Maximum-norm stability and maximal $L^{p}$ regularity of FEMs for parabolic equations with Lipschitz continuous coefficients

Abstract

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximum-norm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in $L^\infty ( Q_T)$ and $L^p((0,T);L^q(\Omega ))$, $1<p,q<\infty$. The maximal $L^p$ regularity of the parabolic finite element equation is also established.

Appendix: Interior estimates of parabolic equations on \(\widetilde{Q}_j'\)

Lemma 4.1

Suppose that $\widetilde{\phi }$ is the solution of

$$\begin{aligned} \partial _t\widetilde{\phi }-\nabla _{\widetilde{x}}\cdot ( \widetilde{a}\nabla _{\widetilde{x}}\widetilde{\phi }) +\widetilde{c}\widetilde{\phi }=0\quad \text{ in }~~\widetilde{Q}_j', \end{aligned}$$

(4.14)

with the Neumann boundary condition $\widetilde{a}\nabla _{\widetilde{x}}\widetilde{\phi }\cdot \mathbf{n}=0$ on $\widetilde{\Omega }_j'\cap \partial \widetilde{\Omega }$ and the initial condition $\widetilde{\phi }(\widetilde{x},0)=0$ in $\widetilde{Q}_j'$, then we have

$$\begin{aligned}&{|\;\!\!|\;\!\!|}\partial _{\widetilde{t}}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j}+{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{2,\widetilde{Q}_j} +{|\;\!\!|\;\!\!|}\partial _{\widetilde{t}}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{2,\widetilde{Q}_j} +{|\;\!\!|\;\!\!|}\partial _{\widetilde{t}\widetilde{t}}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{2,\widetilde{Q}_j} \le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'} ,\\&\Vert \partial _{\widetilde{x}_i}\widetilde{\phi }\Vert _{L^\infty (\widetilde{Q}_j)} +\Vert \partial _{\widetilde{x}_i}\widetilde{\phi }\Vert _{C^{\alpha ,\alpha /2}(\overline{\widetilde{Q}}_j)} + \Vert \partial _{\widetilde{x}_i}\partial _{\widetilde{x}_l}\widetilde{\phi }\Vert _{L^{\infty ,p_1}(\widetilde{Q}_j)} \le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'} . \end{aligned}$$

Proof

Without loss of generality, we can assume that the Eq. (4.14) holds in $\widetilde{Q}_j'''$ with the boundary condition on $\widetilde{\Omega }_j'''\cap \partial \widetilde{\Omega }$.

Let $\widetilde{\omega }(x,t)$ be a smooth function which equals $1$ in $\widetilde{Q}_{j}$ and equals zero outside $\widetilde{Q}_j'$, with $|\nabla \widetilde{\omega }|\le C$ and $|\partial _t\widetilde{\omega }|\le C$, and let $\widetilde{\chi }(x,t)$ be a smooth function which equals $1$ in $\widetilde{Q}_{j}'$ and equals $0$ outside $\widetilde{Q}_{j}''$, with $|\nabla \widetilde{\chi }|\le C$ and $|\partial _t\widetilde{\chi }|\le C$ (thus $\widetilde{\chi }=1$ on the support of $\widetilde{\omega }$). Since $\cup _{k\le j}\widetilde{\Omega }_k'' \cup \widetilde{\Omega }_*$ is of unit size, there exists a smooth subdomain $\widetilde{D}\subset \widetilde{\Omega }$ such that $\widetilde{D}$ has unit size and contains $\cup _{k\le j}\widetilde{\Omega }_k'' \cup \widetilde{\Omega }_*$ ($\partial \widetilde{D}$ may contain a part of the boundary of $\partial \widetilde{\Omega }$). Then $\widetilde{D}\times (0,16)$ contains $\widetilde{Q}_j''$ and $\widetilde{\omega }\widetilde{\phi }$ vanishes outside $\widetilde{D}$.

