Problem 5 of chapter 6 of Evans PDE 1st.

Question

I don't know how to start it. I try to compute the $Lv$ , but nothing I get.Maybe, I think some hint is suitable for me . Thanks.

Answer 1

If $Lv\leq 0$ for $\lambda$ large enough, the maximum principle implies that $\|v\|_{L^\infty(U)} \leq \|v\|_{L^\infty(\partial U)}$. Therefore, $$ \big\||Du|^2\big\|_{L^\infty(U)} \leq \|v\|_{L^\infty(U)} \leq \big\||Du|^2+\lambda u^2\big\|_{L^\infty(U)} \leq \big\||Du|^2+\lambda u^2\big\|_{L^\infty(\partial U)} \leq \big| \| |Du|^2 \big\|_{L^\infty(\partial U)} + \lambda \big\| |u|^2 \big\|_{L^\infty(\partial U)} \leq C^2(\|Du\|_{L^\infty(\partial U)}+\|u\|_{L^\infty(\partial U)})^2. $$