I'm considering the diffraction problem described in section 3.16 of "Linear and quasilinear elliptic equations" of Ladyzhenskaya and Uraltseva (1968). Let $\Omega$ be an open bounded subset in $\mathbb{R}^n$, and suppose the surface $\Gamma$ partition $\Omega$ into sub-regions $\Omega\_1$ and $\Omega\_2$ where $\Omega = \Omega\_1 \cup \Omega\_2$. For simplicity, I assume $\Gamma$ to be the surface $\{x\_n=0\}$. Now suppose $u(x)$ satisfies the following condition in $\Omega\_1$ and $\Omega\_2$
\begin{equation}
Lu \equiv \partial\_{x\_i} (a^{ij}(x) \partial\_{x\_j}u) + b^i(x) \partial\_{x\_i}u + c(x) u = f(x) \quad (16.1)
\end{equation}
and the conditions
\begin{equation}
u|\_{\partial\Omega} = 0, \quad [u]|\_\Gamma = 0, \quad \left.\left[ \frac{\partial u}{\partial N} \right]\right|\_\Gamma = 0 \quad (16.3)
\end{equation}
on $\partial\Omega$ and $\Gamma$.


Here the symbol $[v]|\_\Gamma$ denotes the jump in the function $v$ as it crosses $\Gamma$. Also, $\displaystyle \left.\frac{\partial u}{\partial N} \right|\_\Gamma = a^{ij} (\partial\_{x\_j}u) \cos(n, x\_i)$, where $n$ is the normal to $\Gamma$.


$u \in H^1\_0(\Omega)$ is a weak solution of the problem (16.1) and (16.3) if $u$ satisfies
\begin{equation}
\int\_\Omega \left[ a^{ij}(\partial\_{x\_j}u) (\partial\_{x\_i}v) -b^i (\partial\_{x\_i}u) v - cuv \right] \,dx = - \int\_\Omega fv \,dx \quad (16.5)
\end{equation}
for all $v \in H^1\_0(\Omega)$. Assume boundedness of the coefficients $a^{ij}, \partial\_{x\_k}a^{ij}, b^i, c$, the uniformly elliptic condition, and $f \in L^{q/2}(\Omega)$ for $q > n$.


Suppose we already showed $u \in H^2(\Omega\_i)$. In (16.5), take $v(x) = \partial\_{x\_s} \eta$ for $s \le n-1$, where $\eta$ is a sufficiently smooth function with compact support in $K\_\rho$, where $K\_\rho$ is a sphere inside $\Omega$ with center on $\Gamma$. Let $\Gamma\_1 := K\_\rho \cap \Gamma$.


Substituting into (16.5) and integrating by parts we get
\begin{equation}
\int\_\Omega \left[ a^{ij} (\partial\_{x\_s}\partial\_{x\_j}u) (\partial\_{x\_i}\eta) + \partial\_{x\_s} a^{ij} (\partial\_{x\_j}u) (\partial\_{x\_i}\eta) + b^i (\partial\_{x\_i}u) (\partial\_{x\_s}\eta) + cu(\partial\_{x\_s}\eta) \right] \,dx = \int\_\Omega f (\partial\_{x\_s} \eta) \,dx. \quad (16.13)
\end{equation}
The author stated "On the basis of the conditions (16.13), we have $\left.[\partial\_{x\_s}u] \right|\_{\Gamma\_1} = 0$ for $s \le n-1$.


I don't understand why it is the case. I'm new to this type and really want to understand it.


Thank you.


EDIT: Following John Jiang's comment, I did the computation out. Substituting $v = \partial\_{x\_s} \eta$ into (16.5) and integrating by parts the first term, we get (justifying derivatives of $u$ makes sense)
\begin{equation}
\int\_{K\_\rho} \left[ -\partial\_{x\_s}(a^{ij}\partial\_{x\_j}u)(\partial\_{x\_i}\eta) - b^i(\partial\_{x\_i}u)(\partial\_{x\_s}\eta) - cu(\partial\_{x\_s}\eta) \right] \,dx + \int\_{\Gamma\_1} \big[a^{ij}(\partial\_{x\_j}u)(\partial\_{x\_i}\eta)\big]\Big|\_\Gamma \,dS = -\int\_{K\_\rho} f(\partial\_{x\_s}\eta) dx.
\end{equation}
The integral over $\Gamma\_1$ vanishes by the hypothesis that $\displaystyle \left.\left[ \frac{\partial u}{\partial N} \right]\right|\_\Gamma = 0$. However, I haven't seen why the tangential derivatives of $u$ across $\Gamma\_1$, $[\partial\_{x\_s}u]|\_{\Gamma\_1}$ must be 0.



Why are the tangential derivatives in this diffraction problem zero?
You can exchange the partial derivatives on $\nu = \partial\_{x\_s} \eta$. Taking one partial of $u$ is allowed in the whole $\Omega$ since $u \in H^1\_0(\Omega) \subset W^1\_0(\Omega)$. So $\partial\_{x\_s} u$ makes sense there. Mixed second partials of $u$ are justified by equation 16.10, which you didn't mention; it's established by assuming $u$ to be a solution of the original equation 16.1 satisfying condition 16.3. I believe 16.10 alone tells you there is no gap of $u$ across $\Gamma$. This is of course in the almost sure sense since $u$ is an equivalence class of functions. Any gap of positive measure in $\Gamma$ would show up as a boundary term upon integration by parts. For the simplest situation refer to [this](https://www.math.ucdavis.edu/~hunter/pdes/ch3.pdf). I encourage you to write down the details carefully.


