ALESSIO FIGALLI AND ROBIN NEUMAYER

Transcription

1 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ALESSIO FIGALLI AND ROBIN NEUMAYER Abstact. We pove a stong fom of the quantitative Sobolev inequality in R n fo p, whee the deficit of a function u Ẇ 1,p contols u v L p fo an extemal function v in the Sobolev inequality. 1. Intoduction Given n and 1 p < n, the Sobolev inequality povides a contol of the L nom of a function in tems of a suitable L p nom of its gadient. Moe pecisely, setting p := np/(n p), one defines the homogeneous Sobolev space Ẇ 1,p as the space of functions in R n such that u L p and u L p. Then the following holds: u L p S p,n u L p u Ẇ 1,p. (1.1) Thoughout the pape, all the integals and function spaces will be ove R n, so we will omit the domain of integation when no confusion aises. It is well known that the optimal constant in (1.1) is given by ) 1/n, S p,n = πn 1/p ( n p p 1 ) (p 1)/p ( Γ(n/p)Γ(1 + n n/p) Γ(1 + n/)γ(n) and that equality is attained in (1.1) if and only if u belongs to the family of functions cv λ,y (x) = cλ n/p v 1 (λ(x y)), c R, λ R +, y R n, whee κ 0 v 1 (x) :=, (1.) (1 + x p )(n p)/p see [31, 1] and [13] (hee κ 0 is chosen so that v 1 L p = 1, theefoe cv λ,y L p = c, and p := p/(p 1) denotes the Hölde conjugate of p). In othe wods, M := {cv λ,y : c R, λ R +, y R n } (1.3) is the (n + )-dimensional manifold of extemal functions in the Sobolev inequality (1.1). To quantify how close a function u Ẇ 1,p is to achieving equality in (1.1), we define its deficit to be the p-homogeneous functional δ(u) := u p L p S p p,n u p L p. By (1.1), the deficit is nonnegative and equals zeo if and only if v M. In [5], Bezis and Lieb aised the question of stability fo the Sobolev inequality, that is, whethe the deficit contols an appopiate distance between a function u Ẇ 1,p and the family of extemal functions. This question was fist answeed in the case p = by Bianchi and Egnell in [3]: thee, they showed that the deficit of a function u contols the L distance between the gadient of u and 1

2 FIGALLI AND NEUMAYER the gadient of closest extemal function v. The esult is optimal both in the stength of the distance and the exponent of decay. Howeve, thei poof is vey specific to the case p =, as it stongly exploits the Hilbet stuctue of Ẇ 1,. Late on, in [10], Cianchi, Fusco, Maggi, and Patelli consideed the case 1 < p < n and povided a stability esult in which the deficit contols the L p distance between u and some v M. Thei poof uses a combination of symmetization techniques and tools fom the theoy of mass tanspotation. Moe ecently, in [1], Figalli, Maggi, and Patelli used eaangement techniques and mass tanspotation theoy to show that, in the case p = 1, the deficit contols the appopiate notion of distance of u fom M at the level of gadients (see also [, 8] fo patial esults when p = 1). As in [3], the distance consideed in [1] is the stongest that one expects to contol and the exponent of decay is shap. In view of [3] and [1], one may expect that, fo all 1 < p < n, the deficit contols the L p distance between u and v fo some v M; this would answe the question of Beizis and Lieb in the affimative with the deficit contolling the stongest possible notion of distance in this setting. The main esult of this pape shows that, in the case p, this esult is indeed tue. Moe pecisely, ou main esult states the following: Theoem 1.1. Let p < n. Thee exists a constant C > 0, depending only on p and n, such that fo all u Ẇ 1,p, fo some v M. u v p L C δ(u) + C u p 1 p L u v p Lp (1.4) As a consequence of Theoem 1.1 and the main esult of [10] (see Theoem 5.5 below), we deduce the following coollay, poving the desied stability at the level of gadients: Coollay 1.. Let p < n. Thee exists a constant C > 0, depending only on p and n, such that fo all u Ẇ 1,p, ( ) u ζ v L p C δ(u) u L p u p (1.5) L p ( ) fo some v M, whee ζ = p p 3 + 4p 3p+1 n. The topic of stability fo functional and geometic inequalities has geneated much inteest in ecent yeas. In addition to the afoementioned papes, esults of this type have been addessed fo the isopeimetic inequality [3, 0, 11], log-sobolev inequality [6, 4, 17], the highe ode Sobolev inequality [4, ], the factional Sobolev inequality [7], the Moey-Sobolev inequality [9] and the Gagliado-Nienbeg-Sobolev inequality [6, 9], as well as fo numeous othe geometic inequalities. Aside fom thei intinsic inteest, stability esults have applications in the study of geometic poblems (see [18, 19, 1]) and can be used to obtain quantitative ates of convegence fo diffusion equations (as in [6]). Fo the emainde of the pape, we will always assume that p < n. Acknowledgments: A. Figalli is patially suppoted by NSF Gants DMS and DMS R. Neumaye is suppoted by the NSF Gaduate Reseach Fellowship unde Gant DGE Both authos wamly thank Fancesco Maggi fo useful discussions egading this wok.

3 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 3. Theoem 1.1: idea of the poof As a stating point to pove stability of (1.1) at the level of gadients, one would like to follow the agument used to pove the analogous esult in [3]. Howeve, this appoach tuns out to be sufficient only in cetain cases, and additional ideas ae needed to conclude the poof. Indeed, a Taylo expansion of the deficit δ(u) and a spectal gap fo the lineaized poblem allow us to show that the second vaiation is stictly positive, but in geneal we cannot absob the highe ode tems. Let us povide a few moe details to see to what extent this appoach woks, whee it beaks down, and how we get aound it..1. The expansion appoach. The fist idea of the poof of Theoem 1.1 is in the spiit of the stability esult of Bianchi and Egnell in [3]. Ultimately, this appoach will need modification, but let us sketch how such an agument would go. In ode to intoduce a Hilbet space stuctue to ou poblem, we define a weighted L -type distance of a function u Ẇ 1,p to M at the level of gadients. To this end, fo each v = cv λ,y M, we define A v (x) := (p ) v p ˆ ˆ + v p Id, ˆ = x y x y, (.1) whee (a b)c := (a c)b. Then, with the notation A v [a, a] := a T A v a fo a R n, we define the weighted L distance of u to M by { ( ) } 1/ d(u,m) := inf A v [ u v, u v] : v M, v L p = u L p { ( ) } (.) 1/ = inf A cvλ,y [ u cv λ,y, u cv λ,y ] : λ R+, y R n, c = u L p. Note that A v [ u v, u v] = v p u v + (p ) v p u v. A few emaks about this definition ae in ode. Remak.1. The motivation to define d(u, M) in this way instead of, fo instance, { ( inf v p u v ) } 1/ : v M, v L p = u L p, will become appaent in Section 3. This choice, howeve, is only technical, as v p u v A v [ u v, u v] (p 1) v p u v. Remak.. One could altenatively define the distance in (.) without the constaint c = u L p, instead also taking the infimum ove the paamete c. Up to adding a small positivity constaint to ensue that the infimum is not attained at v = 0, this definition woks, but ultimately the cuent pesentation is moe staightfowad. Remak.3. The distance d(u, M) has homogeneity p/, that is, d(cu, M) = c p/ d(u, M). In Poposition 4.1(1), we show that thee exists δ 0 = δ 0 (n, p) > 0 such that if δ(u) δ 0 u p L p, (.3)

