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 $f_1,\dots, f_n$, 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:
$$
\begin{cases}
\nabla^2\Psi(x,y,z)=0\\
\\
\Psi(x, y, 0)=f(x, y)
\end{cases}\\
$$
whose solution has the form
$$
\begin{split}
\implies \Psi(x,y,z) &=\frac{1}{(2\pi)^2}\int dx^{\prime}\int dy^{\prime}\int d k_x \int d k_y\\
&\qquad\exp\left(-z\sqrt{k_x^2+k_y^2}+ik_x(x-x^{\prime})+ik_y(y-y^{\prime})\right)
f(x^{\prime},y^{\prime}),
\end{split}$$
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 construct an 'equivalent source' term that encodes all the constraints such as
$$
\begin{split}
\nabla^2\Psi(x,y,z)&=J(x,y,z)[f]\\
\implies \Psi(x,y,z)&=\big({\nabla^2}\big)^{-1}J(x,y,z)[f]
\end{split}\quad?
$$
Or for a general operator,
$$
\Psi=L^{-1}J[f]\quad?
$$