I know too little about solutions for PDEs in general, so I would be grateful if anyone has any idea if there is a solution.
I am trying to find a function $h$ of $n$ coordinates defined on a cube of the form $U=(-\varepsilon,\varepsilon)^n$ satisfying the following PDE
$$\partial_{ss}^2h-\sum_k a^k\partial_kh+\sum_{i,j\neq s}\alpha^i\alpha^j\left(\partial_{ij}^2h-\sum_{k}b^k\partial_kh\right)=f,$$ where $a^k$, $b^k$ and $\alpha^i$ are of class $C^\infty$ and $s$ is a fixed coordinate. We also have $a^k(0)=b^k(0)=0$.
Now I state my wild conditions:
- Typically, I need $f$ to be a smooth bump function such that $f(0)=1$.
- $h(0)=0$.
- $\partial_k h(0)=0$ for all $k=1,...,n$.
- $\partial_{ij}^2h(0)=0$, for all $(i,j)\neq(s,s)$.
- $h|_{\partial U}=0$.
I may be asking too many conditions on $h$, but even if I drop some conditions I don't know enough about PDE to have a clue if there exists such a function, but it is linear and with very regular coefficients, so I hope there is some theorem that implies existence. I would appreciate any hint or suggestion.
A PDE of second order is characterized by its symbol, the monomial form of the highest derivatives replaced by vector components in the sense of a Fourier transform.
With a product metric $g_{ij} dx^i dx^j = (\alpha_i \ dx^i ) \ ( \alpha_j dx^j) = dy^i dy^j $ and undefined signature this is no well defined PDE.
If $s$ is supposed to be a time variable and the space part matrix of gthe metrics $g_{ij}$ is positive definite, then its a wave equation; a well defined problem by spatial boundary conditions and two start conditions, that provide the field and its velocity, say, at s=0.