Integrating the Eq. (4.14) against $\widetilde{\omega }^2\widetilde{\phi }$, we derive the basic local energy estimate

$$\begin{aligned} \Vert \widetilde{\omega }\widetilde{\phi }\Vert _{L^{2}((0,16);H^1(\widetilde{D}))} \le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'} . \end{aligned}$$

(4.15)

To present further estimates for $\widetilde{\phi }$, we consider $\widetilde{\omega }\widetilde{\phi }$, which is the solution of

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} \partial _{\widetilde{t}}(\widetilde{\omega }\widetilde{\phi }) -\nabla _{\widetilde{x}} \cdot (\widetilde{a} \nabla _{\widetilde{x}} (\widetilde{\omega }\widetilde{\phi })) =\widetilde{\chi }\widetilde{\phi }\partial _{\widetilde{t}}\widetilde{\omega }-\widetilde{a}\nabla _{\widetilde{x}} \widetilde{\omega }\cdot \nabla _{\widetilde{x}} (\widetilde{\chi }\widetilde{\phi })&{}\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad -\nabla _{\widetilde{x}} \cdot (\widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \widetilde{\omega }) \quad &{} \text{ in }~ \widetilde{D}\times (0,16) ,\\ \widetilde{a} \nabla _{\widetilde{x}} (\widetilde{\omega }\widetilde{\phi })\cdot \widetilde{\mathbf{n}}= \widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \widetilde{\omega }\cdot \widetilde{\mathbf{n}} \quad &{}\text{ on }~ \partial \widetilde{D}\times (0,16) ,\\ \widetilde{\omega }\widetilde{\phi }=0 \quad &{}\text{ on }~ \widetilde{D}\times \{0\} . \end{array}\right. \end{aligned}$$

Let $\widetilde{W}$ be the solution of

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} \nabla _{\widetilde{x}} \cdot (\widetilde{a} \nabla _{\widetilde{x}} \widetilde{W}) =\nabla _{\widetilde{x}} \cdot (\widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \widetilde{\omega }) &{} \text{ in }~ \widetilde{D}\times (0,16) ,\\ \widetilde{a} \nabla _{\widetilde{x}} \widetilde{W}\cdot \widetilde{\mathbf{n}}= \widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \widetilde{\omega }\cdot \widetilde{\mathbf{n}} &{}\text{ on }~ \partial \widetilde{D}\times (0,16) , \end{array}\right. \end{aligned}$$

which satisfies the basic $W^{2,p}$ estimate

$$\begin{aligned}&\Vert \widetilde{W}\Vert _{W^{2,p}(\widetilde{D})} \le C\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{W^{1,p}(\widetilde{D})} . \end{aligned}$$

Since $\partial _{\widetilde{t}}\widetilde{W}$ is the solution of

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} \nabla _{\widetilde{x}} \cdot (\widetilde{a} \nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{W}) =\nabla _{\widetilde{x}} \cdot (\widetilde{a}\partial _{\widetilde{t}}(\widetilde{\chi }\widetilde{\phi })\nabla _{\widetilde{x}} \widetilde{\omega }+\widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{\omega }) \quad &{} \text{ in }~ \widetilde{D}\times (0,16) ,\\ \widetilde{a} \nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{W}\cdot \widetilde{\mathbf{n}}= (\widetilde{a}\partial _{\widetilde{t}}(\widetilde{\chi }\widetilde{\phi })\nabla _{\widetilde{x}} \widetilde{\omega }+\widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{\omega }) \cdot \widetilde{\mathbf{n}} \quad &{}\text{ on }~ \partial \widetilde{D}\times (0,16) , \end{array}\right. \end{aligned}$$

we can estimate $\Vert \partial _{\widetilde{t}}\widetilde{W} \Vert _{L^{p}(\widetilde{D})}$ via a duality argument, by defining $\phi$ as the solution of

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} -\nabla _{\widetilde{x}} \cdot (\widetilde{a} \nabla _{\widetilde{x}} \psi ) =\varphi \quad &{} \text{ in }~ \widetilde{D} ,\\ \widetilde{a} \nabla _{\widetilde{x}} \psi \cdot \widetilde{\mathbf{n}}=0 \quad &{}\text{ on }~ \partial \widetilde{D} , \end{array}\right. \end{aligned}$$

