Partial regularity of solutions to a linear elliptic system with quadratic Jacobian structure

Abstract

In this short note, we prove interior regularity of weak solutions to a class of linear elliptic system with quadratic Jacobian structures in higher dimensions, extending the result of Hajlasz, Strzelecki and Zhong (2008) in dimension two.

Introduction

In this short note, we consider the (partial) regularity of weak solutions to the following elliptic system

$$-\Delta u=\Omega \cdot \nabla u\text{in }{B}^n,$$

where $B^n=\{x\in {\mathbb{R}}^n:\left|x\right|<1\}$, $n\ge 2$, $u\in W^{1,2}(B^n,{\mathbb{R}}^m)$ ($m>1$) is a weak solution, and $\Omega =\left({\Omega }_j^i\right)\in L^2(B^n,M_m\otimes {\mathbb{R}}^n)$ is a matrix-valued function with entries in ${\mathbb{R}}^n$. In coordinates, (1.1) reads as

$$-\Delta u^i={\Omega }_j^i\cdot \nabla u^j\text{in }{B}^n,i=1,\dots ,m.$$

Summation convention over repeated index is used throughout.

System (1.2) can be viewed as a “linearization” of the nonlinear elliptic system

$$-\Delta u=f(x,u,\nabla u)$$

in $B^n$, where $f(x,z,p)$ is quadratic in $p$, i.e. 

$$\left|f(x,z,p)\right|\le C{\left|p\right|}^2\text{for all }(x,z,p)\in {\mathbb{R}}^n\times {\mathbb{R}}^m\times {\mathbb{R}}^{nm}$$

holds for some constant $C>0$.

System (1.3) includes many important geometric partial differential equations such as the equation of harmonic mappings

$$-\Delta u=\sum _{l=1}^{n}A\left(u\right)({\partial }_{x_l}u,{\partial }_{x_l}u),$$

where $A\left(u\right)$ is the second fundamental form of the target manifold of $u$ in ${\mathbb{R}}^m$, and the equation of prescribed mean curvature

$$-\Delta u=-2H\left(u\right){\partial }_{x}u\wedge {\partial }_{y}u.$$

Thus the regularity theory of solutions to system (1.3) is of fundamental importance. A remarkable feature is that the system lies on the borderline of the usual elliptic regularity theory. To see it, let us consider the planar case $n=2$. If $f(x,u,\nabla u)\in L^p$ for some $p>1$, then the elliptic regularity theory implies that $u\in W^{2,p}$, which in turn implies the continuity of the solutions in dimension two. But, due to the growth condition (1.4), it is clear that if $u\in W^{1,2}$ is a solution of (1.3), then $\Delta u\in L^1$, which only implies $\nabla u\in L^p$ for any $p<2$. Thus we did not gain any improvement on the regularity of $u$ from the usual elliptic regularity theory.

Similarly, for the linear system (1.1) if we merely assume $\Omega \in L^2(B^2,M_m\otimes {\mathbb{R}}^2)$ (i.e. $n=2$), then weak solutions are not necessarily continuous (or even bounded), see [1] for counterexamples. Thus, in order to gain continuity of weak solutions of (1.1), extra conditions have to be imposed on $\Omega$. For example, it was observed in [2, Theorem VII.4] that when $n=2$, weak solutions are continuous provided that $\Omega$ belongs to the Lorentz space $L^{2,1}$, which is a proper subspace of $L^2$.

In the ground-breaking work [1] of Rivière, he considered the case $n=2$ and additionally assumed that $\Omega \in L^2(B^2,M_m\otimes {\mathbb{R}}^2)$ is antisymmetric, i.e., ${\Omega }_j^i=-{\Omega }_i^j$ for all $i,j\in \{1,2,\dots ,m\}$. Under this extra assumption, he successfully extended the conservation law of Hélein [3] to (1.1) (not necessarily with a divergence-free $\Omega$), by which he proved the continuity of weak solutions of (1.1). Counterexamples (see [1]) shows that the antisymmetry of $\Omega$ is crucial. Shortly after [1], Rivière and Struwe [4] considered the regularity problem of (1.1) in dimensions $n\ge 3$. The well-known example of Rivière [5] implies that in this case even a partial regularity theory cannot be expected in general. As a consequence, to assure that solutions of (1.1) are continuous when $n\ge 3$, Rivière and Struwe [4] assumed, except that $\Omega \in L^2(B^n,M_m\otimes {\mathbb{R}}^n)$ is antisymmetric, that $u$ and $\Omega$ satisfy the following Morrey-type smallness condition for some $ϵ=ϵ(m,n)>0$:

$$\underset{x\in B,r>0}{sup}{\left(\frac{1}{r^{n-2}}{\int }_{B_r\left(x\right)\cap B}{|\nabla u|}^2+{\left|\Omega \right|}^2\right)}^{1∕2}<ϵ.$$

Their results apply in particular to the harmonic mapping equations and the equations of prescribed mean curvature.

