Structure of sets which are well approximated by zero sets of harmonic polynomials

Abstract

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how "degree k points" sit inside zero sets of harmonic polynomials in R n of degree d (for all n ≥ 2 and 1 ≤ k ≤ d) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree k points" (k ≥ 2) without proving uniqueness of blowups or aid of PDE methods such as monotonicity formulas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of k. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro. 18 6. Dimension bounds in the presence of good topology 20 7. Boundary structure in terms of interior and exterior harmonic measures 27 Appendix A. Local set approximation 29 Appendix B. Limits of complimentary NTA domains 35 References

Nodal sets for broken quasilinear partial differential equations with Dini coefficients

“…In connection to our structure theorem (Theorem 1.2), we would also like to address a recent article [16] that studies the structure of a set that can be approximated by the nodal sets of harmonic polynomials. We believe that our nodal/singular set can also be analysed by their approach, as our solution after adjusting the break (i.e., u þ À jðzÞu À ) can be approximated by homogeneous harmonic polynomials (Lemma 3.3 and Lemma 5.8) at each vanishing point.…”

Geometric Measure Theory–Recent Applications

“…The proof of Theorem 1 uses tools from the theory of non-tangentially accessible domains (NTA) introduced by Jerison and Kenig [37], the monotonicity formula of Alt, Caffarelli, and Friedman [2], the theory of tangent measures introduced by Preiss [56], and the blow up techniques for harmonic measures at infinity for unbounded NTA domains due to Kenig and Toro [40,41]. For additional results along these lines see [11][12][13][14]29].…”

Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data

“…(i) There exist d 0 ≥ 1 depending on at most n and the NTA constants of Ω + and Ω − such that ∂Ω is locally bilaterally well approximated (in a Reifenberg sense) by zero sets of harmonic polynomials p : [BET17] (also see [BL15]). In fact, we proved in [BET17] that (ii) and (iii) hold on any closed set satisfying the bilateral approximation in (i).…”