A priori estimate for diffraction problem for linear elliptic PDEs

I am looking for a reference to show how to obtain a priori estimate of the solution $u\in H^1$ and $u\in C^{2,\alpha}$ to the diffraction problem of linear elliptic equation.

I looked at Ladyzhenskaya's book (section 3.16), but it only has a priori estimate for $u\in H^2$ and confirms the existence of a Holder solution. Gilbarg and Trudinger book has both estimates but not for the diffraction problem.

Edit: The equation I'm considering is \begin{equation} \begin{cases} Lu & = & f & \mbox{in } \Omega \\ u & = & 0 & \mbox{on } \partial\Omega \\ [u]\vert_\Gamma & = & 0 \\ \partial_s(\alpha^{st}\partial_tu) + \left.\left[\displaystyle \frac{\partial u}{\partial N} \right]\right\rvert_\Gamma & = & g \end{cases} \end{equation} I assume $\Gamma$ is $x_n=0$, $[u]$ is a jump condition (I took this notation from Ladyzhenskaya's book), and $\displaystyle \frac{\partial u}{\partial N}$ is the co-normal derivative of $u$.

$\Omega$ is an infinite strip.

$Lu = \partial_i(a^{ij}\partial_j u) + b^i\partial_i u + cu$

$a^{ij}, b^i, c, \alpha^{st}, f, g$ are functions of $x$.

• It might help to state your equation more explicitly. – Sebastian Goette Dec 16 '15 at 19:22
• @SebastianGoette I added the equation. Thank you. – dh16 Dec 16 '15 at 20:14
• It may also help to say what is $\Gamma$, and what is $[u]$ (I assume some sort of jump condition?) – Willie Wong Dec 16 '15 at 20:23
• @WillieWong I added the conditions for $\Gamma$ and the jump condition. I'm sorry I'm new to it. About regularities of coefficients, they must be at least bounded (may be more if needed). – dh16 Dec 16 '15 at 20:33