# 3.2.1: Quasilinear Elliptic Equations

There is a large class of quasilinear equations such that the associated characteristic equation has no solution \(\chi\), \(\nabla\chi\not=0\).

Set

$$

U=\{(x,z,p):\ x\in\Omega,\ z\in\mathbb{R}^1,\ p\in\mathbb{R}\}.

$$

**Definition.** The quasilinear equation (3.2.1) is called *elliptic *if the matrix \((a^{ij}(x,z,p))\) is positive definite for each \((x,z,p)\in U\).

Assume equation (3.2.1) is elliptic and let \(\lambda(x,z,p)\) be the minimum and \(\Lambda(x,z,p)\) the maximum of the eigenvalues of \((a^{ij})\), then

$$

0<\lambda(x,z,p)|\zeta|^2\le\sum_{i,j=1}^na^{ij}(x,z,p)\zeta_i\zeta_j\le \Lambda(x,z,p)|\zeta|^2

$$

for all \(\zeta\in\mathbb{R}\).

**Definition.** Equation (3.2.1) is called *uniformly elliptic* if \(\Lambda/\lambda\) is uniformly bounded in \(U\).

An important class of elliptic equations which are not uniformly elliptic (non-uniformly elliptic) is

\begin{equation}

\label{nonuniform}\tag{3.2.1.1}

\sum_{i=1}^n\frac{\partial}{\partial x_i}\left(\frac{u_{x_i}}{\sqrt{1+|\nabla u|^2}}\right)+\mbox{lower order terms}=0.

\end{equation}

The main part is the minimal surface operator (left hand side of the minimal surface equation). The coefficients \(a^{ij}\) are

$$

a^{ij}(x,z,p)=\left(1+|p|^2\right)^{-1/2}\left(\delta_{ij}-\frac{p_ip_j}{1+|p|^2}\right),

$$

\(\delta_{ij}\) denotes the Kronecker delta symbol. It follows that

$$

\lambda=\frac{1}{\left(1+|p|^2\right)^{3/2}},\ \ \Lambda=\frac{1}{\left(1+|p|^2\right)^{1/2}}.

$$

Thus equation (\ref{nonuniform}) is not uniformly elliptic.

The behavior of solutions of uniformly elliptic equations is similar to linear elliptic equations in contrast to the behavior of solutions of non-uniformly elliptic equations.

Typical examples for non-uniformly elliptic equations are the minimal surface equation and the capillary equation.

### Contributors

Integrated by Justin Marshall.