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Sobolev Inequalities in Familiar and Unfamiliar Settings Sobolev Inequalities in Familiar and Unfamiliar Settings Laurent Salo®-Coste Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial di®erential equations. In fact, they are extremely flexible tools and are useful in many di®erent settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1 Introduction There are few articles that have turned out to be as influential and truly im- portant as S.L. Sobolev 1938 article [93] (the American translation appeared in 1963), where he introduces his famed inequalities. It is the idea of a func- tional inequality itself that Sobolev brings to life in his paper, as well as the now so familiar notion of an a priori inequality, i.e., a functional inequality established under some strong hypothesis and that might be extended later, perhaps almost automatically, to its natural domain of de¯nition. (These ideas are also related to the theory of distributions which did not exist at the time and whose magni¯cent development by L. Schwartz was, in part, anticipated in the work of S.L. Sobolev.) The most basic and important applications of Sobolev inequalities are to the study of partial di®erential equations. Simply put, Sobolev inequalities provide some of the very basic tools in the study of the existence, regularity, and uniqueness of the solutions of all sorts of partial di®erential equations, lin- ear and nonlinear, elliptic, parabolic, and hyperbolic. I leave to others, much better quali¯ed than me, to discuss these beautiful developments. Instead, my aim in this paper is to survey briefly an assortments of perhaps less familiar applications of Sobolev inequalities (and related inequalities) to problems and Laurent Salo®-Coste Cornell University, Mallot Hall, Ithaca, NY 14853, USA, e-mail: [email protected] 299 300 Laurent Salo®-Coste in settings that are not always directly related to PDEs, at least not in the most classical sense. The inequalities introduced by S.L. Sobolev have turned out to be extremely useful flexible tools in surprisingly diverse settings. My hope is to be able to give to the reader a glimpse of this diversity. The reader must be warned that the collection of applications of Sobolev inequalities described below is very much influenced by my own interest, knowledge, and limitations. I have not tried at all to present a complete picture of the many di®erent ways Sobolev inequalities have been used in the literature. That would be a very di±cult task. 2 Moser's Iteration 2.1 The basic technique This section is included mostly for those readers that are not familiar with the use of Sobolev inequalities. It illustrates some aspects of one of the basic techniques associated with their use. To the untrained eyes, the fundamental nature of Sobolev inequalities is often lost in the technicalities surrounding their use. Indeed, outside analysis, Lp spaces other than L1, L2, and L1 still appear quite exotic to many. As the following typical example illustrates, they play a key role in extracting the information contained in Sobolev inequalities. Recall that HÄolder'sinequality states that Z jfgjdx 6 kfkpkgkq as long as 1 6 p; q 6 1 and 1=p + 1=q = 1 (these are called conjugate exponents). A somewhat clever use of this inequality yields θ 1¡θ kfkr 6 kfkskfkt as long as 1 6 r; s; t 6 1 and 1=r = θ=s + (1 ¡ θ)=t. These basic inequalities are used extensively in conjunction with Sobolev inequalities. P 2 n Let ¢ = (@=@xi) be the Laplacian in R . Consider a bounded domain ­ ½ Rn, ¸ > 0, and the (Dirichlet) eigenfunction/eigenvalue problem: ¯ ¯ ¢u = ¡¸u in ­; u @­ = 0: (2.1) Our goal is to show how the remarkable inequality Z 2 n=2 2 supfjuj g 6 An¸ juj dx (2.2) ­ ­ Sobolev Inequalities in Familiar and Unfamiliar Settings 301 (for solutions of (2.1)) follows from the most classical Sobolev inequality, namely, the inequality (2.5) below. For a normalized eigenfunction u with kuk2 = 1 the inequality (2.2) bounds the size of u in terms of the associated eigenvalue. The technique illustrated below is extremely flexible and can be adapted to many situations. 