4 4 FIGALLI AND NEUMAYER then the infimum in d(u, M) is attained. Given a function u Ẇ 1,p satisfying (.3), let v M attain the infimum in (.) and define u v ϕ :=, (u v) L p so that u = v + ɛϕ with ɛ = (u v) L p and ϕ p = 1. Since δ 0 and δ(v) = 0, the Taylo expansion of the deficit of u aound v vanishes both at the zeoth and fist ode. Thus, the expansion leaves us with δ(u) = ɛ p A v [ ϕ, ϕ] ɛ Sp,np(p p 1) v p ϕ + o(ɛ ). (.4) Since v M minimizes the distance between u and M, ɛϕ = u v is othogonal (in some appopiate sense) to the tangent space of M at v, which we shall see coincides with the span the fist two eigenspaces of an appopiate weighted lineaized p-laplacian. Then, a gap in the spectum in this opeato allows us to show that c d(u, M) = c ɛ A v [ ϕ, ϕ] ɛ p A v [ ϕ, ϕ] ɛ Sp,np(p p 1) v p ϕ fo a positive constant c = c(n, p). Togethe with (.4), this implies d(u, M) + o(ɛ ) Cδ(u). Now, if the tem o(ɛ ) could be absobed into d(u, M), then we could use the estimate (.6) below to obtain u v p Cδ(u), which would conclude the poof... Whee this appoach falls shot. The poblem aises exactly when tying to absob the tem o(ɛ ). Indeed, ecalling that ɛ = (u v) L p, we ae asking whethe o( u v L p) d(u, M) v p u v (ecall Remak.1), and unfotunately this is false in geneal. Notice that this poblem neve aises in [3] fo the case p =, as the above inequality educes to which is clealy tue. o( u v L ) u v L,.3. The solution. A Taylo expansion of the deficit will not suffice to pove Theoem 1.1 as we cannot hope to absob the highe ode tems. Instead, fo a function u Ẇ 1,p, we give two diffeent expansions, each of which gives a lowe bound on the deficit, by splitting the tems between the second ode tem and the p th ode tem using elementay inequalities (Lemma 3.). Paiing this with an analysis of the second vaiation, we obtain the following:

5 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 5 Poposition.4. Thee exist constants c 1, C, and C 3, depending only on p and n, such that the following holds. Let u Ẇ 1,p be a function satisfying (.3) and let v M be a function whee the infimum of the distance (.) is attained. Then c 1 d(u, M) C u v p δ(u), (.5) C 3 d(u, M) + 1 u v p δ(u). (.6) 4 Individually, both inequalities ae quite weak. Howeve, as shown in Coollay 4.3, they allow us to pove Theoem 1.1 (in fact, the stonge statement u v p δ(u)) fo the set of functions u such that d(u, M) = A v [ u v, u v] u v p o d(u, M) = A v [ u v, u v] u v p. (.7) We ae then left to conside the middle egime, whee A v [ u v, u v] u v p. We handle this case as follows. Let u t := (1 t)u + tv be the linea intepolation between u and v. Choosing t small enough, u t falls in the second egime in (.7), so Theoem 1.1 holds fo u t. We then must elate the deficit and distance of u t to those of u. While elating the distances is staightfowad, it is not clea fo the deficits whethe the estimate δ(u t ) Cδ(u) holds. Still, we can show that δ(u t ) Cδ(u) + C v p 1 L u v p L p, which allows us to conclude the poof. It is this point in the poof that intoduces that tem u v L p in Theoem 1.1, and fo this eason we ely on the main theoem of [10] to pove Coollay 1.. We note that the application of [10] is not staightfowad, since the function v which attains the minimum in ou setting is a pioi diffeent fom the one consideed thee (see Section 5 fo moe details)..4. Outline of the pape. The pape is stuctued as follows. In Section 3, we intoduce the opeato L v that will be impotant in ou analysis of the second vaiation of the deficit and pove some facts about the spectum of this opeato. We also pove some elementay but cucial inequalities in Lemma 3. and povide othogonality constaints that aise fom taking the infimum in (.). In Section 4, we pove Poposition.4 by exploiting a gap in the spectum of L v and using the inequalities of Lemma 3.. In Section 5, we combine Poposition.4 with an intepolation agument to obtain Theoem 1.1. We then apply the main esult of [10] in ode to pove Coollay 1..

6 6 FIGALLI AND NEUMAYER In Section 6, we pove the compact embedding that shows that L v has a discete spectum and justify the use of Stum-Liouville theoy in the poof of Poposition 3.1. Section 7 is an appendix in which we pove a technical claim. 3. Peliminaies In this section, we state a few necessay facts and tools The tangent space of M and the opeato L v. The set M of extemal functions defined in (1.3) is an (n + )-dimensional smooth manifold except at 0 M. Fo a nonzeo v = c 0 v λ0,y 0 M, the tangent space is computed to be T v M = span {v, λ v, y 1v,..., y nv}, whee y i denotes the ith component of y and λ v = λ λ=λ0 v, y iv = y i y i =y0 i v. Since the functions v = v λ0,y 0 minimize u δ(u) and have v λ0,y 0 L p = 1, by computing the Eule-Lagange equation one discoves that p v = S p p,nv p 1, (3.1) whee the p-laplacian p is defined by p w := div ( w p w). Hence, diffeentiating (3.1) with espect to y i o λ, we see that div (A v (x) w) = (p 1)S p p,nv p w, w span { λ v, y 1v,..., y nv}, (3.) whee A v (x) is as defined in (.1). This motivates us to conside the weighted opeato L v w := div (A v (x) w)v p (3.3) on the space L (v p ), whee, fo a measuable weight ω : R n R, we let ( ) 1/, w L (ω) = w ω L (ω) = {w : R n R : w L (ω) < }. R n Poposition 3.1. The opeato L v has a discete spectum {α i } i=1, with 0 < α i < α i+1 fo all i, and α 1 = (p 1)S p p,n, H 1 = span {v}, (3.4) α = (p 1)S p p,n, H = span { λ v, y 1v,..., y nv}, (3.5) whee H i denotes the eigenspace coesponding to α i. In paticula, Poposition 3.1 implies that T v M = span {H 1 H }. (3.6) The Rayleigh quotient chaacteization of eigenvalues implies that { } Lv w, w Av [ w, w] α 3 = inf = w, w v p w : w span {H 1 H }, (3.7) whee othogonality is with espect to the inne poduct defined by w 1, w := v p w 1 w. (3.8) Note that the eigenvalues of L v ae invaiant unde changes in λ and y.

7 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 7 Poof of Poposition 3.1. The discete spectum of L v follows in the usual way afte establishing the ight compact embedding theoem; we show the compact embedding in Coollay 6. and give details confiming the discete spectum in Coollay 6.3. Since a scaling agument shows that the eigenvalues of L v ae invaiant unde changes of λ and y, it suffices to conside the opeato L = L v fo v = v 0,1, letting A = A v. One easily veifies that v is an eigenfunction of L with eigenvalue (p 1)S p p,n and that λ v and y iv ae eigenfunctions with eigenvalue (p 1)S p p,n, using (3.1) and (3.) epectively. Futhemoe, since v > 0, it follows that α 1 = (p 1)S p p,n is the fist eigenvalue, which is simple, so (3.4) holds. To pove (3.5), we must show that α = (p 1)S p p,n is the second eigenvalue and veify that thee ae no othe eigenfunctions in H. Both of these facts follow fom sepaation of vaiables and Stum-Liouville theoy. Indeed, an eigenfunction ϕ of L satisfies div (A(x) ϕ) + αv p ϕ = 0. (3.9) Assume that ϕ takes the fom ϕ(x) = Y (θ)f(), whee Y : S n 1 R and f : R R. In pola coodinates, div(a(x) ϕ) = (p 1) v p (p 1)(n 1) ϕ + v p ϕ + 1 n 1 v p θj θ j ϕ + (p 1)(p ) v p 4 v v ϕ (3.10) (this computation is given in the appendix fo the convenience of the eade). As v is adially symmetic, that is, v(x) = w( x ), we intoduce the slight abuse of notation by letting v() also denote the adial component: v() = w(), so v () = v and v () = v. Fom (3.10), we see that (3.9) takes fom 0 = (p 1) v p f ()Y (θ) + (p 1)(n 1) v p f ()Y (θ) + 1 v p f() S n 1Y (θ) + (p 1)(p ) v p 4 v v f ()Y (θ) + αv p f()y (θ), which yields the system 0 = S n 1Y (θ) + µy (θ) on S n 1, (3.11) 0 = (p 1) v p f (p 1)(n 1) + v p f µ v p f +(p 1)(p ) v p 4 v v f + αv p f on [0, ). (3.1) The eigenvalues and eigenfunctions of (3.11) ae explicitly known; these ae the spheical hamonics. The fist two eigenvalues ae µ 1 = 0 and µ = n 1. Taking µ = µ 1 = 0 in (3.1), we claim that: - α 1 1 = (p 1)Sp p,n and the coesponding eigenspace is span {v}; - α 1 = (p 1)S p p,n with the coesponding eigenspace span { λ v}. Indeed, Stum-Liouville theoy ensues that each eigenspace is one-dimensional, and that the ith eigenfunction has i 1 inteio zeos. Hence, since v (esp. λ v) solves (3.1) with µ = 0 and α = (p 1)S p p,n (esp. α = (p 1)S p p,n), having no zeos (esp. one zeo) it must be the fist (esp.

