On Local Solutions of Second Order Quasi-linear Elliptic Systems with Arbitrary 1-Jet at a Point

Let $\displaystyle\bm{u}(x)=\big{(}u^{1}(x),u^{2}(x),...,u^{m}(x)\big{)}$ be a $C^{2}$ vector function from an open set in $\mathbb{R}^{n}(n\geq 2)$ to $\mathbb{R}^{m}(m\geq 1)$. We use $L$ to denote an elliptic quasi-linear operator on $\bm{u}$, and write

$$L\bm{u}=\left(Lu^{1},...,Lu^{m}\right),$$

where

$$Lu^{k}=\sum_{i,j=1}^{n}a^{ij}\big{(}x,\bm{u}(x),D\bm{u}(x)\big{)}D_{ij}u^{k},\hskip 14.45377pt1\leq k\leq m,$$

$a^{ij}\in C_{loc}^{1,\alpha}\left(\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{mn}\right)$ ($0<\alpha<1$), and there is a $\lambda>0$ such that

We will prove that any quasi-linear system defined by such an operator $L$ is always locally solvable, given the value and the first derivatives of the solution at any one point in $\mathbb{R}^{n}$. Without loss of generality we can assume the domain of $\bm{u}$ is a ball centered at the origin.

Denote a ball centered at the origin with radius $R$ in $\mathbb{R}^{n}$ as

$$B_{R}=\{x\in\mathbb{R}^{n}\big{|}|x|\leq R\}$$

and denote a ball centered at the origin in $\mathbb{R}^{m}$ as

$$B^{\prime}_{R^{\prime}}=\{y\in\mathbb{R}^{m}\big{|}|y|\leq R^{\prime}\}.$$

The following is our main result.

As will be shown in the proof, there are in fact infinitely many such solutions. An interesting feature of our theorem is that the only assumptions about the coefficient functions are ellipticity and sufficient regularity, therefore it is as general as we can hope for. It is intriguing to know if such a general existence theorem holds for fully nonlinear elliptic systems. The equation $e^{\Delta u}=0$, although being non-elliptic, seems to suggest that general local existence may not be true in the fully nonlinear case.

Example 1. (The Minimal Surface Equation) For a domain $\Omega\subset\mathbb{R}^{n}$ and a smooth function $u$ on $\Omega$, the area of its graph $(x,u(x)),$ where $x\in\Omega$, is $\displaystyle\int_{\Omega}\sqrt{1+|Du|^{2}}\,\,dx.$ Its Euler-Lagrange equation is the minimal surface equation

$$\sum_{i,j=1}^{n}\left(\delta_{ij}-\frac{D_{i}uD_{j}u}{1+|Du|^{2}}\right)D_{i}D_{j}u=0,$$

which is an elliptic quasilinear equation. The graph of $u$ is called a minimal surface, on which the mean curvature is identically 0. Without loss of generality we can assume $0\in\Omega$. By Theorem 1.1, given $u(0)$ and $Du(0)$, this equation always has local solutions near $0$. Therefore, given any point $P=(0,...,0,p)\in\mathbb{R}^{n+1}$ and any normal vector $\eta$, there are infinitely many local minimal surfaces through $P$ with $\eta$ as the normal vector.

Similar existence also holds for the prescribed mean curvature equation

$$\sum_{i,j=1}^{n}\left(\delta_{ij}-\frac{D_{i}uD_{j}u}{1+|Du|^{2}}\right)D_{i}D_{j}u=H(x,u),$$

where $H(x,z)$ is a given function on $\Omega\times\mathbb{R}$.

Example 2. (The Harmonic Map System) Let $(M_{1}^{n},g)$ and $(M_{2}^{m},h)$ be two Riemannian manifolds of dimensions $n$ and $m$ with metrics $g$ and $h$, respectively. A map $u:M_{1}^{n}\to M_{2}^{m}$ is called a harmonic map if it is a critical point of the energy functional. Let $x=(x^{1},...,x^{n})$ be a local coordinate system on $M_{1}$. In a local coordinate system on $M_{2}$, we may write $u(x)=(u^{1}(x),...,u^{m}(x))$, and it satisfies the quasi-linear equations

By Theorem 1.1, given any $p_{1}\in M_{1}$, $p_{2}\in M_{2}$, and a subspace $\Gamma$ of the tangent space $T_{p_{2}}M_{2}$, there exist infinitely many local harmonic maps $u:M_{1}\to M_{2}$ which map $p_{1}$ to $p_{2}$ and $T_{p_{1}}M_{1}$ to $\Gamma$. The theorem also implies the existence of any local objects to be defined by the Laplace-Beltrami operator.

