Uniqueness of solutions to elliptic PDE in $W^{1,1}$

Question

Consider for definiteness the elliptic problem $$-\nabla \cdot a(x) \nabla u(x) = 0, x \in \Omega$$ with boundary conditions $u|_{\partial \Omega} =0 $, where $\Omega$ is some bounded $C^1$ domain in $\mathbb{R}^n$ and $a \in L^\infty(\Omega).$ The usual results says the only solution in $H^1(\Omega)$ is the zero function. More generally though, we can interpret the above equation distributionally for $u \in W^{1,1}(\Omega)$. The question then is whether there exists a non-zero solution to the above problem for $u \in W^{1,1}_0(\Omega)$?

This question can be thought of as asking for a "true" uniqueness result for elliptic problems. $W^{1,1}(\Omega)$ is the largest space which (to me at least) allows a natural interpretation of the above boundary value problem, hence is a natural setting for asking the question of uniqueness.

Answer 1

The problem above does not have a unique solution in $W^{1,1}(\Omega)$. A non-trivial solution was given by James Serrin in [A] (below).

There is an extensive theory of existence, uniqueness and non-uniqueness of weak solutions for the corresponding non-homogeneous problem: $$(*)\qquad \qquad -\mathrm{div}(a \nabla u)=f,$$ which can help understand why the problem is not an easy one. The source $f$ is usually chosen in Lebesgue spaces ($L^p$) or Marcinkiewicz spaces ($L^{p,\infty}$). Marcinkiewicz spaces are occasionally considered more natural, since the fundamental solution to $\Delta u = \delta_0$ ($n\geq 3$) has characteristic growth in a Marcinkiewicz space, but not in any Lebesgue space.

It turns out that the uniqueness of solutions to $(*)$ can fail depending on the summability of $f$. For example:

for every $1\leq m\leq n/2$ there exists a distributional (duality with $\mathcal C_0^\infty(\Omega)$) solution to $(*)$ satisfying $u\in L^{m^{**}}$ and $|\nabla u|\in L^{m^*}$, but this solution is unique only if $m> (2^*)'$ (Hölder conjugate of the Sobolev exponent $2^*$), in which case one additionally has that $u\in W^{1,2}_0$. (For $m> n/2$ one has directly $u\in W^{1,2}_0\cap L^\infty$.)

This problem led to the introduction of so-called SOLAs (Solutions Obtained as Limits of Approximations). Indeed, it turns out that the solution operator to $(*)$ mapping the source $f$ to the corresponding unique solution $u$ (for $m>(2^*)'$) is continuous even below the critical threshold $(2^*)'$, and thus one can show that SOLAs are unique:

if $f_n\in L^\infty$ $L^m$-converges to $f\in L^{m}$ ($1<m<(2^*)'$), then the corresponding solutions $u_n$ $L^{m^{**}}$-converges to the unique SOLA $u$ to $(*)$ with source $f$.

The other possible solutions to $(*)$ with source $f$ are thus “inaccessible“ by source regularization.

[A] Serrin, James, Pathological solutions of elliptic differential equations, Annali della Scuola Normale di Pisa – Classe di Scienze, 3rd series, vol. 18 (1964), pages 385-387