1 In fact, we only assume that u 2 H0 (­), i.e., u is the limit of smooth compactly supported functions in ­ in the norm 0 11=2 Z Xn 2 2 kuk = @ [juj + j@u=@xij ]dxA ­ 1 and that Z Z Xn @u @v dx = ¸ uvdx (2.3) @xi @xi ­ 1 ­ n 1 n 2 P 2 for any v 2 H0 (­). We set ru = (@u=@xi)1 , jruj = j@u=@xij . To avoid 1 additional technical arguments, we assume a priori that u is bounded on ­. For 1 6 p < 1 we take v = juj2p¡1(u=juj) in (2.3). This yields Z Z Z 2p ¡ 1 ¸ juj2p = (2p ¡ 1) juj2p¡2jruj2dx = jrjujpj2dx: (2.4) p2 ­ ­ ­ As our starting point, we take the most basic Sobolev inequality µZ ¶1=qn Z 1 n 2qn 2 2 8 f 2 H (R ); jfj dx 6 Cn jrfj dx; qn = n=(2 ¡ n): (2.5) If kn = (1 + 2=n), then 1=kn = θn=qn + (1 ¡ θn) with θn = n=(n + 2), and HÄolder'sinequality yields Z µZ ¶1=qn µZ ¶2=n jfj2kn dx 6 jfj2qn dx jfj2dx : Together with the previous Sobolev inequality, we obtain Z Z µZ ¶2=n 2(1+2=n) 2 2 2 jfj dx 6 Cn jrfj dx jfj dx : (2.6) For a discussion of this type of \multiplicative" inequality see, for example, [75, Sect. 2.3]. Now, for a solution u of (2.1) the inequalities (2.6) and (2.4) yield Z µZ ¶1+2=n 2p(1+2=n) 2 2p juj dx 6 Cnp¸ juj dx : 302 Laurent Salo®-Coste i This inequality can obviously be iterated by taking pi = (1 + 2=n) , and we get i i µZ ¶1=pi P ¡1 P ¡1 Z (j¡1)pj ¡ ¢ pj 2pi 1 2 1 2 juj dx 6 (1 + 2=n) 3Cn¸ juj dx: 1 P ¡1 2 2 Note that pj = n=2 and lim kjuj kp = ku k1. The desired conclusion 1 p!1 (2.2) follows. 2.2 Harnack inequalities The technique illustrated above is the simplest instance of what is widely known as Moser's iteration technique. In a series of papers [77]{[80], Moser developed this technique as the basis for the study of divergence form uni- n formly elliptic operators in R , i.e., operators of the form (we use @i = @=@xi) X La = @i(ai;j(x)@j) i;j with real matrix-valued function a satisfying the ellipticity condition 8 P 2 <> ai;j(x)»i»j > "j»j ; i;j 8 x 2 ­; P 0 ¡1 0 :> ai;j(x)»i»j 6 " j»jj» j; i;j where " > 0 and the coe±cients ai;j are simply bounded real measurable func- tions. Because of the low regularity of the coe±cients, the most basic question in this context is that of the boundedness and continuity of solutions of the equation Lau = 0 in the interior of an open set ­. This was solved earlier by De Giorgi [34] (and by Nash [81] in the parabolic case), but Moser pro- posed an alternative method, squarely based on the use of Sobolev inequality (2.5). To understand why one might hope this is possible, observe that the argument given in the previous section works without essential changes if, in (2.1), one replaces the Laplacian ¢ by La. Let u be a solution of Lau = 0 in a domain ­, in the sense that for any 1 1 open relatively compact set ­0 in ­, u 2 H (­0) and for all v 2 H0 (­0) Z X @u @v ai;j dx = 0: @xi @xj ­ i;j In [77], Moser observed that the interior boundedness and continuity of such a solution follow from the Harnack inequality that provides a constant C(n; ") Sobolev Inequalities in Familiar and Unfamiliar Settings 303 such that if u as above is nonnegative in ­ and the ball B satis¯es 2B ½ ­, then supfug 6 C(n; ") inffug (2.7) B B (a priori, the supremum and in¯mum should be understood here as essential supremum and essential in¯mum. The ball 2B is concentric with B with twice the radius of B). He then proceeded to prove this Harnack inequality by variations on the argument outlined in the previous section. In his later papers [78]{[80], Moser obtained a parabolic version of the above Harnack inequality. Namely, he proved that there exists a constant C(n; ") such that any nonnegative solution u of the heat equation (@t¡La)u = 0 in a time-space cylinder Q = (s ¡ 4r2; s) £ 2B satis¯es supfug 6 C(n; ") inffug; (2.8) Q Q¡ + 2 2 2 where Q¡ = (s ¡ 3r ; s ¡ 2r ) £ B and Q+ = (s ¡ r =2; s) £ B. Moser's iteration technique has been adapted and used in hundreds of papers studying various PDE problems. Some early examples are [2, 3, 90]. The books [42, 69, 76] contain many applications of this circle of ideas, as well as further references. The survey paper [83] deals speci¯cally with the heat equation and is most relevant for the purpose of the present paper. The basic question we want to explore in the next two subsections is: what exactly are the crucial ingredients of Moser's iteration? This question is moti- vated by our desire to use this approach in other settings such as Riemannian manifolds or more exotic spaces.
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