8 8 FIGALLI AND NEUMAYER second) eigenfunction. Fo µ = n 1, the eigenspace fo (3.11) is n dimensional with n eigenfunctions giving the spheical components of y iv, i = 1,..., n. The coesponding equation in (3.1) gives α 1 = (p 1)Sp,n. p As the fist eigenvalue of (3.1) with µ = µ, α 1 is simple. The eigenvalues ae stictly inceasing, so this shows that α1 3 > (p 1)Sp,n p and α > (p 1)Sp,n, p concluding the poof. The application of Stum-Liouville theoy in the poof above is not immediately justified because ous is a singula Stum-Liouville poblem. The poof of Stum-Liouville theoy in ou setting, that is, that each eigenspace is one-dimensional and that the ith eigenfunction has i 1 inteio zeos, is shown in Section Some useful inequalities. The following lemma contains fou elementay inequalities fo vectos and numbes. This lemma is a key tool fo getting aound the issues pesented in the intoduction; in lieu of a Taylo expansion, these inequalities yield bounds on the deficit by splitting the highe ode tems between the second ode tems and the p th o p th ode tems. Lemma 3.. Let x, y R n and a, b R. The following inequalities hold. Fo all κ > 0, thee exists a constant C = C(p, n, κ) such that ( p x + y p x p + p x p x y + (1 κ) x p y p(p ) + x p 4 (x y) ) C y p. (3.13) Fo all κ > 0, thee exists C = C(p, κ) such that ( p a + b p a p + p a p (p 1) ) ab + + κ a p b + C b p. (3.14) Thee exists C = C(p, n) such that x + y p x p + p x p x y C x p y + y p. (3.15) Thee exists C = C(p) such that a + b p a p + p a p ab + C a p b + b p. (3.16) Poof of Lemma 3.. We only give the poof of (3.13), as the poofs of (3.14)-(3.16) ae analogous. Obseve that if p is an even intege o p is an intege, these inequalities follow (with explicit constants) fom a binomial expansion and splitting the intemediate tems between the second ode and p th o p th ode tems using Young s inequality. Suppose (3.13) fails. Then thee exists κ > 0, {C j } R such that C j, and {x j }, {y j } R n such that x j + y j p x j p < p x j p x j y j + (1 κ) ( p x j p y j + ) p(p ) x j p 4 (x j y j ) C j y j p. If x j = 0, we immediately get a contadiction. Othewise, we divide by x j p to obtain x j + y j p x j p 1 < p x ( j y j x j + (1 κ)p yj x j + (p )(x j y j ) ) y j p x j 4 C j x j p. (3.17)

9 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 9 The left-hand side is bounded below by 1, so in ode fo (3.17) to hold, y j / x j must convege to 0 at a sufficiently fast ate. In this case, y j is much smalle that x j, so a Taylo expansion eveals that the left-hand side behaves like p x j y j x j + p y j p(p ) (x j y j ) ( yj ) + x j x j 4 + o x j, (3.18) which is lage than the ight-hand side, contadicting (3.17). With the same poof, one can show (3.14) with the opposite sign: Fo all κ > 0, thee exists C = C(p, κ) such that ( p a + b p a p + p a p (p 1) ) ab + κ a p b C b p. Theefoe, applying this and (3.14) to functions v and v + ϕ with v p = v + ϕ p, one obtains v p ( vϕ p (p 1) + κ) v p ϕ + C ϕ p. (3.19) 3.3. Othogonality constaints fo u v. Given a function u Ẇ 1,p satisfying (.3), suppose that v = c 0 v λ0,y 0 is a function at which the infimum is attained in (.). Then u p = v p = c p 0, (3.0) and the enegy E(v) = E(λ, y) = A c0 v λ,y [ u c 0 v λ,y, u c 0 v λ,y ], (3.1) aising fom (.) when u is fixed, has a citical point at (λ 0, y 0 ) in the n + 1 paametes λ and y i, i = 1,..., n. In othe wods, 0 = λ λ=λ0 A c0 v λ,y [ u c 0 v λ,y, u c 0 v λ,y ], (3.) 0 = y i y i =y0 i A c0 v λ,y [ u c 0 v λ,y, u c 0 v λ,y ]. We expess u as u = v + ɛϕ, with ϕ scaled such that ϕ p = 1. Computing the deivatives in (3.) gives ɛ A v [ λ v, ϕ] = ɛ (p ) ϕ v p 4 v λ v + ɛ (p ) ϕ v p 4 v λ v, ɛ A v [ y iv, ϕ] = ɛ (p ) ϕ v p 4 v y iv + ɛ (p ) ϕ v p 4 v y iv + ɛ (p ) v p ϕ ϕ y i ˆ, (3.3)

10 10 FIGALLI AND NEUMAYER whee ˆ is as in (.1). Futhemoe, multiplying (3.) by ɛϕ and integating by pats implies Sp,n(p p 1)ɛ v p λ v ϕ = ɛ A v [ λ v, ϕ], Sp,n(p p 1)ɛ v p y iv ϕ = ɛ A v [ y iv, ϕ], so (3.3) becomes [ ɛ v p λ v ϕ =ɛ C 1 [ ɛ v p y iv ϕ =ɛ C 1 ] + v p ϕ ϕ y i ˆ, whee C 1 = (p ) (p 1)S p p,n. ϕ v p 4 v λ v + (p ) ϕ v p 4 v y iv + (p ) A Taylo expansion of the constaint (3.0) implies ɛ v p vϕ = ɛ v p ϕ + o(ɛ ). ] ϕ v p 4 v λ v, (3.4) ϕ v p 4 v y iv (3.5) Howeve, in view of the comments in the intoduction, we cannot geneally absob the tem o(ɛ ), so this is not quite the fom of the othogonality constaint that we need. In its place, using (3.0) and (3.19), we have ɛ v p vϕ ɛ p 1 + κ v p ϕ + Cɛ p ϕ p (3.6) fo any κ > 0, with C= C(p, n, κ). The conditions (3.4), (3.5), and (3.6) show that ϕ is almost othogonal to T v M with espect to the inne poduct given in (3.8). Indeed, dividing though by ɛ, the inne poduct of ϕ with each basis element of T v M appeas on the left-hand side of (3.4), (3.5), and (3.6), while the ight-hand side is O(ɛ). As a esult of (3.6) and ϕ being almost othogonal to T v M, it is shown that ϕ satisfies a Poincaé-type inequality (4.14), which is an essential point in the poof of Poposition.4. Remak 3.3. In [3], the analogous constaints give othogonality athe than almost othogonality; this is easily seen hee, as taking p = makes the ight-hand sides of (3.4) and (3.5) vanish. 4. Poof of Poposition.4 and its consequences We pove Poposition.4 combining an analysis of the second vaiation and the inequalities of Lemma 3.. As a consequence (Coollay 4.3), we show that, up to emoving the assumption (.3), Theoem 1.1 holds fo the two egimes descibed in (.7). To pove Poposition.4, we will need two facts. Fist, we want to know that the infimum in (.) is attained, so that we can expess u as u = v + ɛϕ whee ϕ p = 1, and ϕ satisfies (3.4), (3.5), and (3.6). Second, it will be impotant to know that if δ 0 in (.3) is small enough, then ɛ is small as well. Fo this eason we fist pove the following: Poposition 4.1. The following two claims hold.

