Let $L = L^*$ be a second-order elliptic PDE with smooth and bounded coefficients in some bounded domain $\Omega \subset \mathbb R^d$, $d \geq 3$, with smooth boundary. Let $G(x,y)$ be a Green function for the Dirichlet problem with the following properties: $$ L_x G(x,y) = \delta_y(x), \\ G(x,y) = 0 \; \text{if $x \in \partial \Omega$ or $y \in \partial \Omega$}, \\ |G(x,y)| \leq C|x-y|^{2-d}, \\ |\nabla_x G(x,y)| \leq C|x-y|^{1-d}, \\ |\nabla^2_x G(x,y)| \leq C|x-y|^{-d},\\ \text{$G(x,y)$ is $C^\infty$ away from the diagonal in $\overline \Omega \times \overline \Omega$.} $$ Question: Is it possible to deduce from these formulas that for any $\varphi \in C^2(\overline \Omega)$ satisfying $L\varphi = 0$ in $\Omega$ we have $$ \varphi(x) = \int_{\partial \Omega} \frac{\partial G(x,y)}{\partial \nu_y}\varphi(y) \, dy, $$ where $\nu_y$ is the unit exterior normal, for all $x \in \overline \Omega$? (for $x \in \partial \Omega$ I mean the limit from $\Omega$).
I hope the answer is positive. In M. Taylor's book "Partial Differential Equations, II", p. 36 the author considers a fundamental solution to the Laplace equation in a bounded region on a manifold (I write "a solution" because I don't see there any specification of boundary values for this solution). Any such solution has a form $$ E(x,y) \sim C(d)|x-y|^{2-d} + \cdots, $$ and this allows the author to show that for the single and double layer potentials, defined for a function $f \in C(\partial \Omega)$ as $$ Sf(x) = \int_{\partial \Omega} E(x,y) f(y) \, dy, \\ Df(x) = \int_{\partial \Omega} \frac{\partial E(x,y)}{\partial \nu_y} f(y) \, dy, $$ we have $$ Sf(x-0) = Sf(x), \quad x \in \partial \Omega, \\ Df(x-0) = \frac 1 2 f(x) + Df(x), \quad x \in \partial \Omega, $$ where $x-0$ means a limit from $\Omega$. Now if we specify additionally that $E(x,y) = 0$ for $x \in \partial \Omega$ or $y \in \partial \Omega$ we must have by the property of the Green function for the Dirichlet problem that $$ Df(x) = \frac 1 2 f(x), \quad x \in \partial \Omega, $$ in order to have $Df(x-0) = f(x)$, $x \in \partial \Omega$. Does this equality really hold?