
3.2: Quasilinear Equations of Second Order

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.