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

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

scholarly journals Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

2021 ◽  
Author(s):  
D. Breit ◽  
A. Cianchi ◽  
L. Diening ◽  
S. Schwarzacher
Keyword(s):  
Global Regularity ◽  
Regularity Theory ◽  
Divergence Form ◽  
Dirichlet Problems ◽  
Schauder Estimates ◽  
First Order ◽  
Right Hand ◽  
Regularity Results ◽  
The Right ◽  
Linear Setting

AbstractAn optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the p-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $$\text {BMO}$$ BMO and $${{\,\mathrm{VMO}\,}}$$ VMO spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $$p=2$$ p = 2 , and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.

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