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

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    $\begingroup$ For a general linear differential operator, we don't even know if there are local solutions. $\endgroup$
    – Ben McKay
    Apr 15, 2022 at 15:48
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    $\begingroup$ A simple, but case-by-case approach: Extend your solution $\Psi$ by 0 outside your domain. The derivatives in $L[\Psi]$ produce $\delta$ functions and their derivatives on the boundary. The structure of the coefficients of these $\delta$ functions encode the boundary conditions and the boundary data. Pick a source term $J$ that has $\delta$ functions on the boundary, whose coefficients have the same structure as the above $L[\Psi]$. Then solve $L[\Psi] = J$ with the only boundary condition being that vanishes outside your domain. Keywords: Duhamel's Principle, Multi-Layer Potentials. $\endgroup$ Apr 16, 2022 at 11:01
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    $\begingroup$ It depends on what your assumptions on $L$ are. For example, if $L$ is linear and has constant coefficients, then the existence of $L^{-1}$ was proved by Leon Ehrenpreis. References can be found here: mathoverflow.net/questions/186779/… $\endgroup$
    – Deane Yang
    Apr 16, 2022 at 16:30

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