Estimate for Laplace equation with Neumann boundary on manifold with corner

Question

Let $(M,g)$ be a compact Riemannian manifold with boundary and corner, i.e. locally modelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$.

Consider the Neumann equation, i.e. $$\begin{cases}\Delta u=f&\mbox{in }M,\\ \frac{\partial u}{\partial \nu}=g&\mbox{on }\partial M \end{cases},$$ where the second the equation is defined smooth-wisely on $\partial M$, i.e. defined piece-wisely on $\partial M$.

Q Can we find a solution $u$ satisfying the estimate $$\|u\|_{L^p_{k+2}}\leq C(\|f\|_{L^p_k}+ \|g\|_{L^p_{k+1,\delta}}+\|u\|_{L^p_{k+1}}),$$ where $\|g\|_{L^p_{k+1,\delta}}=\inf\{\|G\|_{L^p_{k+1}(M)}\big| G|_{\partial M}=g \}$.

Any reference is welcome.

Answer 1

First, it is not a Neumann equation, but Laplace equation with Neumann boundary condition.

Let us look at toy-model: domain is just a sector $0< \theta <\alpha$ in $\mathbb{R}^2$ (or looks this way near $0$) and a solution $u(r,\theta)= r^{\pi /\alpha} \cos (\pi \theta/\alpha)$. See that the answer to your question depends on angle $\alpha$.