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It is related to the classic theory of PDE.

Suppose that $U \subset \mathbb{R}^m$ is an open, connected and bounded set, $f : U \to \mathbb{R}$ and $g:\partial U\rightarrow \mathbb{R}$ and $c:\overline{U} \to \mathbb{R}$, with $f, g, c$ continuous functions. Let be following problem

$${}\begin{cases} \Delta v +c(x) v= f& in &U\\ \frac{\partial v}{\partial \nu}=g & in &\partial U\\ \end{cases}$$

where $\nu$ is the outer normal exterior. Prove that, under suitable assumptions, this problem can have infinitely many solutions in $C^2(U)\cap C^1(\overline{U})$. In the solution, I must use steps one and two below in order to obtain infinite solutions for the problem above!!

My attempt to solve was based on three steps:

  1. Replace the hypothesis in the problem above for the corresponding Dirichlet problem.

  2. I applied the Schauder Theorem of existence to prove that exists a unique solution.

  3. Now I am struggling to conclude that from steps one and two I will obtain infinite solutions for the problem in this question.

Would you give me some hints? Someone told me that I have to consider the Fredhom Alternative with a fixed point different from identity, but, I couldn't get this.

Rodrigo de Azevedo
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Silvinha
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  • There is a boundary (i.e. ${\alpha, \beta} \times \mathbb{R}^{n-1}$). The problem assumes a maximum exists, so it suffices to exclude interior maxima. – fwd Oct 31 '23 at 21:25
  • Yes, $U$ is not bounded, and neither is it's boundary. $U$ is an infinite strip, but that shouldn't be a problem because we are told a priori that a maximum exists on $\overline{U}$ – fwd Oct 31 '23 at 21:35
  • Please clarify the symbols:

    $\overline{U}, C^2 (U), C(\overline{U}), \Delta f.$

    The closure, the set of continuously twice differentiable functions, the set of continuous functions ...?

     – Amir Oct 31 '23 at 21:53
  • There is a general result that the maximum of a convex function over a closed convex set can be found on the boundary of the convex set (if there is any). Your problem statement may need some changes. – Amir Oct 31 '23 at 22:06
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    Google "concave minimization"; for example, see the following book chapter:

    https://link.springer.com/chapter/10.1007/978-1-4615-2025-2_3

     – Amir Oct 31 '23 at 22:10
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    The very first sentence of the question does not a good title make. It would be much more useful to have the title give a brief overview of the spirit of the question – FShrike Oct 31 '23 at 22:18
  • I don't know any pde theory, sorry – FShrike Nov 04 '23 at 11:53
  • @FShrike oh, ok. Thanks for answer back – Silvinha Nov 04 '23 at 22:15
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    If you have two solutions $v$ and $w$, then $u=v-w$ satisfies your problem with $f=0$ and $g=0$. If you assume that $\int_U c(x)=0$, then any constant function satisfies this problem with $f=0$ and $g=0$. Hence, if you find one solution $v$ then any function $v+const$ will be a solution. I don't know how to use Steps 1 -3 to find a solution. Where did you get this problem? What are you allowed to use? – Gio67 Nov 06 '23 at 14:04
  • @Gio67 It is a pleasure to see that you came here. Well, maybe is the translation. The original is: Hint: Set under which conditions in U, f, and c it is possible to apply Schauder's Theorem in problem (P), replacing the Neumann condition with the corresponding Dirichlet condition. Justify why this procedure generates infinite solutions. (I do not know where the questions is from, because I took from an past exam made for a professor) – Silvinha Nov 06 '23 at 17:19
  • @Gio67 I forgot to say that I am searching for classic solutions, not weak solutions... That means that I am only allowed to use classic theory, – Silvinha Nov 07 '23 at 15:10

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