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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 …

In your simplified case, I don't see how $A(x,0) = 1$. In fact, the overall factor of $t$ should for the solution to vanish for all $x$ at $t=0$. Actually, I think there is no solution to your boundar …

The idea of the weights used in the ADN definition of the principal symbol is quite natural in the context of graded vector spaces or more generally graded modules. In the equation $u = Dv$, $u=(u_1,\ …

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 …

You talk about the non-separability of the $\Box^2 \phi = 0$ equation, which I don't understand. Each plane wave $e^{ik\cdot x} = \prod_{j=0}^{d-1} e^{i k_j x^j}$ is already in separated form with the …

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 …

Recall that a first order system $A_{ij}^\mu \partial_\mu \phi^j + B_{ij} \phi^j = 0$ (the right hand-side could also be inhomogeneous) is symmetric hyperbolic when there exists at least one covector …

Your conjecture is almost correct. But consider the extreme case when your equation has no integrability conditions, so that $\operatorname{coker}(\sigma(\rho_{q+1} P)) = 0$, whence $\rho^{(1)}_{q+1}( …

What you need are some estimates on the norm of the resolvent, $N(\omega,\lambda) = \| (\tilde{L}_\omega - \lambda)^{-1} \|$, as a function of $\omega$. In general, if $N(\omega,\lambda)$ grows too qu …

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 …

Is it not simply a conversion of an iterated homogeneous equation into a non-iterated non-homogeneous equation? By hypothesis, $L_m u = v$, where $L_k v = 0$. Every such $v$ can be obtained from some …

Check out Section II.1.7 of Tikhonov & Samarskii's text Equations of Mathematical Physics for a nice discussion of the physical interpretation of Dirichlet, Neumann and Robin boundary conditions for t …