Interior Calderón–Zygmund estimates for solutions to general parabolic equations of p-Laplacian type

Abstract

We study general parabolic equations of the form $u_t = \text{ div }\,\mathbf {A}(x,t, u,D u) +\text{ div }\,(|\mathbf {F}|^{p-2} \mathbf {F})+ f$ whose principal part depends on the solution itself. The vector field $\mathbf {A}$ is assumed to have small mean oscillation in x, measurable in t, Lipschitz continuous in u, and its growth in Du is like the p-Laplace operator. We establish interior Calderón–Zygmund estimates for locally bounded weak solutions to the equations when $p>2n/(n+2)$. This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto (Degenerate parabolic equations, Springer, New York, 1993) and the maximal function free approach by Acerbi and Mingione (Duke Math J 136(2):285–320, 2007).

Appendix A: A comparison principle

Let $\Omega _T :=\Omega \times (0,T)$ with $T>0$ and $\Omega$ being a bounded domain in $\mathbb {R}^n$, $n\ge 2$. Let ${\mathbb K}\subset \mathbb {R}$ be an open interval, $\mathbf {A}: \Omega _T \times \overline{{\mathbb K}}\times \mathbb {R}^n \longrightarrow \mathbb {R}^n$, $\Phi : \Omega _T \times \overline{{\mathbb K}}\rightarrow \mathbb {R}^n$ and $g,\, b: \Omega _T \times \overline{{\mathbb K}}\rightarrow \mathbb {R}$ be Carathéodory maps. We assume that there exist constants $\Lambda >0$ and $1< p<\infty$ such that the following conditions are satisfied for a.e. $z\in \Omega _T$ and all $\xi ,\eta \in \mathbb {R}^n$:

$$\begin{aligned}&\big \langle \mathbf {A}(z,u,\xi ) -\mathbf {A}(z,u,\eta ), \xi -\eta \big \rangle \ge 0\quad \, \, \forall u\in \overline{{\mathbb K}}, \end{aligned}$$

(A.1)

$$\begin{aligned}&u\in \mathbb {R}\mapsto g(z,u) \text{ is } \text{ monotone } \text{ nondecreasing }. \end{aligned}$$

(A.2)

Also, there exist functions $K\in L^{p'}(\Omega _T)$ and $k\in L^1(0,T)$ such that

$$\begin{aligned}&|\mathbf {A}(z,u_1,\xi )-\mathbf {A}(z,u_2,\xi )| \le |u_1 - u_2| \Big (\Lambda |\xi |^{p-1} + K(z)\Big ), \end{aligned}$$

(A.3)

$$\begin{aligned}&|\Phi (z,u_1)-\Phi (z,u_2)| \le |u_1 - u_2| K(z),\quad \text{ and } \quad |b(z,u_1)-b(z,u_2)| \le |u_1 - u_2| k(t) \end{aligned}$$

(A.4)

for all $u_1, u_2\in \overline{{\mathbb K}}$ with $|u_1-u_2|$ sufficiently small.

Definition A.1

A map $u\in C((0,T); L^2(\Omega ))\cap L^p(0,T; W^{1,p}(\Omega ))$ is called a weak solution to

$$\begin{aligned} u_t = \text{ div }\,\mathbf {A}(z, u,D u) +\text{ div }\,\Phi (z,u) -g(z,u) +b(z,u) \quad \text {in}\quad \Omega _T \end{aligned}$$

(A.5)

if $u(z)\in \overline{{\mathbb K}}\,\,$ for a.e. $z\in \Omega _T$ and

$$\begin{aligned} \int _{\Omega _T} u \varphi _t\,dz = \int _{\Omega _T} \langle \mathbf {A}(z, u,D u), D \varphi \rangle \,dz +\int _{\Omega _T} \langle \Phi (z, u), D \varphi \rangle \,dz + \int _{\Omega _T} [g(z,u)-b(z,u)] \varphi \,dz \end{aligned}$$

for every $\varphi \in C_0^\infty (\Omega _T)$.

The following result shows that equation (A.5) admits a comparison principle.

Proposition A.2

(comparison principle). Assume that $\mathbf {A}$, $\Phi$, g and b satisfy conditions (A.1)–(A.4). Let u and v be weak solutions to (A.5) such that $u\le v$ on $\partial _p \Omega _T$. Then

$$\begin{aligned} u \le v \quad \text{ a.e. } \text{ in }\quad \Omega _T. \end{aligned}$$

Remark A.3

By inspecting the arguments below, ones see that we in fact only need to assume that u is a weak subsolution and v is a weak supersolution.

