Barriers for a class of geometric evolution problems

Giovanni Bellettini; Matteo Novaga

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Comparison results between minimal barriers and viscosity solutions for geometric evolutions

Giovanni Bellettini, Matteo Novaga (1998)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Uniqueness of minimal projections onto two-dimensional subspaces

Boris Shekhtman, Lesław Skrzypek (2005)

Studia Mathematica

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We prove that minimal projections from $L_{p}$ (1 < p < ∞) onto any two-dimensional subspace are unique. This result complements the theorems of W. Odyniec ([OL, Theorem I.1.3], [O3]). We also investigate the minimal number of norming points for such projections.

On the convergence of minimal boundaries with obstacles

E. Barozzi, I. Tamanini (1985)

Rendiconti del Seminario Matematico della Università di Padova

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Constants of strong uniqueness of minimal norm-one projections

A. Micek (2011)

Banach Center Publications

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In this paper we calculate the constants of strong uniqueness of minimal norm-one projections on subspaces of codimension k in the space $l_{\infty}^{(n)}$ . This generalizes a main result of W. Odyniec and M. P. Prophet [J. Approx. Theory 145 (2007), 111-121]. We applied in our proof Kolmogorov’s type theorem (see A. Wójcik [Approximation and Function Spaces (Gdańsk, 1979), PWN, Warszawa / North-Holland, Amsterdam, 1981, 854-866]) for strongly unique best approximation.

Noninvertible minimal maps

Sergiĭ Kolyada, L&#039;ubomír Snoha, Sergeĭ Trofimchuk (2001)

Fundamenta Mathematicae

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For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and $f^{- 1} (A)$ share with A those topological properties which describe how large a set is. Using...

Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations

Guy Barles, Alessio Porretta (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider a class of stationary viscous Hamilton-Jacobi equations aswhere $λ \geq 0$ , $A (x)$ is a bounded and uniformly elliptic matrix and $H (x, ξ)$ is convex in $ξ$ and grows at most like ${| ξ |}^{q} + f (x)$ , with $1 < q < 2$ and $f \in L^{N / q^{'}} (Ω)$ . Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate, ${(1 + | u |)}^{\bar{q} - 1} u \in H_{0}^{1} (Ω)$ , for a certain (optimal) exponent $\bar{q}$ . This completes the...

Some remarks on uniqueness and regularity of Cheeger sets

V. Caselles, M. Novaga, A. Chambolle (2010)

Rendiconti del Seminario Matematico della Università di Padova

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Induced subsystems associated to a Cantor minimal system

Heidi Dahl, Mats Molberg (2009)

Colloquium Mathematicae

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Let (X,T) be a Cantor minimal system and let (R,) be the associated étale equivalence relation (the orbit equivalence relation). We show that for an arbitrary Cantor minimal system (Y,S) there exists a closed subset Z of X such that (Y,S) is conjugate to the subsystem (Z,T̃), where T̃ is the induced map on Z from T. We explore when we may choose Z to be a T-regular and/or a T-thin set, and we relate T-regularity of a set to R-étaleness. The latter concept plays an important role in the...

Hölder regularity of two-dimensional almost-minimal sets in $ℝ^{n}$

Guy David (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

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We give a different and probably more elementary proof of a good part of Jean Taylor’s regularity theorem for Almgren almost-minimal sets of dimension $2$ in $ℝ^{3}$ . We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor’s result to almost-minimal sets of dimension $2$ in $ℝ^{n}$ , and give the expected characterization of the closed sets $E$ of dimension $2$ in $ℝ^{3}$ that are minimal, in the sense that $H^{2} (E ∖ F) \leq H^{2} (F ∖ E)$ for every closed set $F$ such that there is a bounded set $B$ so...

Cylinder cocycle extensions of minimal rotations on monothetic groups

Mieczysław K. Mentzen, Artur Siemaszko (2004)

Colloquium Mathematicae

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The main results of this paper are: 1. No topologically transitive cocycle $ℝ^{m}$ -extension of minimal rotation on the unit circle by a continuous real-valued bounded variation ℤ-cocycle admits minimal subsets. 2. A minimal rotation on a compact metric monothetic group does not admit a topologically transitive real-valued cocycle if and only if the group is finite.