A relaxed partitioning disk for strictly convex domains (Q2846600)

This is the author's Ph.D. Thesis defended at the Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau, in April 2013. The author obtains some results concerning the connectedness of area minimizing disks for a relaxed isoperimetric problem with respect to a strictly convex body \(\Omega\) in \(\mathbb{R}^3\). Denote by \(S=\partial \Omega\) the supporting manifold for the class \(\mathcal{C} (S)=\{X\in W^{1,2}(D, \mathbb{R}^3): X(w)\in S, \;\text{for. a.e.} \;w\in \partial D\}\) where \(D\subset \mathbb{R}^2\) is the open unit disk. A collection of maps \(\vec {X}\in \mathcal{C}(S,\kappa)\cup \mathcal{C}^+(S,\kappa)\) with nonzero volume \(\kappa\) and \(E(\vec{X})\leq I_S(\kappa)\) is called a disk partitioning surface. An element \(X\in \mathcal{C}(S,\kappa)\) with nonzero volume \(\kappa\) and minimal energy \(E(X)=I_S(\kappa)\) is called a partitioning disk. The main result of this thesis is the Theorem 1.3. At a convex \(C^{3,\alpha}\)-container \(\Omega \simeq B\) with strictly convex boundary \(S\): (1) Simply partitioning disks are the only disk-type partitioning surfaces. The free boundary is regular and meets \(S\) orthogonally from inside, (2) Partitioning disks with small enclosed volume are embedded in \(\overline{\Omega}\) with isoperimetric ratio \(I_S(\kappa)/|\kappa|^{2/3}\sim \sigma_\mathbb{H}:=(18\pi)^{1/3}\). They parametrize the boundary \(\partial E\) of the \(BV\) solution \(E\subset \Omega\). (3) The isoperimetric profile \(I_S^{3/2}\) is a (strictly) concave function of the enclosed volume on \((0,\mathcal{L}^3(\Omega))\). In general we have \(\Omega \simeq B\) for any \(C^{3,\alpha}\) container. (4) Simple partitioning disks with regular boundary are immersions. \textrm The following topics are considered: 2) Analytic regularity, 3) Second variation, 4) Outward maps, 5) Small enclosed volume, 6) Concave profile and simplicity, 7) Branch points. At the end the terms and definitions used in the thesis are shortly presented. The list of references contains 39 titles.