Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem (Q6570533)

In this paper, the authors address the Radon-Nikodym derivative \(h=\frac{d\omega^-}{d\omega^+}\) of harmonic measures, giving examples of domains with \(\log h\in C(\partial \Omega)\) whose boundaries have points with non-unique blow-ups.\N\NMore precisely, let \(\Omega^+=\Omega\subset\mathbb{R}^n\) and \(\Omega^-=\mathbb{R}^n\setminus\overline{\Omega}\) be complementay unbounded domains. Assume that \(\Omega^{\pm}\) belong to the class of NTA domains in the sense of \textit{D. S. Jerison} and \textit{C. E. Kenig} [Adv. Math. 46, 80--147 (1982; Zbl 0514.31003)], and let \(\omega^{\pm}\) be harmonic measures on \(\Omega^{\pm}\) with finite poles \(X^{\pm}\) or with poles at infinity. Assume \(\omega^+\ll \omega^-\ll \omega^+\). \textit{C. Kenig} and the third author [J. Reine Angew. Math. 596, 1--44 (2006; Zbl 1106.35147)] and the first author [Rev. Mat. Iberoam. 27, No. 3, 841--870 (2011; Zbl 1242.28003)] allow one to conclude that if \(\log h\in \text{VMO}(d\omega^+)\) or \(\log h\in C(\partial \Omega)\), then\N\[\N\partial\Omega=\Gamma_1\cup\ldots\cup \Gamma_{d_0},\N\]\Nwhere geometric blow-ups of \(\partial\Omega\) centered at \(Q\in\Gamma_d\) are zero sets \(\Sigma_p\) of homogeneous harmonic polynomials \(p\) of degree \(d\). Following [the second author, Ann. Sci. Éc. Norm. Supér. (4) 49, No. 4, 859--905 (2016; Zbl 1366.35245); the first author et al., Rev. Mat. Iberoam. 36, No. 5, 1375--1408 (2020; Zbl 1460.31012)], one knows that if \(\log h\in C^{0,\alpha}\), then blow-ups are unique. In this paper, the authors present examples where \(\log h\in C(\partial \Omega)\) and the boundary has non-unique blow-ups.\N\NThe examples arise from oscillating or rotating slowly a blow-up limit by an infinite amount. The domains constructed have locally finite perimeter and Ahlfors regular boundaries. Moreover, the boundaries of the domains are smooth outside of a single point. The key idea consists of starting with a blow-up domain \(\Omega_p^{\pm}=\{X\in\mathbb{R}^n \ : \ \pm p(X)>0\}\) associated to a harmonic homogeneous polynomial \(p\) of degree \(d\) with \(\log h\equiv 0\) and \(0\in\Gamma_d\). The domain is deformed near the origin by introducing rotations/oscillations at each small scale. This is done in such a way that the magnitude of the oscillation at scale \(r\) vanishes as \(r\rightarrow 0\).