Bull. Sci. math. 136 (2012) 521–573 www.elsevier.com/locate/bulsci Hitchhiker’s

Bull. Sci. math. 136 (2012) 521–573 www.elsevier.com/locate/bulsci Hitchhiker’s guide to the fractional Sobolev spaces ✩ Eleonora Di Nezza a,∗, Giampiero Palatucci a,b,1, Enrico Valdinoci a,c,2 a Dipartimento di Matematica, Università di Roma “Tor Vergata” – Via della Ricerca Scientifica, 1, 00133 Roma, Italy b Dipartimento di Matematica, Università degli Studi di Parma, Campus – Viale delle Scienze, 53/A, 43124 Parma, Italy c Dipartimento di Matematica, Università degli Studi di Milano – Via Saldini, 50, 20133 Milano, Italy Received 22 June 2011 Available online 29 December 2011 Abstract This paper deals with the fractional Sobolev spaces Ws,p. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains. © 2011 Elsevier Masson SAS. All rights reserved. MSC: primary 46E35; secondary 35S30, 35S05 Keywords: Fractional Sobolev spaces; Gagliardo norm; Fractional Laplacian; Nonlocal energy; Sobolev embeddings Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 2. The fractional Sobolev space Ws,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 ✩Part of this paper has been used in a few lectures held at the University of Rome “Tor Vergata”. * Corresponding author. E-mail addresses: dinezza@mat.uniroma2.it (E. Di Nezza), giampiero.palatucci@unimes.fr (G. Palatucci), enrico@math.utexas.edu (E. Valdinoci). 1 The author has been supported by Istituto Nazionale di Alta Matematica “F. Severi” (Indam) and by ERC grant 207573 “Vectorial problems”. 2 The author has been supported by the ERC grant “ϵ Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities” and the FIRB project “A&B Analysis and Beyond”. 0007-4497/$ – see front matter © 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.bulsci.2011.12.004 522 E. Di Nezza et al. / Bull. Sci. math. 136 (2012) 521–573 3. The space H s and the fractional Laplacian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 4. Asymptotics of the constant C(n,s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 5. Extending a Ws,p(Ω) function to the whole of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 6. Fractional Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 7. Compact embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 8. Hölder regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 9. Some counterexamples in non-Lipschitz domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 1. Introduction These pages are for students and young researchers of all ages who may like to hitchhike their way from 1 to s ∈(0,1). To wit, for anybody who, only endowed with some basic undergraduate analysis course (and knowing where his towel is), would like to pick up some quick, crash and essentially self-contained information on the fractional Sobolev spaces W s,p. The reasons for such a hitchhiker to start this adventurous trip might be of different kind: (s)he could be driven by mathematical curiosity, or could be tempted by the many applications that fractional calculus seems to have recently experienced. In a sense, fractional Sobolev spaces have been a classical topic in functional and harmonic analysis all along, and some important books, such as [58,88] treat the topic in detail. On the other hand, fractional spaces, and the corresponding nonlocal equations, are now experiencing impressive applications in different sub- jects, such as, among others, the thin obstacle problem [85,68], optimization [37], finance [26], phase transitions [2,14,86,40,45], stratified materials [81,23,24], anomalous diffusion [67,96,64], crystal dislocation [90,47,8], soft thin films [56], semipermeable membranes and flame propa- gation [15], conservation laws [9], ultra-relativistic limits of quantum mechanics [41], quasi- geostrophic flows [63,27,21], multiple scattering [36,25,49], minimal surfaces [16,20], materials science [4], water waves [79,98,97,32,29,72,33,34,31,30,42,50,73,35], elliptic problems with measure data [70,53], non-uniformly elliptic problems [39], gradient potential theory [71] and singular set of minima of variational functionals [69,55]. Don’t panic, instead, see also [84,85] for further motivation. For these reasons, we thought that it could be of some interest to write down these notes – or, more frankly, we wrote them just because if you really want to understand something, the best way is to try and explain it to someone else. Some words may be needed to clarify the style of these pages have been gathered. We made the effort of making a rigorous exposition, starting from scratch, trying to use the least amount of technology and with the simplest, low-profile language we could use – since cap- ital letters were always the best way of dealing with things you didn’t have a good answer to. Differently from many other references, we make no use of Besov spaces3 or interpolation techniques, in order to make the arguments as elementary as possible and the exposition suitable for everybody, since when you are a student or whatever, and you can’t afford a car, or a plane fare, or even a train fare, all you can 3 About this, we would like to quote [52], according to which “The paradox of Besov spaces is that the very thing that makes them so successful also makes them very difficult to present and to learn”. E. Di Nezza et al. / Bull. Sci. math. 136 (2012) 521–573 523 do is hope that someone will stop and pick you up, and it’s nice to think that one could, even here and now, be whisked away just by hitchhiking. Of course, by dropping fine technologies and powerful tools, we will miss several very important features, and we apologize for this. So, we highly recommend all the excellent, clas- sical books on the topic, such as [58,88,1,91,92,99,78,89,66,59], and the many references given therein. Without them, our reader would remain just a hitchhiker, losing the opportunity of per- forming the next crucial step towards a full mastering of the subject and becoming the captain of a spaceship. In fact, compared to other Guides, this one is not definitive, and it is a very evenly edited book and contains many passages that simply seemed to its editors a good idea at the time. In any case, of course, we know that we cannot solve any major problems just with potatoes – it’s fun to try and see how far one can get though. In this sense, while most of the results we present here are probably well known to the experts, we believe that the exposition is somewhat original. These are the topics we cover. In Section 2, we define the fractional Sobolev spaces W s,p via the Gagliardo approach and we investigate some of uploads/Finance/ hitcghiker-x27-s-guide.pdf

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