It can be proved that $K_{M}$ is upper semicontinuous on $TM$, and therefore it is a Finsler metric on $M$. The Kobayashi metric is a well studied, important tool in several complex variables and complex geometry, so a natural question is whether a similar concept can be introduced on Riemannian manifolds. Theorem 1.1 paves the way for answering this question by establishing a well-defined, real version of Kobayashi metric as follows.

Again we use $D$ to denote the unit disk in $\mathbb{R}^{2}$, and let $D_{R}$ be a disk of radius $R$ in $\mathbb{R}^{2}$ centered at the origin with coordinates $(x,y$). Let $(M,g)$ be a Riemannian manifold with metric $g$. We consider harmonic maps $u:D_{R}\to M$ that are conformal at $0$, so

$$g\left(\frac{\partial u}{\partial x}(0),\frac{\partial u}{\partial x}(0)\right)=g\left(\frac{\partial u}{\partial y}(0),\frac{\partial u}{\partial y}(0)\right)\hskip 7.22743pt\text{and}\hskip 7.22743ptg\left(\frac{\partial u}{\partial x}(0),\frac{\partial u}{\partial y}(0)\right)=0.$$

For any non-zero vector $X\in T_{p}M$, we can find another vector $Y$ satisfying $g(X,X)=g(Y,Y)$ and $g(X,Y)=0$. Then by Theorem 1.1, there is a harmonic map $u$ on a local disk $D_{R},R>0$, such that $u(0)=p$, $\displaystyle\frac{\partial u}{\partial x}(0)=X$, and $\displaystyle\frac{\partial u}{\partial y}(0)=Y$. This function $u$ is conformal at $0$ by the choice of $X$ and $Y$. Therefore in (1.2) the infimum of $\displaystyle\frac{1}{R}$ is always a finite number, and consequently the Kobayashi metric is well defined on Riemannian manifolds:

We would like to point out that the condition of conformality at 0 excludes geodesics from the admissible harmonic maps. For any geodesic $\gamma(t)$ on $M$ such that $\gamma(0)=p$ and $\gamma^{\prime}(0)=X$, the function $\displaystyle u(x,y)=\gamma(x+y)$ defines a harmonic map from a neighborhood of the origin to $M$, with $u(0)=p$ and $\displaystyle\frac{\partial u}{\partial x}(0)=X$. However, since $\displaystyle\frac{\partial u}{\partial y}(0)=X$ too, $u$ is not conformal at $0$.

Let $f:B_{R}\to\mathbb{R}$ be a function defined on $B_{R}$.

Proof: Define

$$\varphi(t)=f(x+th),\hskip 14.45377pt\text{where}\hskip 14.45377pth=x^{\prime}-x.$$

Then

After subtracting the second derivatives term, we have

which can be written as

$$\Big{|}f(x^{\prime})-\sum_{l=0}^{2}\frac{1}{l!}\sum_{|\beta|=l}\partial^{\beta}f(x)(x^{\prime}-x)^{\beta}\Big{|}\leq\frac{1}{2}\left(\sum_{|\beta|=2}H_{\alpha}[\partial^{\beta}f]\right)|x^{\prime}-x|^{2+\alpha}.$$

Proof: Let $f\in C_{0}^{2,\alpha}(B_{R})$, by definition $f(0)=0$ and $\displaystyle\partial_{i}f(0)=0$ for all $i=1,...,n$.

For any $x\in B_{R}$, define

$$\varphi(t)=f(tx),$$

then $\varphi(0)=0$ and $\varphi^{\prime}(0)=0$. Therefore,

It follows easily from (3) that $\|x_{i}\|_{\alpha}=3R$ for all $i=1,...n$. Then for any $\beta=(\beta_{1},...,\beta_{n})$ with $|\beta|=2$, by (4) we have

$$\|x^{\beta}\|_{\alpha}=\|x_{1}^{\beta_{1}}\cdots x_{n}^{\beta_{n}}\|_{\alpha}\\ \leq(3R)^{2}.$$

Now applying (4) to (7), we obtain

	$\displaystyle\|f\|_{\alpha}$	$\displaystyle\leq$	$\displaystyle\sum_{|\beta|=2}\|\partial^{\beta}f\|_{\alpha}\|x^{\beta}\|_{\alpha}$
		$\displaystyle\leq$	$\displaystyle(3R)^{2}\sum_{|\beta|=2}\|\partial^{\beta}f\|_{\alpha}$
		$\displaystyle\leq$	$\displaystyle(3nR)^{2}\|f\|^{(2,\alpha)}.$

Proof: Let $f\in C_{0}^{2,\alpha}(B_{R})$. If $|\beta|=l<2$, then $\partial^{\beta}f\in C_{0}^{2-l,\alpha}(B_{R})$.