Proof

For $\varepsilon >0$ small, we define $h_\varepsilon (s)$ as in (5.16). Let us denote $\Omega _t =\Omega \times (0,t)$. By using $h_\varepsilon (u-v)$ as a test function in the equations for u and v and subtracting the resulting expressions, we obtain:

$$\begin{aligned}&\int _{\Omega }\Big (\int _0^{(u-v)_+(x,t)} h_\varepsilon (s)\, ds \Big )dx +\int _{\Omega _t}h_\varepsilon '(u-v)\langle \mathbf {A}(z, u,D u)-\mathbf {A}(z, u,D v) , Du -Dv\rangle \, dz\\&+\int _{\Omega _t}\Big [g(z,u)-g(z,v)\Big ] h_\varepsilon (u-v) \, dz\\&= \int _{\Omega _t}h_\varepsilon '(u-v)\langle \mathbf {A}(z, v,D v)-\mathbf {A}(z, u,D v) , Du -Dv\rangle \, dz\\&\qquad -\int _{\Omega _t}h_\varepsilon '(u-v)\langle \Phi (z, u)-\Phi (z, v) , Du -Dv\rangle \, dz +\int _{\Omega _t}\Big [b(z,u)-b(z,v)\Big ] h_\varepsilon (u-v) \, dz \end{aligned}$$

for all $t\in (0,T)$. Since the second and third terms on the left-hand side are nonnegative thanks to (A.1)–(A.2), we deduce that

$$\begin{aligned}&\int _{\Omega }\Big (\int _0^{(u-v)_+(x,t)} h_\varepsilon (s)\, ds \Big )dx \nonumber \\&\le \frac{1}{\varepsilon } \int _0^t \int _{\Omega \cap \{0<u-v<\varepsilon \}}(u-v)\Big (\Lambda |Dv|^{p-1} + 2 K\Big ) |Du -Dv|\, dz\nonumber \\&\quad +\int _{\Omega _t}k(\tau ) |u-v| h_\varepsilon (u-v) \, dx d\tau \nonumber \\&\le \int _0^t \int _{\Omega \cap \{0<u-v<\varepsilon \}}\Big (\Lambda |Dv|^{p-1} + 2K\Big ) |Du -Dv|\, dz +\int _{\Omega _t}k(\tau ) |u-v| h_\varepsilon (u-v) \, dx d\tau . \end{aligned}$$

(A.6)

As $\varepsilon \rightarrow 0^+$, we have

$$\begin{aligned} \int _{\Omega }\Big (\int _0^{(u-v)_+(x,t)} h_\varepsilon (s)\, ds \Big )dx \longrightarrow \int _{\Omega } (u-v)_+(x,t)\,dx. \end{aligned}$$

Moreover, the first term on the right-hand side tends to zero and the last term tends to

$$\begin{aligned} \int _{\Omega _t}k(\tau ) |u-v| \mathrm sgn^+(u-v) \, dx d\tau =\int _0^t\int _{\Omega } k(\tau ) (u-v)_+ \, dx d\tau . \end{aligned}$$

Thus if we denote $m(\tau ) :=\int _{\Omega } (u-v)_+(x,\tau )\,dx$, then by letting $\varepsilon \rightarrow 0^+$ in (A.6) we obtain

$$\begin{aligned} m(t) \le \int _0^t k(\tau ) m(\tau )\, d\tau \quad \forall t\in (0,T). \end{aligned}$$

Therefore, it follows from the Grönwalls inequality that $m(t)\le 0$ for every $t\in (0,T)$. We then conclude that $\int _{\Omega _T} (u-v)_+(x,t)\,dx dt=0$, and hence $u\le v$ for a.e. in $\Omega _T$. The proof is complete. $\square$

Remark A.4

We note that the comparison principle in Proposition A.2 together with the standard method for proving existence using Galerkin approximation (see [16, pp. 466–475] and [19, 25]) ensures that: for any $u\in L^\infty (Q_3)\cap L^p(-9,9; W^{1,p}(B_3))$ satisfying $u(z)\in \overline{{\mathbb K}}\,$ for a.e. $z\in Q_3$, the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{lcll} v_t &{}=&{}\text{ div }\,\mathbf {A}(z, v,D v) \quad &{}\text {in}\quad Q_3, \\ v &{} =&{} u\quad &{}\text {on}\quad \partial _p Q_3 \end{array}\right. \end{aligned}$$

has a weak solution in the sense of Definition A.1.