11 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 11 (1) Thee exists δ 0 = δ 0 (n, p) > 0 such that if δ(u) δ 0 u p L p, (4.1) then the infimum in (.) is attained. In othe wods, thee exists some v M with v p = u p such that A v [ u v, u v] = d(u, M). () Fo all ɛ 0 > 0, thee exists δ 0 = δ 0 (n, p, ɛ 0 ) > 0 such that if u Ẇ 1,p satisfies (4.1), then ɛ := u v L p < ɛ 0 whee v M is a function that attains the infimum in (.). Poof. We begin by showing the following fact, which will be used in the poofs of both pats of the poposition: fo all γ > 0, thee exists δ 0 = δ 0 (n, p, γ) > 0 such that if δ(u) δ 0 u p L p, then inf{ u v L p : v M} γ u L p. (4.) Othewise, fo some γ > 0, thee exists a sequence {u k } Ẇ 1,p such that u k L p = 1 and δ(u k ) 0 while inf{ u k v L p : v M} > γ. A concentation compactness agument as in [7, 30] ensues that thee exist sequences {λ k } and {y k } such that, up to a subsequence, λ n/p k u k (λ k (x y k )) conveges stongly in Ẇ 1,p to some v M. Since ( γ < u k [λ )] L ] n/p k v + y k = [λ n/p λ k u k (λ k ( y k )) v 0 k p L p this gives a contadiction fo k sufficiently lage, hence (4.) holds. Poof of (1). Suppose u satisfies (4.1), with δ 0 to be detemined in the poof. Up to multiplication by a constant, we may assume that u L p = 1. By the claim above, we may take δ 0 small enough so that (4.) holds fo γ as small as needed. The infimum on the left-hand side of (4.) is attained. Indeed, let {v k } be a minimizing sequence with v k = c k v λk,y k. The sequences {c k }, {λ k }, {1/λ k }, and {y k } ae bounded: if λ k o λ k 0, then fo k lage enough thee will be little cancellation in the tem u v k p, so that u v k p 1 u p, contadicting (4.). The analogous agument holds if y k o c k. Thus {c k }, {λ k }, {1/λ k }, and {y k } ae bounded and so, up to a subsequence, (c k, λ k, y k ) (c 0, λ 0, y 0 ) fo some (c 0, λ 0, y 0 ) R R + R n. Since the functions cv λ,y ae smooth, decay nicely, and depend smoothly on the paametes, we deduce that v k c 0 v λ0,y 0 = ṽ in Ẇ 1,p (actually, they also convege in C k fo any k), hence ṽ attains the infimum. To show that the infimum is attained in (.), we obtain an uppe bound on the distance by using v = ṽ/ ṽ L p as a competito. Indeed, ecalling Remak.1, it follows fom Hölde s inequality that d(u, M) (p 1) v p u v (p 1)S p,n (p )/p u v /p L. p

12 1 FIGALLI AND NEUMAYER Notice that, since u L p = 1, it follows by (4.1) that u L p Sp p,n povided δ 0 1/. Hence, since v L p 1 v u L p S p n,p v u L p, it follows by (4.) and the tiangle inequality that u v L p C(n, p) γ, theefoe d(u, M) C(n, p) γ /p. (4.3) Hence, if {v k } is a minimizing sequence fo (.) with v k = v λk,y k (so that v k p = u p = 1), the analogous agument as above shows that if eithe of the sequences {λ k }, {1/λ k }, o {y k } ae unbounded, then d(u, M) 1, contadicting (4.3) fo γ sufficiently small. This implies that v k v λ0,y 0 in Ẇ 1,p, and by continuity v λ0,y 0 attains the infimum in (.). Poof of (). We have shown that (4.) holds fo δ 0 sufficiently small. Theefoe, we need only to show that, up to futhe deceasing δ 0, thee exists C = C(p, n) such that u v 0 L p C inf{ u v L p : v M}, whee v 0 M is the function whee the infimum is attained in (.). Suppose fo the sake of contadiction that thee exists a sequence {u j } such that δ(u j ) 0 and u j L p = 1 but u j v j p j u j v j p, (4.4) whee v j, v j M ae such that A vj [ u j v j, u j v j ] = d(u j, M) and { } u j v j p = inf u j v j p : v M. Since δ(u j ) 0, the same concentation compactness agument as above implies that thee exist sequences {λ j } and {y j } such that, up to a subsequence, λ n/p j u j (λ j (x y j )) conveges in Ẇ 1,p to some v M with v L p = 1. By an agument analogous to that in pat (1), we detemine that v j v in C k and v j v in C k fo any k. Let u j v j u j v j φ j = and φj =. u j v j L p u j v j L p Then (4.4) implies that 1 = φ j p j In paticula, φ j 0 in L p. Now define ψ j = φ j φ v j v j j =. u j v j L p φ j p. (4.5) Fo any η > 0, (4.5) implies that 1 η ψ j L p 1 + η fo j lage enough. In paticula, { ψ j } is bounded in L p and so ψ j ψ in L p fo some ψ Ẇ 1,p.

13 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 13 We now conside the finite dimensional manifold M := {v v : v, v M}. Since v j, v j v, the sequences {λ j }, {1/λ j }, {y j }, { λ j }, {1/ λ j } and {ȳ j } ae contained in some compact set, and thus all noms of v j v j ae equivalent: fo any nom on M thee exists µ > 0 such that Dividing (4.6) by u j v j L p µ v j v j L p v j v j 1 µ v j v j L p. (4.6) gives µ(1 η) µ ψ j L p ψ j 1 µ ψ i L p 1 + η µ. (4.7) Taking the nom = C k, the uppe bound in (4.7) and the Azelà-Ascoli theoem imply that ψ j conveges, up to a subsequence, to ψ in C k. The lowe bound in (4.7) implies that ψ C k 0. To get a contadiction, we use the minimality of v j fo d(u j, M) to obtain v j p φ j + (p ) v j p φj v j p φ j + (p ) v j p φ j = v j p φ j + v j p φ j ψ j + v j p ψ j ( ) + (p ) v j p φj + v j p φj ψ j + v j p ψ j. Since v j p φ j v j p φ j 0 (4.8) and v j p φj v j p φj 0, (4.8) implies that 0 lim j + (p ) v j p φ j ψ j + lim j ( lim v j p φj ψ j + lim j j v j p ψ j v j p ψ j ). (4.9) Howeve, since φ j 0 in L p, lim j In addition, the tems v j p φ j ψ j = 0 and lim j v j p ψ j and v j p φj ψ j = 0. v j p ψ j convege to something stictly positive, as ψ j ψ 0 and v j v with v(x) 0 fo all x 0. This contadicts (4.9) and concludes the poof. The following Poincaé inequality will be used in the poof of Poposition.4:

