Well-posed Elliptic boundary value problems

Question

I've been taught that elliptic boundary value problems (BVPs) are well-posed i.e. a solution exists, it is unique, and it depends continuously on the boundary data. Why then does the following 2d elliptic BVP $$ \nabla^{2}f=-k^{2}f\space\space\space\space\space r<a,\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space f=0\space\space\space\space\space r=a $$ have the solution $f\propto J_{0}(kr)$ whenever $ka$ is a zero of the Bessel function. This Bessel function is well behaved at the origin, so why does this not violate the uniqueness of elliptic BVPs? Thanks in advance for any help.

Answer 1

I've been taught that elliptic boundary value problems (BVPs) are well-posed i.e. a solution exists, it is unique, and it depends continuously on the boundary data.

This is not true. What we instead have is the Fredholm alternative; consider for instance an second order elliptic operator of the form $$ Lu = -\operatorname{div} A \nabla u + B \nabla u + cu. $$ We will assume the coefficients are suitably regular and that $A$ is suitably elliptic in a bounded domain $\Omega \subset \Bbb R^n$ with nice boundary.

Then the Fredholm alternative asserts that for boundary value problems of the form below: $$\begin{cases} Lu = -\operatorname{div} A \nabla u + B \nabla u + cu = f & \text{ in } \Omega, \\ u = g & \text{ on } \partial \Omega, \end{cases} $$ We have exactly one of the following holds:

• A unique solution $u$ exists for each $f$ and $g.$
• There exists a non-trivial solution to the above with $f=0$ and $g=0.$ Moreover the set of such solutions forms a finite dimensional vector space.

Note I'm being a bit vague about which function space I'm working with; you can take everything (including coefficients of $L$ and the boundary) to be smooth and classical for instance.

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The reason this holds is because if we consider only the leading order term $$ L_0u = -\operatorname{div} A \nabla u, $$ then boundary value problems for $L_0$ is always well-posed. However in suitable scales ($H^s$ or $C^{k,\alpha}$ for instance) the addition of the extra lower order terms introduces a compact perturbation, so the abstract Fredholm alternative applies.

You can find a detailed discussion in any elliptic PDE text; for second order equations you can see for instance Evan's PDE book (Chapter 6) or Gilbarg & Trudinger (Sections 6.3, 8.1).