In [6], Hajlasz, Strzelecki and Zhong considered system (1.1) in dimension two with a different extra assumption on $\Omega$. More precisely, they proved the following result.

Theorem 1.1 Theorem 1.2, [6]

Assume that for all $1\le i,j,k\le m$,

$${\Omega }_j^i=h_{jk}^i{\nabla }^{\perp }{v}^k,$$

where $h_{jk}^i\in L^{\infty }\cap W^{1,2}\left(B^2\right)$ and $(v^1,\dots ,v^m)\in W^{1,2}(B^2,{\mathbb{R}}^m)$. Then every solution of (1.1) is locally Hölder continuous.

Note that in the above theorem, $\Omega$ is not antisymmetric but with better regularity, which is the price that one has to be paid for the lack of antisymmetry.

Motivated by the regularity theory of Rivière and Struwe [4] for (1.1) in dimensions $n\ge 3$, it is natural to extend Theorem 1.1 to higher dimensions. This is the goal of the present work. Our main result reads as follows.

Theorem 1.2

Let $n\ge 2$ and $u\in W^{1,2}(B^n,{\mathbb{R}}^m)$ a solution to system (1.1) with $\Omega$ given by

$${\Omega }_j^i=h_{jk}^i{\mathit{v}}^k,$$

where $h_{jk}^i\in L^{\infty }\cap W^{1,n}\left(B^n\right)$ for all $1\le i,j,k\le m$ and ${\mathit{v}}^k\in L^2(B^n,{\mathbb{R}}^n)$ satisfies

$$\mathrm{div}{\mathit{v}}^k=0.$$

Then there exists a constant $\delta >0$, depending only on $n$, $m$ and the functions $h_{jk}^i$, such that $u$ is locally Hölder continuous in $B_r\left(a\right)$, provided $\mathit{v}=({\mathit{v}}^{\mathbf{1}},\dots ,{\mathit{v}}^m)$ satisfies the Morrey-type smallness growth condition

$$\underset{B_{\rho }\left(z\right)\subset B_{4r}\left(a\right)}{sup}{\rho }^{2-n}{\int }_{B_{\rho }\left(z\right)}{\left|\mathit{v}\right|}^2<{\delta }^2$$

whenever $B_{4r}\left(a\right)⋐B^n$.

Remark 1.3

We have a couple of remarks about Theorem 1.2:

• Note that when $n=2$, we can always choose some constant $r_0$ sufficiently small such that (1.8) holds for all $r<r_0$ at every point. This implies everywhere continuity of $u$ and thus recovers Theorem 1.1.

• If, in particular, $\rho \mapsto {\rho }^{2-n}{\int }_{B_{\rho }\left(z\right)}{\left|\mathit{v}\right|}^2$ is monotonically increasing for $\rho >0$ and for every $z\in B_1$, then (1.8) is satisfied outside a closed set $E$ satisfying ${\mathcal{H}}^{n-2}\left(E\right)=0$. This implies $u\in C^{0,\alpha }(B_1\setminus E)$.

• As in the case of [6], the Laplace operator in Theorem 1.2 can be replaced by a more general elliptic operator in divergence form. However, this does not change the essential difficulty caused by the right-hand side of (1.2) and thus we do not consider the more general situation for the sake of simplicity.

• The proof also works without change for the case where ${\Omega }_j^i=h_{jk}^i{\mathit{v}}_i^k$ with $h_{jk}^i\in L^{\infty }\cap W^{1,n}\left(B^n\right)$ and ${\mathit{v}}_i^k\in L^2(B^n,{\mathbb{R}}^n)$ satisfies $\mathrm{div}{\mathit{v}}_i^k=0$ for all $1\le i,j,k\le m$.

We shall follow the approach of Hajlasz, Strzelecki and Zhong [6] and the idea goes essentially back to Lewis [7]. The advantage of this approach is that the proof is elementary without delicate and deep analytic or geometric tools. In particular, it does not involve conservation law, Uhlenbeck Gauge decomposition, Hodge decomposition etc.