Well, since no one has given a proper answer I will describe the regularity for the two equations I am familiar with. Maybe someone else knows how to answer in the case of the wave equations. For the sake of simplicity, lets assume Dirichlet boundary conditions in both cases, although this does not really affect the answer.
Poisson's Equation $\Delta u = f$:
This is an elliptic PDE. The usual way to attack the regularity of these is to use the Elliptic Regularity Theorem. A very basic form of this says, for example:
If $\Omega$ is a "nice" open set in $R^n$, $f \in H^k(\Omega)$ and $u$ satisfies $\Delta u = f$, then $u \in H^{k+2}(\Omega)$.
This implies a variety of generalizations to more general equations. For example, if $\Psi \in L^{\infty}(\Omega)$, $f \in L^2(\Omega)$ and $u \in L^2(\Omega)$ satisfies $\Delta u + \Psi u = f$, then we know that $\Delta u = f - \Psi u$. But the RHS is in $L^2(\Omega)$ by the assumptions on $u$, $f$ and $\Psi$. Thus $u \in H^2(\Omega)$. Also notice that the need to know a priori that $u \in L^2(\Omega)$ is not really a restriction since it is trivial by Lax-Milgram that $u \in H^1(\Omega)$. Moreover, if $f \in C^{\infty}(\Omega)$, then $f \in H^k(\Omega)$ for every $k > 0$. Thus if $\Delta u = f$ we know that $u \in H^{k+2}$ for every $k > 0$. Then the Sobolev Embedding Theorem implies that $u$ is also smooth. In general, you should expect solutions to be classical if $f \in H^k(\Omega)$ with $k + 2$ large enough so that the Sobolev Embedding implies that $u \in C^2_b(\Omega)$ These are the type of things to keep in mind when dealing with elliptic PDE.
Heat equation $u_t = \Delta u$, $u(0, x) = f(x)$:
This is a prime example of an evolution problem (it is also parabolic). These tend to have the property that the operator on the RHS ($\Delta$ in this case) generates a continuous (sometimes even analytic) semigroup (I recommend Roger & Renardy's book for an accessible introduction). It turns out that $\Delta$ generates an analytic semigroup $e^{t\Delta}$ on $L^2(\Omega)$ so the time regularity comes pretty much for free. Moreover, if $T(t)$ is an analytic semigroup with generator $A$ then $T(t)$ maps into the domain of $A^k$ for every $k > 0$. In the case of $A = \Delta$, this means that $e^{t\Delta}$ maps into $H^k(\Omega)$ for every even $k > 0$. Thus by applying the Sobolev Embedding Theorem, $e^{t\Delta}$ maps into $C^{\infty}(\Omega)$. Therefore, for every $f\in L^2(\Omega)$, the solution $u(t) = e^{t\Delta}f$ is a strong solution on $(0, \infty) \times \Omega$ and is immediately smoothed. So for this type of equation you get both existence and regularity if you can show that the operator on the RHS generates an analytic semigroup. A similar approach works for the Schrodinger equation, which is hyperbolic, so the main thing to notice here is the "evolution form" $u_t = Au$. So for these the solution will almost always be classical
I have never studied the wave equation, so in this case I have no idea.
You can pick the space of test functions $v$ to be the space $$H^1_{\Gamma_1}(U) = \{v \in H^1(U)\colon v = 0 \text{ on } \Gamma_1 \}, $$
which is a Hilbert space with respect to the inner product in $H^1$. Actually, you can verify that in $H^1_{\Gamma_1}$ the norm $$ \|v\|_{H^1_{\Gamma_1}(U)} = \|Dv\|_{L^2(U)}$$ is equivalent to the $H^1$ norm (by Poincaré inequality). Then the second integral equals $0$ since $v \in H^1_{\Gamma_1}(U).$
Best Answer
I believe the papers by Gary Lieberman will answer your question: one is Mixed boundary value problem for elliptic and parabolic differential equations of second order. Another one is Optimal holder regularity for mixed boundary value problems. Thanks