and considering

$$\begin{aligned} \big |\big ( \partial _{\widetilde{t}}\widetilde{W}, \varphi \big )\big |&= \big |\big (\widetilde{a} \nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{W}, \nabla _{\widetilde{x}} \psi \big )\big |\\&=\big |\big (\widetilde{a}\partial _{\widetilde{t}}\widetilde{\phi }\nabla _{\widetilde{x}} \widetilde{\omega }+\widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{\omega }, \nabla _{\widetilde{x}}\psi )\big |\\&= \big |\big (\nabla _{\widetilde{x}}\cdot (\widetilde{a}\nabla _{\widetilde{x}} \widetilde{\phi })-\widetilde{c}\widetilde{\phi }, \widetilde{a}\nabla _{\widetilde{x}}\widetilde{w} \cdot \nabla _{\widetilde{x}}\psi \big ) + \big (\widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{\omega }, \nabla _{\widetilde{x}}\psi )\big | \\&= \big |-\big (\widetilde{a}\nabla _{\widetilde{x}} \widetilde{\phi },\nabla _{\widetilde{x}}( \widetilde{a}\nabla _{\widetilde{x}}\widetilde{w} \cdot \nabla _{\widetilde{x}}\psi )\big ) -\big (\widetilde{c}\widetilde{\phi }, \widetilde{a}\nabla _{\widetilde{x}}\widetilde{w} \cdot \nabla _{\widetilde{x}}\psi \big )\\&\quad + \big (\widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{\omega }, \nabla _{\widetilde{x}}\psi )\big | \\&= \big |-\big (\widetilde{a}\nabla _{\widetilde{x}} (\widetilde{\chi }\widetilde{\phi }) ,\nabla _{\widetilde{x}}( \widetilde{a}\nabla _{\widetilde{x}}\widetilde{w} \cdot \nabla _{\widetilde{x}}\psi )\big ) -\big (\widetilde{c}\widetilde{\chi }\widetilde{\phi }, \widetilde{a}\nabla _{\widetilde{x}}\widetilde{w} \cdot \nabla _{\widetilde{x}}\psi \big )\\&\quad + \big (\widetilde{a}\widetilde{\chi }\widetilde{\phi }\nabla _{\widetilde{x}} \partial _{\widetilde{t}}\widetilde{\omega }, \nabla _{\widetilde{x}}\psi )\big | \\&\le C\Vert \widetilde{\phi }\widetilde{\chi }\Vert _{W^{1,p}(\widetilde{D})} \Vert \psi \Vert _{W^{2,p'}(\widetilde{D})}\\&\le C\Vert \widetilde{\phi }\widetilde{\chi }\Vert _{W^{1,p}(\widetilde{D})} \Vert \varphi \Vert _{L^{p'}(\widetilde{D})}, \end{aligned}$$

which implies

$$\begin{aligned} \Vert \partial _{\widetilde{t}}\widetilde{W} \Vert _{L^{p}(\widetilde{D})} \le C\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{W^{1,p}(\widetilde{D})} . \end{aligned}$$

Therefore, we have

$$\begin{aligned} \Vert \widetilde{W}\Vert _{L^p((0,16);W^{2,p}(\widetilde{D}))} +\Vert \partial _{\widetilde{t}}\widetilde{W}\Vert _{L^p((0,16);L^p(\widetilde{D}))} \le C\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{L^p((0,16);W^{1,p}(\widetilde{D}))} . \end{aligned}$$

(4.16)

Then we consider $\widetilde{Z}=\widetilde{\omega }\widetilde{\phi }-\widetilde{W}$, which is the solution of

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} \partial _{\widetilde{t}}\widetilde{Z} -\nabla _{\widetilde{x}} \cdot (\widetilde{a} \nabla _{\widetilde{x}} \widetilde{Z}) =-\partial _{\widetilde{t}}\widetilde{W} +\widetilde{\chi }\widetilde{\phi }\partial _{\widetilde{t}}\widetilde{\omega }-\widetilde{a}\nabla _{\widetilde{x}} \widetilde{\omega }\cdot \nabla _{\widetilde{x}} (\widetilde{\chi }\widetilde{\phi }) \quad &{} \text{ in }~ \widetilde{D}\times (0,16) ,\\ \widetilde{a} \nabla _{\widetilde{x}} \widetilde{Z}\cdot \widetilde{\mathbf{n}}= 0 \quad &{}\text{ on }~ \partial \widetilde{D}\times (0,16) ,\\ \widetilde{Z}=0 \quad &{}\text{ on }~ \widetilde{D}\times \{0\} , \end{array}\right. \end{aligned}$$

and obeys the $L^p((0,16);W^{2,p}(\widetilde{D}))$ estimate (see Lemma 2.1)

$$\begin{aligned}&\Vert \partial _{\widetilde{t}}\widetilde{Z}\Vert _{L^p(\widetilde{D}\times (0,16))} +\Vert \widetilde{Z}\Vert _{L^p((0,16);W^{2,p}(\widetilde{D}))} \nonumber \\&\le C\Vert \partial _{\widetilde{t}}\widetilde{W}\Vert _{L^p((0,16);L^{p}(\widetilde{D}))} +C\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{L^p((0,16);W^{1,p}(\widetilde{D}))} \nonumber \\&\le C\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{L^p((0,16);W^{1,p}(\widetilde{D}))} . \end{aligned}$$

