The Lamm–Rivière system I: $L^p$ regularity theory

Abstract

Motivated by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues of the fourth order Lamm–Rivière system

$$\begin{aligned} \Delta ^{2}u=\Delta (V\cdot \nabla u)+\mathrm{div}(w\nabla u)+(\nabla \omega +F)\cdot \nabla u+f \end{aligned}$$

in dimension four, with an inhomogeneous term f which belongs to some natural function space. We obtain optimal higher order regularity and sharp Hölder continuity of weak solutions. Among several applications, we derive weak compactness for sequences of weak solutions with uniformly bounded energy, which generalizes the weak convergence theory of approximate biharmonic mappings.

Appendices

Appendix A: A note on Calderón–Zygmund estimate

The aim of this section is to prove that the Calderón–Zygmund estimate is locally uniform with respect to p.

Proposition A.1

For each $\delta \in (0,\frac{1}{2})$, there exists $C=C_{\delta ,n}$ such that for any $p\in [1+\delta ,\frac{1+\delta }{\delta }]$, the following Calderón–Zygmund estimate holds

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{p,B_{{1}/{2}}}\le C_{\delta ,n}\left( \Vert \Delta u\Vert _{p,B_{1}}+\Vert u\Vert _{p,B_{1}}\right) \end{aligned}$$

(A.1)

for all $u\in W^{2,p}(B_1)$.

Proof

We first prove a global version. That is, for each $\delta \in (0,\frac{1}{2})$, there exists $C=C_{\delta ,n}$ such that for any $p\in (1+\delta ,\frac{1+\delta }{\delta })$,

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{L^p({\mathbb {R}}^n)}\le C_{\delta ,n}\left( \Vert \Delta u\Vert _{L^p({\mathbb {R}}^n)}+\Vert u\Vert _{L^p({\mathbb {R}}^n)}\right) \end{aligned}$$

(A.2)

for all $u\in W^{2,p}({\mathbb {R}}^n)$.

By the well-known estimates for Calderón–Zygmund operators (see e.g. [8]), we have

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{L^{1,\infty }({\mathbb {R}}^{n})}\le C_{1,n}\Vert \Delta u\Vert _{L^{1}({\mathbb {R}}^n)} \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{L^2({\mathbb {R}}^n)}\le C_{2,n}\Vert \Delta u\Vert _{L^2({\mathbb {R}}^n)}. \end{aligned}$$

By [7, Corollary 9.10], we can take $C_{2,n}=1$. For $1<p<2$ and $u\in W^{2,p}({\mathbb {R}}^{n})$, the Marcinkiewicz interpolation theorem (see e.g. [7, Theorem 9.8] or [8, Theorem 1.3.2]) implies

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{L^p({\mathbb {R}}^n)}\le C_{p,n}\Vert \Delta u\Vert _{L^p({\mathbb {R}}^n)}, \end{aligned}$$

where

$$\begin{aligned} C_{p,n}=2\left( \frac{p}{(p-1)(2-p)}\right) ^{1/p}\left( C_{1,n}\right) ^{\theta } \left( C_{2,n}\right) ^{1-\theta } \end{aligned}$$

and $\theta =\frac{2}{p}-1$. Thus, for any given $\delta \in (0,\frac{1}{2})$ and $1<p\le 1+\delta$, we have $2-p>1/2$ and $\theta \in (\frac{1}{3},1)$. Consequently, we infer that

$$\begin{aligned} C_{p,n}\le \frac{C(n)}{p-1}. \end{aligned}$$

Fix $\delta \in (0,1/2)$. We apply the Riesz-Thorin interpolation theorem (see e.g. [8, Theorem 1.3.4] with $p_{0}=q_{0}=1+\delta$, $p_{1}=q_{1}=2$), to obtain, for any $1+\delta \le p\le 2$,

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{L^p({\mathbb {R}}^n)}\le C_{1+\delta ,n}^{\theta }C_{2,n}^{1-\theta }\Vert \Delta u\Vert _{L^p({\mathbb {R}}^n)}\le C_{\delta ,n}\Vert \Delta u\Vert _{L^p({\mathbb {R}}^n)}, \end{aligned}$$

where in the last inequality we used the fact that $\theta =\frac{2(p-1-\delta )}{p(1-\delta )}\le \frac{2}{1+\delta }$.

For $2\le p\le \frac{1}{1-\frac{1}{1+\delta }}=\frac{1+\delta }{\delta }$, we conclude by duality that

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{L^p({\mathbb {R}}^n)}\le C_{\delta ,n}\Vert \Delta u\Vert _{L^p({\mathbb {R}}^n)} \end{aligned}$$

holds for all $u\in W^{2,p}({\mathbb {R}}^{n})$. This proves (A.2).

