we obtain

$$\begin{aligned} \Vert u\Vert _{{\mathcal {H}}_{p,w}^{1}({\mathcal {Q}})}=\Vert ve^{\lambda _{0}t}\Vert _{{\mathcal {H}}_{p,w}^{1}({\mathcal {Q}})}\le N\Vert f\Vert _{L_{p,w}({\mathcal {Q}})}. \end{aligned}$$

The theorem is proved. $\square$