I am self-learning PDE using Evan's text. The question is:
Suppose an open set $U\in\mathbb{R}^n$ is subdivided by a smooth hypersurface $\Gamma$ into the subregions $V^+$ and $V^-$. Let $\nu$ denote the unit normal to $\Gamma$, point into $V^+$. Assume that $u$ is a viscosity solution of $$H(Du)=0\quad \mathrm{in}\,U$$ and that u is smooth in $\bar{V}^+$ and $\bar{V}^-$. Write $u_{\nu}^+$ for the limit of $Du\cdot\nu$ along $\Gamma$ from within $V^+$, and write $u_{\nu}^-$ for the limit from within $V^-$. Prove that along $\Gamma$ we have the inequalities $$H(\lambda Du^-+(1-\lambda)Du^+)\ge0\quad\mathrm{if}\quad u_{\nu}^-\leq u_{\nu}^+$$ and $$H(\lambda Du^-+(1-\lambda)Du^+)\le0\quad\mathrm{if}\quad u_{\nu}^+\leq u_{\nu}^-$$ for each $0\le\lambda\le1$, where $Du^{\pm}$ denote the gradients along $\Gamma$ from $V^{\pm}$.
My opinion is to find a smooth function $v$, whose gradient on $\Gamma$ equals to $\lambda Du^-+(1-\lambda)Du^+$. If $u-v$ has a maximum (minimum) at a point $x_0\in\Gamma$, then $H(\lambda Du^-+(1-\lambda)Du^+)\le0(\ge0)$. However I have no idea how to find the smooth function $v$.