Now we can prove the local Calderón–Zygmund estimate (A.1).

For any given $u\in W^{2,p}(B_{1})$, we extend u to ${\mathbb {R}}^n$ as zero outside $B_1$ and choose $\eta \in C_{0}^{\infty }(B_{1})$ such that $\eta \equiv 1$ on $B_{\frac{1}{2}}$, $0\le \eta \le 1$ on ${\mathbb {R}}^n$ and $\max \{\Vert \nabla \eta \Vert _{L^\infty ({\mathbb {R}}^n)}, \Vert \Delta \eta \Vert _{L^\infty ({\mathbb {R}}^n)} \}\le C_n$. Then for any $p\in [1+\delta ,\frac{1+\delta }{\delta }]$, we apply the previous global estimate to find a constant $C_{\delta ,n}>0$ such that

$$\begin{aligned} \Vert \nabla ^{2}(\eta u)\Vert _{L^p({\mathbb {R}}^n)}\le C_{\delta ,n}\Vert \Delta (\eta u)\Vert _{L^p({\mathbb {R}}^n)}. \end{aligned}$$

As a consequence, we have

$$\begin{aligned} \begin{aligned} \Vert \nabla ^{2}u\Vert _{p,B_{1/2}}&\le C_{\delta ,n}\left( \Vert \eta \Delta u\Vert _{L^p({\mathbb {R}}^n)}+2\Vert \nabla \eta \cdot \nabla u\Vert _{L^p({\mathbb {R}}^n)}+\Vert \Delta \eta u\Vert _{L^p({\mathbb {R}}^n)}\right) \\&\le 2C_{\delta ,n}C_n\left( \Vert \Delta u\Vert _{p,B_{1}}+\Vert \nabla u\Vert _{p,B_{1}}+\Vert u\Vert _{p,B_{1}}\right) . \end{aligned} \end{aligned}$$

(A.3)

On the other hand, by the interpolation inequality for Sobolev spaces (see e.g. [7, Theorem 7.28]), there exists $C=C(n)$ such that for any $\epsilon >0$,

$$\begin{aligned} \Vert \nabla u\Vert _{p,B_{1}}\le \epsilon \Vert \Delta u\Vert _{p,B_{1}}+\frac{C(n)}{\epsilon }\Vert u\Vert _{p,B_{1}}. \end{aligned}$$

Substituting the above inequality with $\epsilon =1$ into (A.3), we finally obtain

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{p,B_{1/2}}\le C_{\delta ,n}\left( \Vert \Delta u\Vert _{p,B_{1}}+\Vert u\Vert _{p,B_{1}}\right) . \end{aligned}$$

The proof is complete. $\square$

Appendix B: A slightly improved Calderón–Zygmund estimate

In this section, we prove the following proposition which states a slightly improved Calderón–Zygmund estimate. It seems very possible that this proposition was already established in some literature. As we did not find a precise reference at hand, we present a detailed proof here for the reader’s convenience.

Proposition B.1

For each $\delta \in (0,\frac{1}{2})$, there exists $C=C_{\delta ,n}$ such that for any $p\in [1+\delta ,n-\delta ]$, the following Calderón–Zygmund estimate holds

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{p,B_{1/2}}\le C_{\delta ,n}\left( \Vert \Delta u\Vert _{p,B_{1}}+\Vert u\Vert _{1,B_{1}}\right) \end{aligned}$$

(B.1)

for all $u\in W^{2,p}(B_1)$.

In our case, $n=4$ and $1<p<4/3$, so $1<p/(2-p)<\gamma<2p/(2-p)<4$. Thus, there exists a constant C independent of $\gamma \in (\frac{p}{2-p},\frac{2p}{2-p})$ such that

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{\gamma ,B_{1/2}}\le C\left( \Vert \Delta u\Vert _{\gamma ,B_{1}}+\Vert u\Vert _{1,B_{1}}\right) . \end{aligned}$$

(B.2)

Proof

We first recall the following results from [2, Chapter 5].

• P1. There exists $C_{n}>0$ depending only on n such that there exists an extension operator $E:W^{1,p}(B_{1})\rightarrow W_{0}^{1,p}(B_{2})$ for all $1\le p<\infty$ satisfying

  $$\begin{aligned} \Vert Eu\Vert _{W^{1,p}({\mathbb {R}}^n)}\le C_{n}\Vert u\Vert _{W^{1,p}(B_1)}. \end{aligned}$$

• P2. Let $1<p<n$ and $u\in W^{1,p}(B_{1})$ for $B_{1}\subset {\mathbb {R}}^{n}$. Then

  $$\begin{aligned} \Vert u\Vert _{p}\le \Vert u\Vert _{1}^{\theta }\Vert u\Vert _{p^{*}}^{1-\theta }, \end{aligned}$$

  where $\frac{1}{p}=1+(1-\theta )\big (\frac{1}{p^*}-1\big )$ or equivalently $\theta =\frac{p}{np-n+p}$.

