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Mathematics LibreTexts

3.2: Quasilinear Equations of Second Order

  • Page ID
    2143
  • [ "article:topic" ]

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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.

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