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# 3.2: Quasilinear Equations of Second Order

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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

• Integrated by Justin Marshall.