By the Sobolev embedding theorem and property P1, we have

$$\begin{aligned} \Vert u\Vert _{p^{*},B_{1}}\le \Vert Eu\Vert _{p^{*},B_{2}}\le C_{p}\Vert \nabla (Eu)\Vert _{p,B_{2}}\le C_{p}C_{n}(\Vert u\Vert _{p,B_{1}}+\Vert \nabla u\Vert _{p,B_{1}}). \end{aligned}$$

Here $C_{p}$ is the best Sobolev constant satisfying

$$\begin{aligned} C_{p}\le \frac{n-1}{\sqrt{n}}\frac{p}{n-p}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert u\Vert _{p, B_1}\le \Vert u\Vert _{1, B_1}^{\theta }\left( C_{p}C_{n}(\Vert u\Vert _{p,B_{1}}+\Vert \nabla u\Vert _{p,B_{1}})\right) ^{1-\theta }. \end{aligned}$$

Next we apply the following interpolation theorem (see e.g. [2, Theorem 5.2]): there exists $C_{n}>0$ such that for all $1\le p<\infty$ and $\epsilon >0$,

$$\begin{aligned} \Vert \nabla u\Vert _{p,B_{1}}\le \epsilon \Vert \Delta u\Vert _{p,B_{1}}+\frac{C(n)}{\epsilon }\Vert u\Vert _{p,B_{1}}, \end{aligned}$$

Taking $\epsilon =1$, we obtain

$$\begin{aligned} \Vert u\Vert _{p, B_1}\le \Vert u\Vert _{1, B_1}^{\theta }\left( C_{p}C_{n}(\Vert u\Vert _{p,B_{1}}+\Vert \Delta u\Vert _{p,B_{1}})\right) ^{1-\theta }. \end{aligned}$$

Since $a^{\theta }b^{1-\theta }\le \epsilon ^{-\frac{1-\theta }{\theta }}a+\epsilon b$, we have

$$\begin{aligned} \Vert u\Vert _{p, B_1}\le \epsilon ^{-\frac{1-\theta }{\theta }}\Vert u\Vert _{1, B_1}+\epsilon C_{p}C_{n}(\Vert u\Vert _{p,B_{1}}+\Vert \Delta u\Vert _{p,B_{1}}). \end{aligned}$$

Take $\epsilon =1/(2C_{p}C_{n})$ yields

$$\begin{aligned} \Vert u\Vert _{p}\le (2C_{p}C_{n})^{\frac{1-\theta }{\theta }}\Vert u\Vert _{1}+\frac{1}{2}\Vert \Delta u\Vert _{p,B_{1}}. \end{aligned}$$

Note that $1/n\le \theta \le 1$ for all $1<p<n$, and $\theta \rightarrow 1$ as $p\rightarrow 1$, $\theta \rightarrow 1/n$ as $p\rightarrow n$. Thus, $0\le 1-\theta \le \frac{1-\theta }{\theta }\le n(1-\theta )\le n$ and so

$$\begin{aligned} (2C_{p}C_{n})^{\frac{1-\theta }{\theta }}\le 2^{n}C_{n}^{n}\left( \frac{p}{n-p} \right) ^{\frac{1-\theta }{\theta }} \end{aligned}$$

is locally uniformly bounded for $p\in [1,n)$.

Finally, combining the above estimate with Proposition A.1, we obtain

$$\begin{aligned} \Vert \nabla ^{2}u\Vert _{p,B_{1/2}}\le C_{\delta ,n}C_{p}\left( \Vert \Delta u\Vert _{p,B_{1}}+\Vert u\Vert _{1,B_{1}}\right) \end{aligned}$$

with a constant $C_{\delta ,n}C_{p}$ uniformly bounded by $C(\delta ,n)$ for all $p\in [1+\delta ,n-\delta ]$. $\square$

Appendix C: A regularity result

The following lemma is a special case of Theorem 3.31 of [3]. We sketch the proof for readers’ convenience.