(4.17)

(4.16)–(4.17) imply

$$\begin{aligned}&\Vert \partial _{\widetilde{t}}(\widetilde{\omega }\widetilde{\phi }) \Vert _{L^p((0,16);L^p(\widetilde{D}))} +\Vert \widetilde{\omega }\widetilde{\phi }\Vert _{L^p((0,16);W^{2,p}(\widetilde{D}))} \le C\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{L^p((0,16);W^{1,p}(\widetilde{D}))} . \end{aligned}$$

(4.18)

In particular, by setting $p=2$ in the last inequality and using (4.15), we obtain

$$\begin{aligned}&\Vert \partial _{\widetilde{t}}(\widetilde{\omega }\widetilde{\phi }) \Vert _{L^2((0,16);L^2(\widetilde{D}))} +\Vert \widetilde{\omega }\widetilde{\phi }\Vert _{L^2((0,16);H^2(\widetilde{D}))} \le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j''} . \end{aligned}$$

(4.19)

Since

$$\begin{aligned} L^q((0,16);W^{2,q}(\widetilde{D}))\cap W^{1,q}((0,16);L^{q}(\widetilde{D}))\hookrightarrow L^p((0,16);W^{1,p}(\widetilde{D})) \end{aligned}$$

for $p>2$ and $q=3dp/(3d+p)<p$, (4.18) also implies

$$\begin{aligned}&\Vert \partial _{\widetilde{t}}(\widetilde{\omega }\widetilde{\phi }) \Vert _{L^p((0,16);L^p(\widetilde{D}))} +\Vert \widetilde{\omega }\widetilde{\phi }\Vert _{L^p((0,16);W^{2,p}(\widetilde{D}))} \\&\le C(\Vert \partial _{\widetilde{t}}(\widetilde{\chi }\widetilde{\phi }) \Vert _{L^q((0,16);L^q(\widetilde{D}))} +\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{L^q((0,16);W^{2,q}(\widetilde{D}))} ) , \end{aligned}$$

and an iteration of this inequality gives (with a change of indices)

$$\begin{aligned}&\Vert \partial _{\widetilde{t}}(\widetilde{\omega }\widetilde{\phi }) \Vert _{L^p((0,16);L^p(\widetilde{D}))} +\Vert \widetilde{\omega }\widetilde{\phi }\Vert _{L^p((0,16);W^{2,p}(\widetilde{D}))} \nonumber \\&\le C(\Vert \partial _{\widetilde{t}}(\widetilde{\chi }\widetilde{\phi }) \Vert _{L^2((0,16);L^2(\widetilde{D}))} +\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{L^2((0,16);H^2(\widetilde{D}))} ) \nonumber \\&\le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'''} , \end{aligned}$$

(4.20)

where we have used (4.19) in the last step. Since $\partial _{\widetilde{t}}\widetilde{\phi }$ and $\partial _{\widetilde{t}\widetilde{t}}\widetilde{\phi }$ satisfies the same equation as $\widetilde{\phi }$ in $Q_j'''$, the last inequality still holds if $\widetilde{\phi }$ is replaced by $\partial _{\widetilde{t}}\widetilde{\phi }$ or $\partial _{\widetilde{t}\widetilde{t}}\widetilde{\phi }$ (with a change of indices, replacing $\widetilde{Q}_j'''$ by $\widetilde{Q}_j'$), i.e.

$$\begin{aligned}&\Vert \partial _{\widetilde{t}}(\widetilde{\omega }\partial _{\widetilde{t}}\widetilde{\phi })\Vert _{L^{p}(\widetilde{D}\times (0,16))} +\Vert \widetilde{\omega }\partial _{\widetilde{t}}\widetilde{\phi }\Vert _{L^{p}((0,16);W^{2,p}(\widetilde{D}))} \le C{|\;\!\!|\;\!\!|}\partial _{\widetilde{t}}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'} , \end{aligned}$$

(4.21)

$$\begin{aligned}&\Vert \partial _{\widetilde{t}}(\widetilde{\omega }\partial _{\widetilde{t}\widetilde{t}}\widetilde{\phi })\Vert _{L^{p}(\widetilde{D}\times (0,16))} +\Vert \widetilde{\omega }\partial _{\widetilde{t}\widetilde{t}}\widetilde{\phi }\Vert _{L^{p}((0,16);W^{2,p}(\widetilde{D}))} \le C{|\;\!\!|\;\!\!|}\partial _{\widetilde{t}\widetilde{t}}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'} . \end{aligned}$$

