# Estimate for Laplace equation with Neumann boundary on manifold with corner

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.

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 $$\alpha$$.