Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Question

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is open. We also apply the Dirichlet boundary condition on $\partial \Omega$. Let us say that apriori $u$ is known to be in $L^2(\Omega)$.

Consider a ball $B_{2r}(x)$, where $x \in \overline{\Omega}$ and call $B_2 = B_{2r}(x) \cap \overline{\Omega}$. Also call $B_1 = B_{r}(x) \cap \overline{\Omega}$. Do we have estimates of the form $$\Vert u\Vert_{C^{2, \alpha}(B_1)} \leq C\Vert u\Vert_{L^2(B_2)}?$$ Do we have analogous estimates for Neumann boundary conditions? I will be very grateful for a reference.

Edit: Is $u \in L^2$ enough to start with, or do we need stronger apriori conditions on $u$?

Answer 1

1. One needs to start with a 'energy-weak solutions', i.e. $u \in W^{1,2}$, in the general case, even for interior regularity. If $L$ is the Laplacian, you can use Weyl's lemma to first prove that $u \in L^2$ implies $u \in W_{loc}^{1,2}$.

2. Although the estimate you are looking for looks like a local estimate ( as you are only asking for estimate on $B_{1}$ to be controlled by quantities on $B_{2}$ ), since you are not assuming $B_{1}$ is not compactly contained in $\Omega,$ what you need is actually a boundary estimate and as such does not hold without more assumption on the boundary traces of $u$.

3. Since as I mentioned in (2), it is a boundary estimate, one needs to start with at least $u \in W^{1,2}$ so that the Dirichlet and Neumann traces are defined ( in $W^{\frac{1}{2},2}(\partial\Omega)$ and $W^{-\frac{1}{2},2}(\partial\Omega)$ respectively ). But to obtain such an estimate, you need the traces to be $C^{2,\alpha},$ since otherwise $u$ can never be $C^{2,\alpha}$ up to the boundary, as mentioned by Andrews comment above.

4. Now if you assume the boundary traces of $u$ are at least $C^{2,\alpha}$, the coefficients of $L$ are at least $C^{2,\alpha}$ and $\partial\Omega$ is at least of class $C^{2,\alpha},$ then one can obtain an estimate $$\lVert u \rVert_{C^{2,\alpha}(B_{1})} \leq C \left( \lVert u \rVert_{L^{2}(B_{2})} + \lVert u_{0} \rVert_{C^{2,\alpha}(\partial\Omega \cap B_{2})}\right), $$ where $u_{0} = u|_{\partial \Omega \cap B_{2}}$ and the constant $C$ depends on the dimension $n,$ the $C^{2,\alpha}$ norm of the coefficients of $L$ and the $C^{2,\alpha}$ norm of the mapping that locally gives the boundary $\partial\Omega.$ The same estimate holds for Neumann BVP ( one needs Neumann trace to be only $C^{1,\alpha}$ ).

5. Thus by (4), you would obtain your estimate if the boundary values are 0.

In Gianquinta and Martinazzi's book: http://www.springer.com/gp/book/9788876424427

You can read section 5.4 on Schauder estimates there. Specifically Theorem 5.19 and 5.20. These give the gradient estimates, but can easily be iterated to give hessian estimates. On the other hand, the estimate in Theorem 5.20 is somewhat misleading, as it does not have the $\lVert u \rVert_{L^{2}}$ term. The term in general can not be dropped, unless you have uniqueness of solutions. If $L$ has lower order terms, uniqueness might not be ensured.

Remark about my previous answer: I do not know if the question has been edited in the meantime, but as the question stands now, my previous answer seems flat out wrong. I can not remember what I was thinking when I wrote this. May be I missed that you assumed smooth coefficients of $L$. Just for the sake of records, I keep it here unchanged.
" If the coefficients are not constant, such an estimate can not be expected to hold even with Dirichlet boundary data for domains with smooth boundary... for constant coefficient equations, they certainly hold for the Dirichlet BC on a smooth enough domain. You can see Gianquinta and Martinazzi's book: http://www.springer.com/gp/book/9788876424427 "