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# 3.2.1: Quasilinear Elliptic Equations

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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

\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.

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.