I am recently reading Poisson equations in Strauss's Partial Differential Equations: an Introduction. I found that solving the Poisson's equation $\Delta u=f$ in ${\mathbb R}^2$ and ${\mathbb R}^3$ by separating the variables depends heavily on the "shape" of the domain. The geometries should be rectangles, cubes, circles, wedges or annuli, if one wants to use the method of separating variables.
There are two ways to generalize the equations. On the one hand, one may want to see the equations in ${\mathbb R}^n$. One the other hand, it may be natural to ask what will happen if we consider $\Omega\in{\mathbb R}^n$ only to be a bounded domain with smooth boundary or even unbounded without any special geometry.
The generalized PDE above can be summarized as following (take the Dirichlet Problem for example):
Given functions $f$ on $\Omega$ and $g$ on $\partial \Omega$, find a function $u$ on $\overline{\Omega}$ satisfying
$$\Delta u=f\quad\text{on}~\Omega,\qquad u=g\quad \text{on}~ \partial \Omega$$
where $\Omega$ is a domain in ${\mathbb R}^n$ with smooth boundary $\partial\Omega$.
For understanding the more general PDEs, I am trying to read Folland's Partial Differential Equations. After a glimpse at this book, I am totally confused — I don't see the bridge between solving the simpler PDE (as in Strauss's book) and solving the more difficult ones (as in Folland's book, I'll call them advanced PDEs here), which I quoted above.
Here is my first question:
What's the relationship between methods solving simpler PDEs (i.e., with domain of special shape) and those solving the advanced PDEs?
Besides, I cannot summarize the procedure to solve the advanced PDE as I do for the simpler one:
(i) Look for separated solutions of the PDE
(ii) Put in the homogeneous boundary conditions to get the eigenvalues. This is the step that requires the special geometry.
(iii) Sum the series.
(iv) Put in the inhomogeneous initial or boundary conditions.
There are many approaches to the Dirichlet problems, as indicated in Folland's book: Dirichlet's principle, layer potentials, $L^2$ estimates, etc. (I only know the names.) The book sketches yet another approach — on the formal level — using the notion of Green's function.
Here comes my second question:
What are the procedures solving the advanced PDEs (Poisson equations) using Green's function? It seems that the first step is to find the Green's function $G(x,y)$ on $\Omega\times \overline{\Omega}$, and there will be no hope to find the Green's function explicitly unless one has the domain of special shape. (Then how can one solve the PDE?)
Best Answer
Partial answer:
According to Folland's book, the Green's function for the bounded domain $\Omega\subset{\mathbb R}^n$ with smooth boundary $S$ is the real function $G(x,y)$ on $\Omega\times\overline{\Omega}$ determined by the following properties:
i. $G(x,\cdot)-N(x,\cdot)$ is harmonic on $\Omega$ and continuous on $\overline{\Omega}$, where $N$ is defined as following: $$N(x,y)=\tilde{N}(x-y)$$ where $$\tilde{N}(x)=\frac{|x|^{2-n}}{(2-n)\omega_n}\quad (n>2);\qquad \tilde{N}(x)=\frac{1}{2\pi}\log|x|\quad(n=2).$$ ii. $G(x,y)=0$ for each $x\in\Omega$ and $y\in S$.
As the book says, if we can find the Green's function, we obtain simple formulas for the solution of the Dirichlet problem. So the first job to "solve" the PDE is constructing the Green's function on $\Omega$ which can be written "explicitly" when $\Omega$ is of the "special shape". The existence of the Green's function is guaranteed by the following theorem:
Once we find the Green's function, we can write the solution in terms of integral.
Thanks to Willie's comment, the phrase "solving a PDE" is somehow inaccurate. In Folland's book, I think, the author mainly means the existence of solutions of a PDE when talking about "solving" PDEs.