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 and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.