Lemma C.1

Let ${\varvec{f}}\in L^{p}(B_{1})$ and $v\in W^{1,p}(B_{1})$ for some $1<p<2$. Then there exists a unique $h\in W^{1,p}(B_{1})$ solving equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta h=\mathrm{div}\,{\varvec{f}} &{} \text {in }B_{1},\\ h=v &{} \text {on }\partial B_{1}. \end{array}\right. } \end{aligned}$$

(C.1)

Moreover, there exists a constant $C=C(n,p)>0$, such that

$$\begin{aligned} \Vert h\Vert _{W^{1,p}(B_{1})}\le C\left( \Vert {\varvec{f}}\Vert _{L^{p}(B_{1})}+\Vert v\Vert _{W^{1,p}(B_{1})}\right) . \end{aligned}$$

Proof

First assume ${\varvec{f}},v\in C^{2}({\bar{B}}_{1})$ such that there exists a solution h of equation (C.1). Let ${\bar{h}}=h-v\in W_{0}^{1,p}(B_{1})$. Then

$$\begin{aligned} -\Delta {\bar{h}}=\mathrm{div}\,({\varvec{f}}+\nabla v) \end{aligned}$$

in $B_{1}$. Put $F=|\nabla {\bar{h}}|^{p-2}\nabla {\bar{h}}\in L^{p^{\prime }}(B_{1})$. Note $p^{\prime }>2$. Hodge decomposition gives a function $\varphi \in W_{0}^{1,p^{\prime }}(B_{1})$ and a function $G\in L^{p^{\prime }}(B_{1})$ satisfying $\mathrm{div}\,G=0$ in $B_{1}$, and

$$\begin{aligned} \Vert \nabla \varphi \Vert _{L^{p^{\prime }}(B_{1})}+\Vert G\Vert _{L^{p^{\prime }}(B_{1})}\le C(n,p)\Vert F\Vert _{L^{p^{\prime }}(B_{1})}=C(n,p)\Vert \nabla {\bar{h}}\Vert _{L^{p^{\prime }}(B_{1})}^{p-1}. \end{aligned}$$

Thus, using this Hodge decomposition and Hölder’s inequality, we obtain

$$\begin{aligned} \begin{aligned} \Vert \nabla {\bar{h}}\Vert _{L^{p}(B_{1})}^{p}&=\int _{B_{1}}\nabla {\bar{h}}\cdot F\\&=\int _{B_{1}}\nabla {\bar{h}}\cdot \nabla \varphi =-\int _{B_{1}}\left( {\varvec{f}}+\nabla v\right) \cdot \nabla \varphi \\&\le \left( \Vert {\varvec{f}}\Vert _{L^{p}(B_{1})}+\Vert \nabla v\Vert _{L^{p}(B_{1})}\right) \Vert \nabla \varphi \Vert _{L^{p^{\prime }}(B_{1})}\\&\le C(n,p)\left( \Vert {\varvec{f}}\Vert _{L^{p}(B_{1})}+\Vert \nabla v\Vert _{L^{p}(B_{1})}\right) \Vert \nabla {\bar{h}}\Vert _{L^{p^{\prime }}(B_{1})}^{p-1}. \end{aligned} \end{aligned}$$

This implies $\Vert \nabla {\bar{h}}\Vert _{L^{p}(B_{1})}\le C(n,p)\left( \Vert {\varvec{f}}\Vert _{L^{p}(B_{1})}+\Vert \nabla v\Vert _{L^{p}(B_{1})}\right)$, which in turn gives

$$\begin{aligned} \Vert \nabla h\Vert _{L^{p}(B_{1})}\le C\left( \Vert {\varvec{f}}\Vert _{L^{p}(B_{1})}+\Vert \nabla v\Vert _{L^{p}(B_{1})}\right) \end{aligned}$$

(C.2)

for some $C>0$ depending only on n and p.

Then, using Poincaré’s inequality, we obtain

$$\begin{aligned} \Vert h\Vert _{L^{p}(B_{1})}\le \Vert h-v\Vert _{L^{p}(B_{1})}+\Vert v\Vert _{L^{p}(B_{1})}\le C_{n,p}\Vert \nabla h-\nabla v\Vert _{L^{p}(B_{1})}+\Vert v\Vert _{L^{p}(B_{1})}. \end{aligned}$$

Combining the estimate (C.2) together with the above estimate yields

$$\begin{aligned} \Vert h\Vert _{L^{p}(B_{1})}\le C\left( \Vert {\varvec{f}}\Vert _{L^{p}(B_{1})}+\Vert v\Vert _{W^{1,p}(B_{1})}\right) . \end{aligned}$$

This completes the proof in the case ${\varvec{f}},v\in C^{2}({\bar{B}}_{1})$.

The general case follows from a standard approximation argument. We omit the details. The uniqueness proof is also standard. $\square$