scholarly journals A geometric inequality for convex free boundary hypersurfaces in the unit ball

2017 ◽  
Vol 145 (9) ◽  
pp. 4009-4020 ◽  
Author(s):  
Ben Lambert ◽  
Julian Scheuer
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Geometric Inequality

scholarly journals Free boundary minimal surfaces in the unit ball with low cohomogeneity

2016 ◽  
Vol 145 (4) ◽  
pp. 1671-1683 ◽  
Author(s):  
Brian Freidin ◽  
Mamikon Gulian ◽  
Peter McGrath
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Minimal Surfaces

scholarly journals Regularity and quantification for harmonic maps with free boundary

2017 ◽  
Vol 10 (1) ◽  
pp. 69-82 ◽  
Author(s):  
Paul Laurain ◽  
Romain Petrides
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Harmonic Maps ◽  
Finite Energy ◽  
Riemannian Surfaces

AbstractWe prove a quantification result for harmonic maps with free boundary and finite energy from arbitrary Riemannian surfaces into the unit ball of ${{\mathbb{R}}^{n+1}}$. This generalizes results obtained by Da Lio [1] on the disk.

scholarly journals Large Steklov eigenvalues via homogenisation on manifolds

2021 ◽  
Author(s):  
Alexandre Girouard ◽  
Jean Lagacé
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Lower Bounds ◽  
Upper Bound ◽  
Minimal Surfaces ◽  
Constant Weight ◽  
Steklov Eigenvalue ◽  
Laplace Eigenvalues ◽  
Steklov Eigenvalues ◽  
Steklov Eigenfunctions

AbstractUsing methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of $$8\pi $$ 8 π for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than $$2\pi $$ 2 π . This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois–El Soufi–Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.

scholarly journals An overview of unconstrained free boundary problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140281 ◽  
Author(s):  
Alessio Figalli ◽  
Henrik Shahgholian
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Free Boundary Problems ◽  
Regularity Of Solutions ◽  
Regular Solutions ◽  
Optimal Regularity ◽  
Open Problems ◽  
Boundary Problems ◽  
Matching Problems ◽  
Open Set

In this paper, we present a survey concerning unconstrained free boundary problems of type where B 1 is the unit ball, Ω is an unknown open set, F 1 and F 2 are elliptic operators (admitting regular solutions), and is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.

Mathematical modeling of oscillatory processes with a free boundary

2017 ◽  
Vol 1 (1) ◽  
pp. 102-112
Author(s):  
A.E. Chistyakov ◽  
◽  
E. A. Protsenko ◽  
E.F. Timofeeva ◽  
◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Free Boundary ◽  
Oscillatory Processes

Uniqueness of solution to a free boundary problem from combustion with transport

MAT Serie A ◽  
2001 ◽  
Vol 5 ◽  
pp. 37-41
Author(s):  
Claudia Lederman ◽  
Juan Luis Vázquez ◽  
Noemí Wolanski
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Uniqueness Of Solution ◽  
A Free Boundary Problem
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