부산대학교 도서관

This book is a concise collection of the main results known about fractional Sobolev spaces. The authors first present some well-known facts about standard Sobolev spaces before extending them to their fractional companions. Very often, they also explore various directions for these extensions, which makes this monograph a very complete guide for researchers working in this area. \par Chapter 1 is a reminder of all important facts used all along the book. Notation is also introduced. It is organised in five sections: \roster \item"$\bullet$" Integration, which mainly contains the convergence results from measure theory; \item"$\bullet$" Banach spaces, with facts concerning their dual, singular maps, convexity, the approximation property and the gauge function; \item"$\bullet$" Function spaces, which first presents fundamental facts about the spaces of continuous functions, $k$-times continuously differentiable functions, and Hölder functions. The regularity of the boundary of an open set is also considered. The Morrey and Campanato spaces are then defined. Finally, the (maximal) non-increasing rearrangement of a function is presented in order to consider Banach function spaces and norms. Lorentz spaces and absolutely continuous norms are introduced in this context. \item"$\bullet$" The Palais-Smale condition, about critical and regular points and values for ${G \in C^1(X,\Bbb R)}$, where $X$ is a Banach space; \item"$\bullet$" Inequalities. \endroster \par Chapter 2 is about classical Sobolev spaces, as its name says. These spaces are defined in the two alternative standard ways: using derivatives (in the sense of distribution) or the Bessel operator. Some embeddings and important inequalities, such as the Poincaré inequality, are presented, as well as the behaviour under translation of functions in the Sobolev spaces of order 1. \par Fractional Sobolev spaces $W_p^s(\Omega)$ are defined in Chapter 3 by means of the Gagliardo seminorm. The authors note that, on $\Bbb R^n$, this approach is equivalent to the one using the Bessel operator if and only if the Lebesgue exponent $p=2$. Nevertheless, they justify their choice by the fact that the more explicit norm of $W_p^s(\Omega)$ has numerous advantages. Basic properties of fractional Sobolev spaces are then exposed: \roster \item"$\bullet$" embedding results; \item"$\bullet$" the $(s,p)$-Friedrichs inequality, leading to a discussion concerning the completion of $C_0^\infty(\Omega)$ with respect to various norms; \item"$\bullet$" a Hölder-type inequality, leading to a general $(s,p)$-Poincaré inequality; \item"$\bullet$" the notion of extension domain with characterisations; \item"$\bullet$" non-compactness of the inclusion of $W_p^s(\Omega)$ in $L_p(\Omega)$ when $\Omega$ does not have a smooth enough boundary, in terms of entropy and approximation numbers; \item"$\bullet$" behaviour under translation; \item"$\bullet$" connection with standard spaces when $s$ approaches 0 or 1. \endroster This last point is then understood with terms of interpolation spaces, together with the notion of (quasi-)normal spaces. In the last section of this chapter, fractional Sobolev spaces are connected with fractional powers of the Laplacian. \par The last section of Chapter 3 is then reinforced in Chapter 4, which is mainly devoted to the problem of minimising the fractional Rayleigh quotient in connection with eigenvalues of the fractional $p$-Laplacian $(- \Delta)_p^s$. Among other things, it allows one to present various facts concerning the weak solutions $u$ of the problem $$ (- \Delta)_p^s= \lambda |u|^{p-2} \quad \text{in } \Omega, \ u=0 \text{ in } \Bbb R^n \setminus \Omega, $$ notably concerning the (almost-everywhere) positiveness of such solutions. After that, the spectrum and eigenspaces of this operator are characterised. The chapter ends with extensions of the inequalities of Faber-Krahn type in this context. \par Chapter 5 deals with the classical Hardy inequalities. First, in $\Bbb R^n$, it is written $$ \int_{\Bbb R^n} \frac{|f(x)|^p}{|x|^p} \, dx \leq \Big| \frac{p}{p-n} \Big|^p \int_{\Bbb R^n} |\nabla f(x)|^p \, dx, $$ and holds true for all $f \in C^\infty_0(\Bbb R^n \setminus \{0 \})$ if $n < p < \infty$ and all $f \in C_0^\infty(\Bbb R^n)$ if ${1 \leq p