Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis. The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment. This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the one-variable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several variables. The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces. The book contains over 200 exercises.

A First Course in Sobolev Spaces

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation (−Δ)^sQ+Q−Q^(α+1)=0inR,
( − Δ ) s Q + Q − Q α + 1 = 0 i n R , where 0 < s < 1 and 0 < α < 4s/(1−2s) for s<12 s < 1 2 and 0 < α < ∞ for s≥12 s ≥ 1 2 . Here (−Δ)^s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s=12 s = 1 2 and α = 1 in [5] for the Benjamin–Ono equation.
As a technical key result in this paper, we show that the associated linearized operator L_+ = (−Δ)^s +1−(α+1)Q^α is non-degenerate; i.e., its kernel satisfies ker L_+ = span{Q′}. This result about L_+ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.

/pdf/uniqueness-of-non-linear-ground-states-for-fractional-2yfuffqk3l.pdf

Uniqueness of non-linear ground states for fractional Laplacians in ${\mathbb{R}}$

Omar Antoĺın Camarena pointed out a gap in the proofs in [BT&G, Bro06] of a condition for the Phragmen–Brouwer Property not to hold; this note gives the correction in terms of a result on a pushout of groupoids, and some additional background.

https://gendocs.ru/docs/16/15764/conv_1/file1.pdf

Topology and Groupoids

Topology: A Geometric Account of General Topology Homotopy Types and the Fundamental Groupoid

The Mizar system is one of the pioneering systems aimed at supporting mathematical proof development on a computer that have laid the groundwork for and eventually have evolved into modern interactive proof assistants. We claim that an important milestone in the development of these systems was the creation of organized libraries accumulating all previously available formalized knowledge in such a way that new works could effectively re-use all previously collected notions. In the case of Mizar, the turning point of its development was the decision to start building the Mizar Mathematical Library as a centrally-managed knowledge base maintained together with the formalization language and the verification system. In this paper we show the process of forming this library, the evolution of its design principles, and also present some data showing its current use with the modern version of the Mizar proof checker, but also as a rich corpus of semantically linked mathematical data in various areas including web-based and natural language proof presentation, maths education, and machine learning based automated theorem proving.

/pdf/the-role-of-the-mizar-mathematical-library-for-interactive-3ko53mzzjl.pdf

The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar

(1984). The Jordan Curve Theorem Via the Brouwer Fixed Point Theorem. The American Mathematical Monthly: Vol. 91, No. 10, pp. 641-643.

Ryuji Maehara

Papers

The Jordan Curve Theorem Via the Brouwer Fixed Point Theorem