14 14 FIGALLI AND NEUMAYER Lemma 4.. Thee exists a constant C > 0 such that v p ϕ C v p ϕ (4.10) fo all ϕ Ẇ 1,p. Poof. Let v M and ϕ C0. As v is a local minimum of the functional δ, 0 d dɛ δ(v + ɛϕ) = p v p ϕ + p(p ) v p ϕ ( ɛ=0 p )( Sp,n p( p p 1 v p ) p/p ( ) v p v ϕ + p(p 1) ( ) v p p /p 1 v p ϕ ). Noting that v p ϕ v p ϕ and ( v p ) p/p ( ) v p v ϕ 0, this implies that 0 p(p 1) v p ϕ S p p,np(p 1)( v p ) p /p 1 v p ϕ. Thus (4.10) holds fo ϕ C 0, and fo ϕ Ẇ 1,p by appoximation. We now pove Poposition.4. Poof of Poposition.4. Fist of all, thanks to (.3), we can apply Poposition 4.1(1) to ensue that some v = c 0 v λ0,y 0 M attains the infimum in (.). Also, expessing u as u = v + ɛϕ whee ϕ p = 1, it follows fom Poposition 4.1() and the discussion in Section 3.3 that ɛ can be assumed to be as small as desied (povided δ 0 is chosen small enough) and that ϕ satisfies (3.4), (3.5), and (3.6). Note that, since all tems in (.5) and (.6) ae p-homogeneous, without loss of geneality we may take c 0 = 1. Poof of (.5). The inequalities (3.13) and (3.14) ae used to expand the gadient tem and the function tem in δ(u) espectively, splitting highe ode tems between the second ode and the p th o p th ode tems. Fom (3.13) and fo κ = κ(p, n) > 0 to be chosen at the end of the poof, we have u p v p + ɛp v p v ϕ + ɛ p(1 κ) ( v p ϕ + (p ) v p ϕ ) ɛ p C ϕ p. (4.11) Note that the second ode tem is pecisely 1 ɛ p(1 κ) A v [ ϕ, ϕ]. Similaly, (3.14) gives u p 1 + ɛp v p 1 ϕ + ɛ ( p (p 1) + p κ ) Sp,n p v p ϕ + Cɛ p ϕ p. (4.1) Fom the identity (3.1), the fist ode tem in (4.1) is equal to ɛp v p 1 ϕ = ɛp Sp,n p v p v ϕ. (4.13)

15 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 15 Using (4.13) and ecalling that (p 1)Sp,n p = α (see (3.5)), (4.1) becomes u p 1 + ɛp Sp,n p v p v ϕ + ɛ p (α + κ) Sp,n p v p ϕ + Cɛ p, The following estimate holds, and is shown below: ɛ v p ϕ (1 + κ) ɛ α 3 A v [ ϕ, ϕ] + Cɛ p, (4.14) Philosophically, (4.14) follows fom a spectal gap analysis, using (3.7) and the fact that (3.4), (3.5), and (3.6) imply that ϕ is almost othogonal to H 1 and H. As ɛ may be taken as small as needed, using (4.14) we have u p 1 + p S p p,n ( ɛ v p v ϕ + ɛ (α + κ)(1 + κ) α 3 The function z z p/p is concave, so u p L 1 + p p p ( u p 1): ( Sp,n u p p L p Sp p,n + p ɛ v p v ϕ + ɛ (α + κ)(1 + κ) α 3 Subtacting (4.15) fom (4.11) gives δ(u) ɛ p ( 1 κ (α ) + κ)(1 + κ) α 3 A v [ ϕ, ϕ] + Cɛ p ). A v [ ϕ, ϕ] + Cɛ p ). (4.15) A[ ϕ, ϕ] Cɛ p. Since 1 α α 3 > 0, we may choose κ sufficiently small so that 1 κ (α +κ)(1+κ) α 3 > 0. To conclude the poof of (.5), we need only to pove (4.14). Poof of (4.14). If ϕ wee othogonal to T v M instead of almost othogonal, that is, if the ighthand sides of (3.4), (3.5), and (3.6) wee equal to zeo, then (4.14) would be an immediate consequence of (3.7). Theefoe, the poof involves showing that the eo in the othogonality elations is tuly highe ode, in the sense that it can be absobed in the othe tems. Up to escaling u and v, we may assume that λ 0 = 1 and y 0 = 0. We ecall the inne poduct w, y defined in (3.8) which gives ise to the nom w = ( v p w ) 1/. As in Section 3, we let H i denote the eigenspace of L v in L (v p ) coesponding to eigenvalue α i, so H i = span {Y i,j } N(i), whee Y i,j is an eigenfunction with eigenvalue α i with Y i,j = 1. We expess ɛϕ in the basis of eigenfunctions: N(i) ɛϕ = β i,j Y i,j whee β i,j := ɛ v p ϕy i,j. i=1 We let ɛ ϕ be the tuncation of ɛϕ: ɛ ϕ = ɛϕ N(i) β i,j Y i,j, i=1

16 16 FIGALLI AND NEUMAYER so that ϕ is othogonal to span {H 1 H } and, intoducing the shothand βi := N(i) β i,j, v p (ɛϕ) = v p (ɛ ϕ) + β1 + β. (4.16) Applying (3.7) to ϕ implies that v p (ɛ ϕ) ɛ α 3 L v ϕ, ϕ, which combined with (4.16) gives v p (ɛϕ) ɛ α 3 L v ϕ, ϕ + β 1 + β = 1 α 3 α i βi + β1 + β i=3 ɛ α 3 L v ϕ, ϕ + We thus need to estimate β1 + β. The constaint (3.6) implies ( β1 ɛ p 1 + κ v p ϕ + Cɛ p ( 1 α 1 α 3 ) (β 1 + β ). ϕ p ) (4.17) Cɛ 4( v p ϕ ) + Cɛ p ( ϕ p ). By (4.10), v p ϕ v p ϕ. Futhemoe, both v p ϕ and ϕ p ae univesally bounded, so fo ɛ sufficiently small depending only on p and n and κ, β1 κɛ ( v p ϕ + (p ) v p ϕ ) + Cɛ p. (4.18) α 3 Fo β,1, we notice that Hölde s inequality and (3.4) imply β,1 (C p,n ɛ v p 3 ϕ ) λv λ v v p λ v C p,n λ v v p 4 ɛ ϕ 4 = C p,n ɛ 4 v p 4 ϕ 4, (4.19) whee the final equality follows because the tem v p λ v / λ v is bounded (in fact, it is bounded by α ). Then, using Young s inequality, we get β,1 ɛ κ ( v p ϕ + (p ) v p ϕ ) + C κ,p ɛ p ϕ p. (n + 1)α 3 The analogous agument using (3.5) implies that β,j C p,n ɛ 4 v p 4 ϕ 4 + C p,n ɛ 4 ( v p ϕ ϕ fo j =,..., n + 1. Fo the second tem in (4.0), Hölde s inequality implies that ( ) v p y i ˆ ϕ ϕ v p 4 ϕ 4 v p y i ˆ y v y iv. ) y i ˆ. (4.0) y v

17 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 17 Since we find that v p y i ˆ y iv y i ˆ = xi x x 3, y i ˆ 1 x, conveges, so (4.0) implies that β,j C p,n ɛ 4 v p 4 ϕ 4. Then using Young s inequality just as in (4.19), we find that β,j ɛ κ ( v p ϕ + (p ) v p ϕ ) + C κ,p ɛ p (n + 1)α 3 ϕ p, and thus β ɛ κ ( v p ϕ + (p ) v p ϕ ) + C κ,p ɛ p. (4.1) α 3 Togethe (4.17), (4.18), and (4.1) imply (4.14), as desied. Poof of (.6). The poof of (.6) is simila to, but simple than, the poof of (.5), as no spectal gap o analysis of the second vaiation is needed. The pinciple of the expansion is the same, but now we use (3.15) and (3.16) fo the expansion, putting most of the weight of the highe ode tems on the second ode tem and peseving the positivity of the p th ode tem. Fom (3.15), we have u p v p + pɛ v p v ϕ C ɛ v p ϕ + ɛp ϕ p. (4.) Similaly, (3.16) implies u p 1 + ɛp v p 1 ϕ + C ɛ v p ϕ + ɛ p ϕ p. (4.3) As befoe, the identity (3.1) implies (4.13), so (4.3) becomes u p 1 + ɛp Sp,n p v p v ϕ + C ɛ v p ϕ + ɛ p ϕ p. By the Poincaé inequality (4.10), u p 1 + ɛp Sp,n p v p v ϕ + Cɛ v p ϕ + ɛ p. As in (4.15), the concavity of z z p/p yields Sp,n u p p L p Sp p,n + ɛp v p v ϕ + Cɛ v p ϕ + Cɛ p. (4.4) Subtacting (4.4) fom (4.) gives δ(u) C ɛ v p ϕ + ɛp Cɛp Cd(u, M) + ɛp 4.

