scholarly journals Area‐Minimizing Currents mod 2 Q : Linear Regularity Theory

2020 ◽  
Author(s):  
Camillo De Lellis ◽  
Jonas Hirsch ◽  
Andrea Marchese ◽  
Salvatore Stuvard
Keyword(s):  
Regularity Theory ◽  
Linear Regularity ◽  
Area Minimizing

scholarly journals Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Siran Li
Keyword(s):  
Banach Space ◽  
Discrete Geometry ◽  
Thin Shell ◽  
Euclidean Geometry ◽  
Convex Geometry ◽  
Regularity Theory ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Space Theory ◽  
Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

scholarly journals Regularity theory for nonlinear systems of SPDEs

2014 ◽  
Vol 146 (3-4) ◽  
pp. 329-349 ◽  
Author(s):  
Dominic Breit
Keyword(s):  
Nonlinear Systems ◽  
Regularity Theory

Discs area-minimizing in mean convex Riemannian n-manifolds

2021 ◽  
pp. 1-22
Author(s):  
Ezequiel Barbosa ◽  
Franciele Conrado
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Higher Dimensions ◽  
Dimensional Torus ◽  
Rigidity Result ◽  
Equality Case ◽  
Totally Geodesic ◽  
Zero Degree ◽  
Area Minimizing ◽  
Mean Convex

In this work, we consider oriented compact manifolds which possess convex mean curvature boundary, positive scalar curvature and admit a map to $\mathbb {D}^{2}\times T^{n}$ with non-zero degree, where $\mathbb {D}^{2}$ is a disc and $T^{n}$ is an $n$ -dimensional torus. We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed discs whose boundaries are homotopically non-trivial curves. We also prove a rigidity result for the equality case when the boundary is strongly totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in (2015, J. Geom. Anal., 25, 1001–1017) to higher dimensions.

Concentrating standing waves for Davey–Stewartson systems

2021 ◽  
pp. 1-40
Author(s):  
Yi He ◽  
Xiao Luo
Keyword(s):  
Water Waves ◽  
Elliptic Equations ◽  
Standing Waves ◽  
Regularity Theory ◽  
Local Minima ◽  
Shallow Water Waves ◽  
Penalization Method ◽  
Finite Set ◽  
New Ideas ◽  
Concentrating Solutions

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

Selected Papers on Regularity Theory

2015 ◽  
pp. 143-241
Author(s):  
Šárka Nečasová ◽  
Milan Pokorný ◽  
Vladimír Šverák
Keyword(s):  
Regularity Theory
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