Asked this on math stack exchange awhile ago but it got ignored then deleted.

To solve a differential equation of one variable, you need constraints equal to the number of derivatives.

For a partial differential equations, they are typically solved with functional constraints equal to the number of derivatives (though I'm not sure on this point). My question is if I have a linear operator $L$, with boundary condition functions $f_1 ... f_n$, is there some generalized way to construct the solution $\Psi$ in terms of the operator and boundary conditions?

The typical problems in partial differential equations involve Fourier transforms to invert the operator in a roundabout way, but it's not clear to me what is the sufficient or necessary information need to solve this more generally.

Typical problems such as:

$\nabla^2 \Psi(x,y,z)=0$

$\Psi(x, y, 0)=f(x, y)$

$\implies \Psi(x,y,z)=\frac{1}{(2\pi)^2}\int dx^{\prime}\int dy^{\prime}\int d k_x \int d k_y \exp\left(-z\sqrt{k_x^2+k_y^2}+ik_x(x-x^{\prime})+ik_y(y-y^{\prime})\right) f(x^{\prime},y^{\prime})$

Require reasoning about the specific problems, and are hard for me to generalize in arbitrary coordinate constraints such as:

$\Psi(x,y,z)=f(x-y,z+y)$

Is there a way to make an 'equivalent source' that encodes all the constraints?

$\nabla^2\Psi(x,y,z)=J(x,y,z)[f]$

$\implies \Psi(x,y,z)={\nabla^2}^{-1}J(x,y,z)[f]$

Or for a general operator:

$\Psi=L^{-1}J[f]$