= Averaged linearization of a nonlinear divergence-form equation
{title2=$A_h=\int_0^1DF((1-t)p_-+tp_+)\,dt$}
For a $C^1$ flux $F$ and a <weak solution> of $\operatorname{div}F(Du)=0$, <discrete integration by parts> and the <fundamental theorem of calculus along a line segment> show that $w_h=\delta_{\ell,-h}u$ solves $\operatorname{div}(A_hDw_h)=0$, with $p_-=Du(x-he_\ell)$ and $p_+=Du(x)$. Coercive quadratic-form bounds and <operator norm> bounds for $DF$ on all segments between these <gradients> pass to $A_h$. This produces a <divergence-form elliptic operator> without initially assuming second <partial derivatives> of $u$.
Back to article page
