Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

3.2: Quasilinear Equations of Second Order

Here we consider the equation
\begin{equation}
\label{quasilin}
\sum_{i,j=1}^na^{ij}(x,u,\nabla u)u_{x_ix_j}+b(x,u,\nabla u)=0
\end{equation}
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