18 18 FIGALLI AND NEUMAYER The final inequality follows fom Remak.1 and once moe taking ɛ is as small as needed. This concludes the poof of (.6). Coollay 4.3. Suppose u Ẇ 1,p is a function satisfying (.3) and v M is a function whee the infimum in (.) is attained. Thee exist constants C, c and c, depending on n and p only, such that if Av [ u v, u v] Av [ u v, u v] C o c u v p, (4.5) u v p then c u v p δ(u). Poof. Let C = C c 1 and let c = 1 8C 3 whee c 1, C and C 3 ae as defined in Poposition.4. Fist suppose that u satisfies the fist condition in (4.5). Then in (.5), we may absob the tem C u v p into the tem c 1 d(u, M), giving us c 1 d(u, M) δ(u). Given this contol, we may bootstap using (.6) to gain contol of the stonge distance: 1 u v p δ(u) + C 3 d(u, M) Cδ(u). 4 Similaly, if u satisfies the second condition in (4.5), then we may absob the tem C 3 d(u, M) into the tem 1 4 u v p in (.6), giving us 1 u v p δ(u) Poofs of Theoem 1.1 and Coollay 1. Coollay 4.3 implies Theoem 1.1 fo the functions u Ẇ 1,p that satisfy (.3) and that lie in one of the two egimes descibed in (.7). Theefoe, to pove Theoem 1.1, it emains to undestand the case when the tems A v [ u v, u v] and u v p ae compaable and to emove the assumption (.3). The following poposition accomplishes the fist. Poposition 5.1. Let u Ẇ 1,p be a function satisfying (.3), and let v M be a function whee the infimum in (.) is attained. If Av [ u v, u v] c C u v p, (5.1) whee c and C ae the constants fom the Coollay 4.3, then u v p Cδ(u) + C v p 1 L u v p Lp (5.) fo a constant C depending only on p and n.

19 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 19 Poof. Suppose u lies in the egime (5.1). Then we conside the linea intepolation u t := tu+(1 t)v and notice that Av [ u t v, u t v] ut v p = t A v [ u v, u v] t p u v p t p c. Hence, thee exists t sufficiently small, depending only on p and n, such that t p c > C. We claim that we may apply Coollay 4.3 to u t. This is not immediate because v may not attain the infimum in (.) fo u t. Howeve, each step of the poof holds if we expand u t aound v. Indeed, keeping the pevious notation of u v = ɛϕ with ϕ p = 1, we have u t v = t ɛϕ. so the othogonality constaints in (3.4), (3.5), and (3.6) still hold fo u t and v by simply multiplying though by t (this changes the constants by a facto of t but this does not affect the poof). Futhemoe, (.3) is used in the poofs of Poposition.4 and (4.14) to ensue that ɛ is a small as needed to absob tems. Since t < 1, if ɛ is sufficiently small then so is t ɛ. With these two things in mind, evey step in the poof of Poposition.4, and theefoe Coollay 4.3 goes though fo u t. Coollay 4.3 then implies that u v p = u t v p Cδ(u t ). Theefoe, (5.) follows if we can show t p δ(u t ) Cδ(u) + C v p 1 L u v p Lp. (5.3) In the diection of (5.3), by convexity and ecalling that v L p = S p,n v L p = S p,n u L p, we have δ(u t ) = t u + (1 t ) v p Sp,n p t u + (1 t )v p L p t u p + (1 t ) v p Sp,n t p u + (1 t )v p (5.4) L p ) = t δ(u) + Sp,n ( v p pl t u + (1 t p )v pl. p Also, by the tiangle inequality, t (u v) + v p L p ( v L p t (u v) L p )p, and by the convexity of the function f(z) = z p, f(z + y) f(z) + f (z)y, and so ( v L p t (u v) L p )p v L p p v p 1 L p u v L p. These two inequalities imply that v p L t u + (1 t p )v p L p p v p 1 L u v p L p. Combining this with (5.4) yields (5.3), concluding the poof. Fom hee, the poof of Theoem 1.1 follows easily: Poof of Theoem 1.1. Togethe, Coollay 4.3 and Poposition 5.1 imply the following: thee exists some constant C such that if u Ẇ 1,p satisfies (.3), then thee is some v M such that u v p Cδ(u) + C v p 1 L p u v L p.

20 0 FIGALLI AND NEUMAYER Theefoe, we need only to emove the assumption (.3) in ode to complete the poof of Theoem 1.1. Howeve, in the case whee (.3) fails, then tivially, inf{ u v p L p : v M} u p L p 1 δ 0 δ(u). Theefoe, by choosing the constant to be sufficiently lage, Theoem 1.1 is poven. We now pove Coollay 1. using the main esult fom [10], which we ecall hee: Theoem 5. (Cianchi, Fusco, Maggi, Patelli, [10]). Thee exists C such that λ(u) ζ u L p C( u L p S p,n u L p ), (5.5) whee λ(u) = inf { u v p Lp / u p L : v M, v p = } ( ). u p and ζ = p 3 + 4p 3p+1 p n Poof of Coollay 1.. As befoe, if (.3) does not hold, then Coollay 1. holds tivially by simply choosing the constant to be sufficiently lage. Now suppose u Ẇ 1,p satisfies (.3). Thee ae two obstuctions to an immediate application of Theoem 5.. The fist is the fact that the deficit in (5.5) is defined as u L p S p,n u L p, while in ou setting it is defined as u p L p S p p,n u p L p. Howeve, this is easy to fix. Indeed, using the elementay inequality we let a = u L p/s p,n u L p a p b p a b a b 1, and b = 1 to get u L p S p,n u L p S p,n u L p u p L p S p p,n u p L p S p p,n u p L p 1 1 δ 0 u p L p S p p,n u p L p u p L p, whee the last inequality follows fom (.3). Theefoe, up to inceasing the constant, (5.5) implies that λ(u) ζ C δ(u) u p. (5.6) L p The second obstuction to applying Theoem 5. is the fact that (5.5) holds fo the infimum in λ(u), while we must contol u v L p fo v attaining the infimum in (.). To solve this issue it is sufficient to show that thee exists some constant C = C(n, p) such that v u p C inf { u v p L : v M, v p = u p } p whee v attains the infimum in (.). The poof of this fact is nealy identical (with the obvious adaptations) to that of pat () of Poposition 4.1, with the only nontivial diffeence being that one must integate by pats to show that the analogue of fist tem in (4.9) goes to zeo. Theefoe, (5.5) implies ( ) u ζ v L p C δ(u) u L p u L p whee v M attains the infimum in (.). Paied with Theoem 1.1, this poves Coollay 1. with ζ = ζ p.