(4.22)

The last three inequalities imply that (with a change of indices)

$$\begin{aligned}&{|\;\!\!|\;\!\!|}\partial _{\widetilde{t}}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j}+{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{2,\widetilde{Q}_j} +{|\;\!\!|\;\!\!|}\partial _{\widetilde{t}}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{2,\widetilde{Q}_j} +{|\;\!\!|\;\!\!|}\partial _{\widetilde{t}\widetilde{t}}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{2,\widetilde{Q}_j} \le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'} , \nonumber \\&\Vert \partial _{\widetilde{x}_i}\partial _{\widetilde{x}_l}\widetilde{\phi }\Vert _{L^{p_1}(\widetilde{Q}_j)}+ \Vert \partial _{\widetilde{x}_i}\partial _{\widetilde{x}_l}\partial _{\widetilde{t}}\widetilde{\phi }\Vert _{L^{p_1}(\widetilde{Q}_j)} \le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'} , \end{aligned}$$

(4.23)

and so

$$\begin{aligned} \Vert \partial _{\widetilde{t}}(\widetilde{\omega }\widetilde{\phi }) \Vert _{L^{p}((0,16);W^{2,p}(\widetilde{D}))}&\le \Vert \widetilde{\phi }\partial _{\widetilde{t}}\widetilde{\omega }\Vert _{L^{p}((0,16);W^{2,p}(\widetilde{D}))} +\Vert \widetilde{\omega }\partial _{\widetilde{t}}\widetilde{\phi }\Vert _{L^{p}((0,16);W^{2,p}(\widetilde{D}))} \\&\le C(\Vert \widetilde{\chi }\widetilde{\phi }\Vert _{L^{p}((0,16);W^{2,p}(\widetilde{D}))} +\Vert \widetilde{\chi }\partial _{\widetilde{t}}\widetilde{\phi }\Vert _{L^{p}((0,16);W^{2,p}(\widetilde{D}))}) \\&\le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j''} , \end{aligned}$$

where we have used (4.20)–(4.21) (replacing $\widetilde{\omega }$ by $\widetilde{\chi }$ in these inequalities). This gives an estimate of $\widetilde{\omega }\widetilde{\phi }$ in terms of the norm of $W^{1,p}((0,16);W^{2,p}(\widetilde{D}))$. When $p_1>d$, we have $W^{1,p_1}((0,16);W^{2,p_1}(\widetilde{D}))\hookrightarrow C^{1+\alpha ,(1+\alpha )/2}(\overline{\widetilde{D}}\times [0,16])$ for some $\alpha >0$, and so

$$\begin{aligned}&\Vert \widetilde{\omega }\widetilde{\phi }\Vert _{C^{1+\alpha ,(1+\alpha )/2}(\overline{\widetilde{D}}\times [0,16])} +\Vert \widetilde{\omega }\widetilde{\phi }\Vert _{L^{\infty }((0,16);W^{2,p_1}(\widetilde{D}))}\\&\le C(\Vert \partial _{\widetilde{t}}(\widetilde{\omega }\widetilde{\phi }) \Vert _{L^{p_1}((0,16);W^{2,p_1}(\widetilde{D}))} +\Vert \widetilde{\omega }\widetilde{\phi }\Vert _{L^{p_1}((0,16);W^{2,p_1}(\widetilde{D}))})\\&\le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j''} . \end{aligned}$$

With a change of indices, the last inequality implies

$$\begin{aligned}&\Vert \partial _{\widetilde{x}_i}\widetilde{\phi }\Vert _{L^\infty (\widetilde{Q}_j)} +\Vert \partial _{\widetilde{x}_i}\widetilde{\phi }\Vert _{C^{\alpha ,\alpha /2}(\overline{\widetilde{Q}}_j)} + \Vert \partial _{\widetilde{x}_i}\partial _{\widetilde{x}_l}\widetilde{\phi }\Vert _{L^{\infty ,p_1}(\widetilde{Q}_j)} \le C{|\;\!\!|\;\!\!|}\widetilde{\phi }{|\;\!\!|\;\!\!|}_{\widetilde{Q}_j'} . \end{aligned}$$

(4.24)

The proof of Lemma 4.1 is completed. $\square$