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Integro-Differential Elliptic Equations
This monograph offers a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. This class of equations finds relevance in fields such as analysis, probability theory, mathematical physics, and in several contexts in the applied sciences. The work gives a detailed presentation of all the necessary techniques, with a primary focus on the main ideas rather than on proving all the results in their greatest generality. The basic building blocks are presented first, with the study of the square root of the Laplacian, and weak solutions to linear equations. Subsequently, the theory of viscosity solutions to nonlinear e...
The obstacle problem
The material presented here corresponds to Fermi lectures that I was invited to deliver at the Scuola Normale di Pisa in the spring of 1998. The obstacle problem consists in studying the properties of minimizers of the Dirichlet integral in a domain D of Rn, among all those configurations u with prescribed boundary values and costrained to remain in D above a prescribed obstacle F. In the Hilbert space H1(D) of all those functions with square integrable gradient, we consider the closed convex set K of functions u with fixed boundary value and which are greater than F in D. There is a unique point in K minimizing the Dirichlet integral. That is called the solution to the obstacle problem.
Analysis at Large
Analysis at Large is dedicated to Jean Bourgain whose research has deeply influenced the mathematics discipline, particularly in analysis and its interconnections with other fields. In this volume, the contributions made by renowned experts present both research and surveys on a wide spectrum of subjects, each of which pay tribute to a true mathematical pioneer. Examples of topics discussed in this book include Bourgain’s discretized sum-product theorem, his work in nonlinear dispersive equations, the slicing problem by Bourgain, harmonious sets, the joint spectral radius, equidistribution of affine random walks, Cartan covers and doubling Bernstein type inequalities, a weighted Prékop...
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Accountability and Regulatory Governance
This collection improves our understanding of the problems associated to accountability in regulatory governance, focusing on audiences, controls and responsibilities in the politics of regulation and through a systematic exploration of the various mechanisms through which accountability in regulatory governance
The Solitary of Juan Fernandez, or the Real Robinson Crusoe
The novel 'The Solitary of Juan Fernandez, or the Real Robinson Crusoe' by X.-B. Saintine unfolds by introducing us to a Scottish woman named Catherine Felton, known as pretty Kitty, who runs an inn called the Royal Salmon in the town of St. Andrew. The story is set in the early 18th century and follows Catherine's life as she navigates the challenges of being an unmarried woman of means in a society that values marriage and domesticity. Eventually, she meets a sailor named Captain Stradling who proposes marriage to her, and she must decide whether to accept his offer or continue running her inn alone. The novel is a fictionalized version of the true story of Alexander Selkirk, a sailor who was marooned on the island of Juan Fernandez for several years and inspired Daniel Defoe's novel 'Robinson Crusoe'.
The Solitary of Juan Fernandez, Or the Real Robinson Crusoe
This young lady, known throughout the neighborhood under the name of pretty Kitty, had contributed not a little, by her personal charms, to the success and popularity of the inn. In her early youth, she had been a lively and piquant brunette, with black, glossy hair, combed over a smooth and prominent forehead, and dark, brilliant eyes, a style of beauty much in vogue at that period. Though tall and slender in stature, she was, as our ancestors would have said, sufficiently en bon point. In fine, Kitty merited her surname, and more than one laird in the neighborhood, more than one great nobleman even,-thanks to the familiarity which reigned among the different classes in Scotland,-had figured occasionally among her customers, caring as little what people might say as did the brave Duke of Argyle, whom Walter Scott has shown as conversing familiarly with his snuff merchant.
Geometric Invariant Theory and Decorated Principal Bundles
The book starts with an introduction to Geometric Invariant Theory (GIT). The fundamental results of Hilbert and Mumford are exposed as well as more recent topics such as the instability flag, the finiteness of the number of quotients, and the variation of quotients. In the second part, GIT is applied to solve the classification problem of decorated principal bundles on a compact Riemann surface. The solution is a quasi-projective moduli scheme which parameterizes those objects that satisfy a semistability condition originating from gauge theory. The moduli space is equipped with a generalized Hitchin map. Via the universal Kobayashi-Hitchin correspondence, these moduli spaces are related to...