21 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 1 6. Spectal Popeties of L v In this section, we give the poofs of the compact embedding theoem and Stum-Liouville theoy that wee postponed in the poof of Poposition 3.1. As in Poposition 3.1, by scaling, it suffices to conside the opeato L = L v whee v = v 0, The discete spectum of L. Given two measuable functions ω 0, ω 1 : Ω R, let W 1, (Ω, ω 0, ω 1 ) := {g : g W 1, (Ω,ω 0,ω 1 ) < }, whee W 1, (Ω,ω 0,ω 1 ) is the nom defined by ( g W 1, (Ω,ω 0,ω 1 ) = g ω 0 + Ω Ω g ω 1 ) 1/. (6.1) The space W 1, 0 (Ω, ω 0, ω 1 ) is defined as the completion of the space C0 (Ω) with espect to the nom W 1, (Ω,ω 0,ω 1 ). The following compact embedding esult was shown in [8]: Theoem 6.1 (Opic, [8]). Let Z = W 1, 0 (R n, ω 0, ω 1 ) and suppose ω i L 1 loc and ω 1/ i i = 0, 1. If thee ae local compact embeddings whee B k = {x : x < k}, and if then Z embeds compactly in L (R n, ω 0 ). L loc, (6.) W 1, (B k, ω 0, ω 1 ) L (B k, ω 0 ), k N, (6.3) lim sup { u L k (R n \B k,ω 0 ) : u Z, u Z 1 } = 0, (6.4) We apply Theoem 6.1 to show that the space embeds compactly into L (R n, v p ). X = W 1, 0 (R n, v p, v p ), (6.5) Coollay 6.. The compact embedding X L (R n, v p ) holds, with X as in (6.5). Poof. Let us veify that Theoem 6.1 may be applied in ou setting, taking ω 0 = v p, ω 1 = v p. In othe wods, we must show that (6.), (6.3) and (6.4) ae satisfied. A simple computation veifies (6.). To show (6.3), we fix δ > 0 small (the smallness depending only on n and p) and show the thee inclusions below: W 1, (B, ω 0, ω 1 ) (1) W 1,(n+δ)/(n+) (B ) () L (B ) (3) L (B, ω 0 ). Since (n/( + n)) =, the Rellich-Kondachov compact embedding theoem implies (), while the inclusion (3) holds simply because v p c n,p, fo x B. In the diection of showing (1), we use this fact and Hölde s inequality to obtain ( ) (n+)/(n+δ) B ( δ)/(n+δ) u C n,p, v p u. (6.6) B B B u (n+δ)/(n+)

22 FIGALLI AND NEUMAYER Futhemoe, since v p = C(1 + x p ) n(p )/p x (p )/(p 1) c n,p, x (p )/(p 1) fo x B, Hölde s inequality implies that ( ) (n+)/(n+δ) ( )( x (p )/(p 1) u B B u (n+δ)/(n+) C n,p, B v p u, B x β ) ( δ)/(n+δ) (6.7) whee β = ( p )( n+δ )( n+ ) p 1 n+ δ. Then the inclusion (1) follows fom (6.6) and (6.7), and thus (6.3) is veified. To show (6.4), let u k be a function almost attaining the supemum in (6.4), in othe wods, fo a fixed η > 0, let u k be such that u k X, u k X 1, and sup { u L (R n \B k,ω 0 ) : u X, u X 1 } u k L (R n \B k,ω 0 ) + η. By mollifying u and multiplying by a smooth cutoff η C0 (Rn \B k ), we may assume without loss of geneality that u k C0 (Rn \B k ). Recalling that v = v 1 with v 1 as in (1.), we have v p u k = κ 0 (1 + x p ) (p )(n p)/p u k κ 0 x (p )(n p)/(p 1) u k (6.8) R n \B k R n \B k R n \B k fo k. We use Hady s inequality in the fom x s u C x s+ u (6.9) R n R n fo u C0 (Rn ) (see, fo instance, [3]). Applying (6.9) to the ight-hand side of (6.8) implies x (p )(n p)/(p 1) u k C x (p )(n p)/(p 1)+ u k (6.10) R n \B k R n \B k and (6.8) and (6.10) combined give v p u k C x (p )(n p)/(p 1)+ u k R n \B k R n \B k = C x (p )(n 1)/(p 1) u k whee the final inequality follows because Thus and (6.4) is poved. R n \B k x p Ck p R n \B k v p u k, v p C x (p )(n 1)/(p 1) fo x R n \B 1. v p u k Ck p u k X, R n \B k Thanks to the compact embedding X L (R n, ω 0 ), we can now pove the following impotant fact:

23 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 3 Coollay 6.3. The opeato L has a discete spectum {α i } i=1. Poof. We show that the opeato L 1 : L (v p ) L (v p ) is bounded, compact, and selfadjoint. Fom thee, one applies the spectal theoem (see fo instance [15]) to deduce that L 1 has a discete spectum, hence so does L. Appoximating by functions in C0 (Rn ), the Poincaé inequality (4.10) holds fo all functions ϕ X, with X as defined in (6.5). Thanks to this fact, the existence and uniqueness of solutions to Lu = f fo f L (v p ) follow fom the Diect Method, so the opeato L 1 is well defined. Self-adjointness is immediate. Fom (4.10) and Hölde s inequality, we have c u X v p u A[ u, u] u X Lu L (v p ). This poves that L 1 is bounded fom L (v p ) to L (v p ), and by Coollay 6. we see that L 1 is a compact opeato. 6.. Stum-Liouville theoy. Multiplying by the integating facto n 1, the odinay diffeential equation (3.1) takes the fom of the Stum-Liouville eigenvalue poblem Lf + αf = 0 on [0, ), (6.11) whee with Lf = 1 w [(P f ) Qf] P () = (p 1) v p n 1, Q() = µ n 3 v p, w() = v p n 1. (6.1) This is a singula Stum-Liouville poblem; fist of all, ou domain is unbounded, and second of all, the equation is degeneate because v (0) = 0. Nonetheless, we show that Stum-Liouville theoy holds fo this singula poblem. Lemma 6.4 (Stum-Liouville Theoy). The following popeties hold fo the singula Stum- Liouville eigenvalue poblem (6.11): (1) If f 1 and f ae two eigenfunctions coesponding to the eigenvalue α, then f 1 = cf. In othe wods, each eigenspace of L is one-dimensional. () The ith eigenfunction of L has i 1 inteio zeos. Note that L has a discete spectum because L does (Coollay 6.3), and that eigenfunctions f of L live in the space Y = W 1, ( 0 [0, ), v p n 1, v p n 1), using the notation intoduced at the beginning of Section 6.1. In any ball B R aound zeo, the opeato L is degeneate elliptic with the matix A bounded by an A -Muckenhoupt weight, so eigenfunctions of L ae Hölde continuous; see [16, 5]. Theefoe, eigenfunctions of L ae Hölde continuous on [0, ).

24 4 FIGALLI AND NEUMAYER Remak 6.5. The function P () as defined in (6.1) has the following behavio: P () (p )(p 1)+n 1 in [0, 1], P () (n 1)/(p 1) as. In paticula, the weight v p n 1 (n 1)/(p 1) goes to infinity as, which implies that 1 f d < fo any f Y. In ode to pove Lemma 6.4, we fist pove the following lemma, which descibes the asymptotic decay of solutions of (6.11). Lemma 6.6. Suppose f Y is a solution of (6.11). Then, fo any 0 < β < n p p 1, thee exist C and 0 such that f() C β and f () C β 1 fo 0. Poof. Step 1: Qualitative Decay of f. Fo any function f Y, f() 0 as. Indeed, nea infinity, v p p 1 behaves like C γ whee γ := n 1 p 1 > 1. Then fo any, s lage enough with < s, ( f() f(s) f ) 1/ ( (t) dt f (t) t γ 1/ dt t dt) γ (6.13) by Hölde s inequality. As both integals on the ight-hand side of (6.13) convege, fo any ɛ > 0, we may take lage enough such that the ight-hand side is bounded by ɛ, so the limit of f() as exists. We claim that this limit must be equal to zeo. Indeed, since Y is obtained as a completion of C0, if we apply (6.13) to a sequence f k C0 ([0, )) conveging in Y to f and we let s, we get ( ) 1/ ( f k () f k (t) t γ 1/, dt t dt) γ thus, by letting k, ( ) 1/ ( f() f (t) t γ 1/. dt t dt) γ Since the ight-hand side tends to zeo as, this poves the claim. Step : Qualitative Decay of f. Fo > 0, (6.11) can be witten as L f := f + af + bf = 0 (6.14) whee a = P Q + wα and b =. P P Fixing ɛ > 0, an explicit computation shows that thee exists 0 lage enough such that and (1 ɛ)(n 1) p 1 1 a (1 + ɛ)(n 1) p 1 µ 1 p 1 + (1 ɛ)c p,nα (3p )/(p 1) b µ 1 p 1 + (1 + ɛ)c p,nα (3p )/(p 1) 1

