$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 3.2: Quasilinear Equations of Second Order

[ "article:topic" ]

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Here we consider the equation

\label{quasilin}
\sum_{i,j=1}^na^{ij}(x,u,\nabla u)u_{x_ix_j}+b(x,u,\nabla u)=0

in a domain $$\Omega\subset\mathbb{R}$$, where $$u:\ \Omega\mapsto\mathbb{R}^1$$. We assume that $$a^{ij}=a^{ji}$$.

As in the previous section we can derive the characteristic equation
$$\sum_{i,j=1}^na^{ij}(x,u,\nabla u)\chi_{x_i}\chi_{x_j}=0.$$
In contrast to linear equations, solutions of the characteristic equation depend on the solution considered.

### Contributors

• Integrated by Justin Marshall.