Application of Calderon-Zygmund theory to Dirichlet problems

Question

Assume $\Omega$ is a bounded open subset of $\mathbb R^n$ with Lipschitz boundary. Let $1< p< \infty$, $f\in L^p(\Omega, \mathbb R^n)$. We want to solve the equation $$\Delta u=\text{div }f$$ in $\Omega$ for $u\in W_0^{1,p}(\Omega)$. In many standard textbooks it's said that this problem is uniquely solvable by Calderon-Zygmund theory and defines a bounded linear operator from $L^p$ to $W_0^{1,p}$, but how to prove it? By Calderon-Zygmund theory, it's easy to find a solution $u$ in $W^{1,p}(\Omega)$, but the difficulty lies in dealing with the boundary condition.

Answer 1

Well, as phrased above this is not uniquely solvable in all cases, Calderon-Zygmund theory notwithstanding. As you observe, there is a close relationship between an inhomogeneous right-hand side and an inhomogeneous Dirichlet problem, using say a global Calderon-Zygmund estimate. The issue is that being on a Lipschitz domain does not ensure that we can easily solve the resulting Dirichlet problem.

To be more precise: when $p = 2$, this is of course solvable. It turns out that if $p$ is close enough to $2$ ($p \in (3/2 - \epsilon, 3 + \epsilon)$ where $\epsilon > 0$ depends on the Lipschitz constant), the required estimate is valid ($ \|u\|_{W^{1, p}_0} \leq C \|f\|_p$). On the other hand, for any $p \notin [3/2, 3]$ there is a Lipschitz domain $\Omega$ and an $f \in C^\infty(\mathbb{R^n})$ such that the (unique, classical) solution $u$ to $\Delta u = f$ with $u = 0$ on $\partial \Omega$ does not belong to $W^{1, p}(\Omega)$. This is the situation in dimension $n \geq 3$; in dimension $2$ replace $[3/2, 3]$ by the slightly better $[4/3, 4]$.

This theorem is due to Jerison and Kenig.

The more common, positive, result that textbooks might mention is that the problem is uniquely solvable if $\Omega$ is smooth enough. For example, $\partial \Omega$ being $C^{1,1}$ is sufficient, and you can find the theory in Gilbarg-Trudinger, Chapter 9. The proof there involves flattening the boundary to get a PDE with sufficiently smooth coefficients, reducing the half-space case to the full-space case with a reflection trick, and proving a Calderon-Zygmund theorem perturbatively for equations with coefficients. Going through this carefully will show that actually $C^1$ is sufficient (for the Laplace equation, possibly not for the more general equations treated there). With even more care and this one can check that for a fixed $p$, a small enough Lipschitz constant depending on $p$ will suffice. Alternatively, this can be obtained from layer potential approaches and is implicit in the Jerison-Kenig paper I linked above.