A singular integral approach to a two phase free boundary problem (Q2814415)

A fundamental question at the intersection of geometric measure theory and boundary value problems is to relate the regularity of the boundary of a domain \(\Omega \subset \mathbb{R}^{n+1}\) to the regularity of the Poisson kernel \(k\) (the kernel of the integral operator that maps a Dirichlet datum on \(\partial \Omega\) to the corresponding solution of the Laplace equation on \(\Omega\)). Assuming that \(\partial \Omega\) has certain geometric measure theoretic properties, one of the most classical result in the field is the fact that the unit normal \(\nu\) to \(\partial \Omega\) is in \(C^{\alpha}\) (for some \(\alpha \in (0,1)\)) if and only if \(\log k\) is. The direct part was proven in [\textit{O. D. Kellogg}, Foundations of potential theory. York: Springer-Verlag 1967 (1967; Zbl 0152.31301)], while the reverse direction was proven in the milestone paper [\textit{H. W. Alt} and \textit{L. A. Caffarelli}, J. Reine Angew. Math. 325, 105--144 (1981; Zbl 0449.35105)] and its improvement [\textit{D. Jerison}, Colloq. Math. 60/61, No. 2, 547--568 (1990; Zbl 0732.35025)].NEWLINENEWLINEBelow the Hölder regularity threshold, this result has analogues with \(C^{\alpha}\) replaced by \(VMO\). In dimension \(n=1\), this goes back to [\textit{Ch. Pommerenke}, Math. Ann. 236, 199--208 (1978; Zbl 0385.30013)]. In higher dimensions, such results have been obtained in an impressive series of papers by \textit{C. E. Kenig} and \textit{T. Toro}, starting with [Ann. Math. (2) 150, No. 2, 369--454 (1999; Zbl 0946.31001)]. The key problem is to determine exactly which geometric measure theoretic properties of \(\partial \Omega\) are needed in these results.NEWLINENEWLINEGiven the current state of knowledge, the natural conjecture is that the appropriate properties are ADR (Ahlfors-David regularity) and two sided NTA (Non-tangentially accessible). Very roughly speaking, ADR means that the restriction of the \(n\) dimensional Hausdorff measure to \(\partial \Omega\) scales approximately like the \(n\) dimensional Lebesgue measure. NTA is a more geometric condition introduced in [\textit{D. S. Jerison} and \textit{C. E. Kenig}, Adv. Math. 46, 80--147 (1982; Zbl 0514.31003)] that guarantees that points on \(\partial \Omega\) can be approached through a region of \(\Omega\) similar to the cones used in the context of Lipschitz domains. Two sided NTA means that both \(\Omega\) and \(\Omega_{\mathrm{ext}}:=\mathbb{R}^{n+1} \backslash \overline{\Omega}\) are NTA. Precise definitions are nicely recalled in Section 1.1 of the paper under review.NEWLINENEWLINEThis conjecture is still open at the time of writing. However, the following slightly weaker result has been obtained in [\textit{C. Kenig} and \textit{T. Toro}, J. Reine Angew. Math. 596, 1--44 (2006; Zbl 1106.35147)]. Assuming ADR and two sided NTA, if the Poisson kernels \(k_{1}\) and \(k_{2}\) corresponding, respectively, to \(\Omega\) and \(\Omega_{\mathrm{ext}}\), are such that \(\log k_{1}, \log k_{2} \in\mathrm{VMO}\), then \(\nu \in \mathrm{VMO}_{\mathrm{loc}}\). The paper under review gives a new proof of this result.NEWLINENEWLINEThe new proof is interesting, and relies on two key ingredients: \(L^{p}\) estimates on the non-tangential maximal function of the gradient of the single layer potential, plus jump relations relating the values of this gradient when one approaches \(\partial \Omega\) through non-tangential approach regions in \(\Omega\) or in \(\Omega_{\mathrm{ext}}\). Both ingredients rely on the fact that, under the ADR and two sided NTA condition, \(\partial \Omega\) is uniformly rectifiable, and therefore the corresponding Riesz transforms are \(L^{p}\) bounded.NEWLINENEWLINEBeyond the insight given by the authors' new approach to Kenig-Toro's result, expert readers will also appreciate the fact that the ADR and two sided NTA condition is slightly weakened (in the case where the Poisson kernels have finite poles) to uniform rectifiability combined with an assumption stating that the difference between \(\partial \Omega\) and its measure theoretic counterpart \(\partial_{*}\Omega\) has \(n\)-dimensional Hausdorff measure \(0\).