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My question is quite simple: Suppose we are given a PDE of with a boundary condition $$ \Delta u + u^2 =0 $$ where $u=u(r,\theta), 0<r<1$ and $u(1,\theta) = \cos\theta$ with $0 \leq \theta \leq 2\pi$.

Then we may perturb the nonlinear term, getting $$ \Delta u + \epsilon u^2 =0 $$ and look for a solution $$ u(r,\theta;\epsilon) = \sum_{n=0}^{\infty} \epsilon^n u_n. $$ As substituting this into the perturbed PDE and considering the coefficients of the powers of $\epsilon$, we can get $u_0,u_1,...$ and so on with some algebraic work.

Then my question:

1) Can we analytically prove (or disprove) that the constructed $u$ is indeed a solution of the original PDE when taking $\epsilon =1$?

1.1) If so, is there any PDE textbook that deals with this type of matter in a systematic manner?

2) If not, should we prove numerically? Then how?

2.1) And for this numerical treatment, any textbook?

3) In general, where can I learn perturbation method properly in a rigorous setting with analysis?

Note on my knowledge: I'm new to PDE, currently learning by a textbook by Farlow and I know measure theory and some basic functional analysis.

[Duplication: mathoverflow]

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I cannot answer your question in full detail, but here are some thoughts.

  1. To learn perturbation analysis I would suggest to start with simpler objects than nonlinear PDE. An excellent and very readable introduction is given Perturbation Methods, although it does not deal with PDE.
  2. To prove analytically that this is indeed solution we need several theorems, the most important one is the differential dependence of the solution on the parameter, in the realm of nonlinear PDE this is dealt with on a case by cases basis. Then you need to worry about the convergence of the obtained series. And you also need to prove that the original problem has a solution.
  3. To indicate that the matter is complicated here, I point out that the nonlinear problem you are talking about has two real solutions, therefore the approach in Farlow will allow you to find only one of them. Moreover, if you replace your boundary condition with $g(\theta)=A\cos \theta$, then for $A>20.65$ nonlinear problem will have no real solutions, so the series of approximations will be misleading.
  4. This means also that the statement that you can solve the nonlinear problem by using $\epsilon=1$ in Farlow's book is false. If the boundary condition is $g(\theta)=0$ then $\Delta u=0$ has unique trivial solution, and $\Delta u+u^2=0$ has two real solutions.
  5. The examples which I cited are taken from the Russian translation of Farlow's book. At the point where Farlow writes about the "rocket" that will bring you to the solution of the nonlinear problem, the editor of the Russian translation made a comment: "This rocket, first, not for all the boundary conditions will fly to $\epsilon=1$ (where for some boundary condition the world of complex solutions starts), and, second, never fly to the second real solution if it exists".
  6. As a conclusion I would recommend to simply ignore this chapter of Farlow's book (a nice one in many other respects) and start with some other problems, where perturbations can be rigorously analyzed.
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  • $\begingroup$ I really thank for you answer. If you don't mind one further question: In 2, you said I need several theorems (especially ones in Nonlinear PDE). So where can I get those? Could you name some papers or textbooks where I can learn these theorems from? $\endgroup$ – jachilles Nov 7 '14 at 16:30
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    $\begingroup$ @julypraise I am afraid I cannot answer your question with one decisive reference. The problem is that for nonlinear PDE each PDE usually has to be treated separately. My suggestion would be for you to start with ODE and study some perturbation problems there. For ODE the theorems of 1) existence, 2) uniqueness, 2) continuous dependence on parameters are relatively straightforward, and you can see how the method works. Then you switch to PDE with the knowledge what kind of theory to look for. $\endgroup$ – Artem Nov 7 '14 at 16:36

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