Regularity for almost minimizers with free boundary

In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_\Omega |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(\Omega)$. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see \cite{AC}, \cite{ACF}, \cite{CJK}, \cite{W}). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers

Free boundary regularity for harmonic measures and Poisson kernels

One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said to be doubling with constant C if C times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center. We prove that if the doubling constant of a measure on \R^{n+1} is close to the doubling constant of the n-dimensional Lebesgue measure then its support is well approximated by n-dimensional affine spaces, provided that the support is relatively flat to start with. Primarily we consider sets which are boundaries of domains in \R^{n+1}. The n-dimensional Hausdorff measure may not be defined on the boundary of a domain in R^{n+1}. Thus we turn our attention to the harmonic measure which is well behaved under minor assumptions. We obtain a new characterization of locally flat domains in terms of the doubling properties of their harmonic measure. Along these lines we investigate how the ``weak'' regularity of the Poisson kernel of a domain determines the geometry of its boundary.Comment: 85 pages, published version, abstract added in migratio

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

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 $\mathbb R^n$ of degree $d$ (for all $n\geq 2$ and $1\leq k\leq d$) 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" ($k\geq 2$) 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.Comment: 40 pages, 2 figures (v2: streamlined several proofs, added statement of Lojasiewicz inequality for harmonic polynomials [Theorem 3.1]

Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions

In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure $\omega^+$ of a domain $\Omega=\Omega^+\subset\mathbb{R}^n$ and the harmonic measure $\omega^-$ of $\Omega^-$, $\Omega^-=$int$(\Omega^c)$, in dimension $n\ge 3$