Similar to the proof of Lemma 2.2, we can show that

	$\displaystyle\|\partial^{\beta}f\|_{\alpha}$	$\displaystyle\leq$	$\displaystyle(3nR)^{2-l}\|\partial^{\beta}f\|^{(2-l,\alpha)}$
		$\displaystyle\leq$	$\displaystyle(3nR)^{2-l}\|f\|^{(2,\alpha)}.$

Therefore

$$\|f\|^{(l,\alpha)}\leq(3nR)^{2-l}\|f\|^{(2,\alpha)}.$$

$\Box$

Proof: By definition $\|\cdot\|^{(2,\alpha)}$ is a semi-norm on $C^{2,\alpha}(B_{R})$. If $\|f\|^{(2,\alpha)}=0$ , then $\partial^{\beta}f=0$ for all $|\beta|=2$ on $B_{R}$, which implies $f$ is a constant or a linear function. If in addition $f\in C_{0}^{2,\alpha}(B_{R})$, then $f$ and its first derivatives all vanish at $0$, so $f$ must be identically $0$. Thus $\|\cdot\|^{(2,\alpha)}$ is a norm on $C_{0}^{2,\alpha}(B_{R})$.

Since $C_{0}^{2,\alpha}(B_{R})$ is a closed subspace of $C^{2,\alpha}(B_{R})$ and $C^{2,\alpha}(B_{R})$ is a Banach space with the $\|\cdot\|_{\alpha}$ norm, we know $C_{0}^{2,\alpha}(B_{R})$ is a Banach space with the $\|\cdot\|_{\alpha}$ norm. Then by Lemma 2.3, $C_{0}^{2,\alpha}(B_{R})$ is also complete under the $\|\cdot\|^{(2,\alpha)}$ norm. Therefore, $C_{0}^{2,\alpha}(B_{R})$ equipped with the $\|\cdot\|^{(2,\alpha)}$ norm is a Banach space.

For an integrable function $f$ on $B_{R}$, the Newtonian potential of $f$ is defined on $\mathbb{R}^{n}$ by

$$\mathcal{N}(f)(x)=\int_{B_{R}}\Gamma(x-y)f(y)dy;$$

it solves the Poisson’s Equation $\Delta u=-f$. In our proof we will consider $\mathcal{N}$ as an operator acting on a function space. The following result is well-known and a proof is given in [1].

Although in [1] Lemma 3.2 was proved in the case $\rho\leq\frac{R}{4}$, the proof actually holds for all $\rho$ without restrictions.

We would like to point out that although the constant $C(n,\alpha)$ is independent of $R$, there is an $R^{\alpha}$ term in the definition of the weighted norm $\|f\|_{\alpha}$, so the $\|\mathcal{N}(f)\|^{(2,\alpha)}$ norm actually does depend on $R$. This shows the advantage of choosing the weighted norm over unweighted norm.

Proof: Recall that by (5),

$$\|\mathcal{N}(f)\|^{(2,\alpha)}=\max_{|\beta|=2}\big{\{}\|\partial^{\beta}\mathcal{N}(f)\|_{\alpha}\big{\}}=\max_{1\leq i,j\leq n}\big{\{}\|\partial_{ij}\mathcal{N}(f)\|_{\alpha}\big{\}}$$

Let

$$\zeta(x)=\int_{B_{R}}\partial_{ij}\Gamma(x-y)\big{(}f(y)-f(x)\big{)}dy.$$

By Lemma 3.1,    $\displaystyle\partial_{ij}\mathcal{N}(f)=\zeta(x)-\frac{\delta_{ij}}{n}f(x)$, and therefore

(8)

$$\|\partial_{ij}\mathcal{N}(f)\|_{\alpha}\leq\|\zeta\|_{\alpha}+\|f\|_{\alpha}.$$

So we only need to bound $\|\zeta\|_{\alpha}$ in terms of $\|f\|_{\alpha}$.

First, for $x\in\mathrm{Int}(B_{R})$,

To compute the Hölder constant of $\zeta$, let $x,x^{\prime}$ be two (distinct) points in $B_{R}$. Let $B_{\rho}(x)$ be the open ball of radius $\rho=2|x-x^{\prime}|$ and centered at $x$.

Next we will estimate each of $I_{1}$, $I_{2}$, $I_{3}$, and $I_{4}$.