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Register a new userFree boundary minimal surfaces and overdetermined boundary value problems
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In this paper we establish a connection between free boundary minimal surfaces in a ball in $mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.
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We adapt the viscosity method introduced by Rivi`ere to the free boundary case. Namely, given a compact oriented surface $Sigma$, possibly with boundary, a closed ambient Riemannian manifold $(mathcal{M}^m,g)$ and a closed embedded submanifold $mathcal{N}^nsubsetmathcal{M}$, we study the asymptotic behavior of (almost) critical maps $Phi$ for the functional begin{align*} &E_sigma(Phi):=operatorname{area}(Phi)+sigmaoperatorname{length}(Phi|_{partialSigma})+sigma^4int_Sigma|{mathrm {I!I}}^Phi|^4,operatorname{vol}_Phi end{align*} on immersions $Phi:Sigmatomathcal{M}$ with the constraint $Phi(partialSigma)subseteqmathcal{N}$, as $sigmato 0$, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection $mathcal{F}$ of compact subsets of the space of smooth immersions $(Sigma,partialSigma)to(mathcal{M},mathcal{N})$, assuming $mathcal{F}$ to be stable under isotopies of this space we show that the min-max value begin{align*} &beta:=inf_{Ainmathcal{F}}max_{Phiin A}operatorname{area}(Phi) end{align*} is the sum of the areas of finitely many branched minimal immersions $Phi_{(i)}:Sigma_{(i)}tomathcal{M}$ with $partial_ uPhi_{(i)}perp Tmathcal{N}$ along $partialSigma_{(i)}$, whose (connected) domains $Sigma_{(i)}$ can be different from $Sigma$ but cannot have a more complicated topology. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_sigma$ and, along the way, we simplify several arguments from the original work. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.
We employ min-max techniques to show that the unit ball in $mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,partial M)$ where $3leq (n+1)leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $C^infty$ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in $M$. If $partial M$ is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.
Given a $C^k$-smooth closed embedded manifold $mathcal Nsubset{mathbb R}^m$, with $kge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $partial S eqemptyset$, we consider $frac 12$-harmonic maps $uin H^{1/2}(partial S,mathcal N)$. These maps are critical points of the nonlocal energy begin{equation}E(f;g):=int_Sbig| ablawidetilde ubig|^2,dtext{vol}_g,end{equation} where $widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $frac 12$-energies at each boundary component of $partial S$ (suitably identified with the circle $mathcal S^1$), plus a quadratic term which is continuous in the $H^s(mathcal S^1)$ topology, for any $sinmathbb R$. We show the $C^{k-1,delta}$ regularity of $frac 12$-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichmuller space of $S$.
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