A FREE BOUNDARY PROBLEM, PROJECTIONS OF RANDOM CANTOR SETS, AND THE GEOMETRY OF CURVES WITH SMALL INTERSECTION WITH MANY LINES

Finding the geometric properties of a set is a very old problem. The present text consists of three chapters where we study such properties with techniques involving Complex and Harmonic Analysis, Probability, and Geometric Measure Theory. We specifically deal with a few considerations of free boundary problems, we calculate the decay rate of the projections of a certain random Cantor set, and we describe the shape of planar graphs which avoid having too many intersections with a positive cone of lines.To begin with, we introduce Schwarz functions; holomorphic functions on open domains $\Omega$ satisfying $S(\zeta)=\overline{\zeta}$ on $\Gamma$, part of $\Omega$'s boundary. Sakai in 1991 gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if $\Omega$ is simply connected and $\Gamma=\partial \Omega\cap D(\zeta,r)$, then $\Gamma$ has to be regular real analytic. Here, we attempt to describe $\Gamma$ when the boundary condition is slightly relaxed. In particular, three different scenarios over a simply connected domain $\Omega$ are treated: when $f_1(\zeta)=\overline{\zeta}f_2(\zeta)$ on $\Gamma$ with $f_1,f_2$ holomorphic and continuous up to the boundary, when $\mathcal{U}/\mathcal{V}$ equals certain real analytic function on $\Gamma$ with $\mathcal{U},\mathcal{V}$ positive and harmonic on $\Omega$ and vanishing on $\Gamma$, and when $S(\zeta)=\Phi(\zeta,\overline{\zeta})$ on $\Gamma$ with $\Phi$ a holomorphic function of two variables. It turns out that the boundary piece $\Gamma$ can be, respectively, anything from real analytic to merely $C^1$, regular except finitely many points, or regular except for a measure zero set.For the second chapter, we consider a model of randomness for self-similar Cantor sets of finite and positive $1$-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands $\delta$-close to a Cantor set of this particular randomness. Two quite different models of randomness for Cantor sets, by Peres and Solomyak, and by Shiwen Zhang, appear to have the same order of decay for the Buffon needle probability: $\frac{c}{\log\frac{1}{\delta}}$. Here, we prove the same rate of decay for a third model of randomness, which asserts a vague feeling that any "reasonable" random Cantor set of positive and finite length will have Favard length of order $\frac{c}{\log\frac{1}{\delta}}$ for its $\delta$-neighbourhood. The estimate from below was obtained long ago by Mattila.In the last chapter, we show the local Lipschitz property for a graph avoiding multiple-point intersection with lines directed in a given cone. The assumption is much stronger than those of the well-known Marstrand's theorem, but the conclusion is much stronger too. Additionally, we find that a continuous curve with a similar property is $\sigma$-finite with respect to Hausdorff length, and we give an estimate on the Hausdorff measure of each "piece".