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Let me start by writing $1/U$ instead of $U$. The case when $U$ or $1/U$ vanishes (or $V$ vanishes) would require special handling of course. Then your constraint is $$ \frac{\omega_u}{\omega_v} = \fr …

Have a look at Ringström's The Cauchy Problem in General Relativity (EMS 2009). He spends several chapters building up the analytical material of the kind that you are asking about.

For normally hyperbolic operators (those whose principal symbol is the same as for the wave operator, but possibly acting on a vector bundle, the theory of fundamental solutions/Green functions (as di …

Section 4 of the following paper considers in some detail the 2D wave equation ($\partial_x\partial_y f = 0$ in your coordinates; not exactly the same but closely related) on compact domains with smoo …

I will add a pessimistic answer. You are right that Choque-Bruhat's (and any related local-in-time) existence result only guarantees that the solution metric exists and is sufficiently regular (includ …

I believe that a zero eigenvalue of $A(u)$ implies that the vector field $\partial_t$ is a characteristic direction. For instance, the equation $u_s + u_y = 0$ takes the form $u_t=0$ after switching t …

Consider a determined linear system of differential order $k$ of the form $\sigma^{i_1\cdots i_k}_{ab}(x) \partial_{i_1} \cdots \partial_{i_k} u^b(x) + l.o.t = 0$. The coefficients $\sigma^{i_1\cdots …

To my knowledge, the principal symbol of a non-linear differential operator is not discussed very often. When I have seen it discussed, the definition basically coincided with your approach 1. For exa …

A pretty extensive reference on Huygens' principle is [G] Günther, Paul MR 946226 Huygens’ principle and hyperbolic equations, Perspectives in Mathematics ISBN: 0-12-307330-8. First, some definition …

It's difficult for me to check the details at the moment, but the book Non-Homogeneous Boundary Value Problems and Applications by Lions and Magenes might also be helpful, perhaps more so for elliptic …

Using your coordinates, we can define a Lorentzian metric $g = - dt^2 + \frac{1}{\alpha(t)} dx^2$. Then your equation takes the form \begin{equation} \square u + v^i \partial_i u + \gamma u = h , \e …

I think you are asking about the possibility of satisfying the desired boundary conditions by each solution of the Helmholtz equation in the product form $X(x) Y(y)$, where $X(x)$ an $Y(y)$ were obtai …