25 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 5 fo 0, whee c n,p is a positive constant depending only on n and p. Asymptotically, theefoe, ou equation behaves like f + n 1 f ( p 1 + cp,n α µ ) f p p 1 = 0. If f is a solution of (6.11), then squaing (6.14) on [ 0, ), we obtain ( (n 1 ) ) f f ( ((1 + p 1 + ɛ ɛ)cp,n α + + µ ) ) f p p 1 C( f + f ). Integating on [R, R + 1] fo R 0 implies R+1 R f C R+1 R f + C R+1 Step 1 and Remak 6.5 ensue that both tems on the ight-hand side go to zeo. Applying Moey s embedding to f η R, whee η R is a smooth cutoff equal to 1 in [R, R + 1], we detemine that f L ([R,R+1]) 0 as R, poving that f () 0 as. Step 3: Quantitative Decay of f and f. Standad aguments (see fo instance [14, VI.6]) show that, also in ou case, the ith eigenfunction f of L has at most i 1 inteio zeos; in paticula, f() does not change sign fo sufficiently lage. Without loss of geneality, we assume that eventually f 0. Taking 0 as in Step and applying the opeato L defined in (6.14) to the function g = C β +c, c > 0, fo 0 gives L g Cβ(β + 1) β (1 ɛ)(n 1) p 1 C β ( β(β + 1) R f. ( (1 + Cβ β ɛ)cp,n α + (1 ɛ)(n 1) β + (1 + ɛ)c ) p,nα p 1 p (3p )/(p 1) µ p 1 + (1 + ɛ)c p,nα c. (3p )/(p 1) ) (C β + c) Fo any 0 < β < (n p)/(p 1), 0 may be taken lage enough (and theefoe ɛ small enough) such that L g < 0 on [ 0, ), so g is a supesolution of the equation on this inteval. Choosing C = f( 0 ) β 0 and c > 0, then (g f)( 0) > 0 and (g f)() c > 0 as. Since L (g f) < 0, we claim that g f > 0 on ( 0, ). Indeed, othewise, g f would have a negative minimum at some ( 0, ), implying that (g f)() 0, (g f) () = 0, and (g f) () 0, focing L (g f) 0, a contadiction. This poves that 0 f g on [ 0, ), and since c > 0 was abitay, we detemine that f C β on [ 0, ). We now deive bounds on f : by the fundamental theoem of calculus and using (6.14) and the bound on f fo 0, we get f () = f C f + C t β C f() + C β + β 1 C β 1.

26 6 FIGALLI AND NEUMAYER With these asymptotic decay estimates in hand, we ae eady to pove Lemma 6.4. Poof of Lemma 6.4. We begin with the following emak about uniqueness of solutions. If f 1 and f ae two solutions of (6.11) and f 1 ( 0 ) = f ( 0 ), f 1( 0 ) = f ( 0 ) fo some 0 > 0, then f 1 = f on [0, ). Indeed, fo > 0, we may expess ou equation as in (6.14). As a and b ae continuous on (0, ), the standad poof of uniqueness fo (non-degeneate) second ode ODE holds. Once f 1 = f on (0, ), they ae also equal at = 0 by continuity. Poof of (1). Suppose α is an eigenvalue of L with f 1 and f satisfying (6.11). In view of the uniqueness emak, if thee exists 0 > 0 and some linea combination f of f 1 and f such that f( 0 ) = f ( 0 ) = 0, then f is constantly zeo and f 1 and f ae linealy dependent. Let [ ] f1 f W () = W (f 1, f )() := det f 1 f () denote the Wonskian of f 1 and f. This is well defined fo > 0 (since f 1 and f ae C thee) and a standad computation shows that (P W ) = 0 on (0, ): indeed, since W = f 1 f f f 1, we get (P W ) = P W + P W = P (f 1 f f f 1 ) + P (f 1 f f f 1), and by adding and subtacting the tem (αw Q)f 1 f it follows that (P W ) = f 1 ( P f + P f + (αw Q)f ) f ( P f 1 + P f + (αw Q)f 1 ) = 0. Thus P W is constant on (0, ). We now show that that P W is continuous up to = 0 and that (P W )(0) = 0. Indeed, (6.11) implies that (P f i) = (Q αw)f i fo i = 1,. The ight-hand side is continuous, so (P f i ) is continuous, fom which it follows easily that P W is also continuous on [0, ). To show that (P W )(0) = 0, we fist pove that (P f i )(0) = 0. Indeed, let c i := (P f i )(0). If c i 0, then keeping in mind Remak 6.5, f i() c i P () c i (p )/(p 1)+n 1 fo 1, (6.15) theefoe R v p f n 1 d 0 R 0 (p )/(p 1)+n 1 f d R 0 d = +, (p )/(p 1)+n 1 contadicting the fact that f Y. Hence, we conclude that lim(p f 0 i )() = 0, and using this fact we obtain (P W )(0) = lim 0 (P f 1f P f f 1 ) = lim 0 (P f 1) lim 0 f lim 0 (P f ) lim 0 f 1 = 0. Theefoe (P W )() = 0 fo all [0, ). Since P () > 0 fo > 0, we detemine that W () = 0 fo all > 0. In paticula, given 0 (0, ), thee exist c 1, c such that c 1 + c 0 and c 1 f 1 ( 0 ) + c f ( 0 ) = 0, c 1 f 1( 0 ) + c f ( 0 ) = 0.

27 GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p 7 Then f := c 1 f 1 + c f solves (6.11) and f( 0 ) = f ( 0 ) = 0. By uniqueness, f 0 fo all t (0, ), and so f 1 = cf. Poof of (). Thanks to ou peliminay estimates on the behavio of f i at infinity, the following is an adaptation of the standad agument in, fo example, [14, VI.6]. Suppose that f 1 and f ae eigenfunctions of L coesponding to eigenvalues α 1 and α espectively, with α 1 < α, that is, (P f i) Qf i + α i wf i = 0. Ou fist claim is that between any two consecutive zeos of f 1 is a zeo of f, including zeos at infinity. Note that (P W ) = P [f 1 f f f 1 ] + P [f 1 f f f 1] = f 1 [(P f ) + (α Q)f ] f [(P f 1) + (α 1 w Q)f 1 ] + (α 1 α )wf 1 f = (α 1 α )wf 1 f. (6.16) Suppose that f 1 has consecutive zeos at 1 and, and suppose fo the sake of contadiction that f has no zeos in the inteval ( 1, ). With no loss of geneality, we may assume that f 1 and f ae both nonnegative in [ 1, ]. Case 1: Suppose that <. Then integating (6.16) fom 1 to implies 0 > (α 1 α ) wf 1 f = (P W )( ) (P W )( 1 ) 1 = P ( )[f 1 ( )f ( ) f 1( )f ( )] P ( 1 )[f 1 ( 1 )f ( 1 ) f 1( 1 )f ( 1 )] = P ( )f 1( )f ( ) + P ( 1 )f 1( 1 )f ( 1 ). The function f 1 is positive on ( 1, ), so f 1 ( 1) 0 and f 1 ( ) 0. Also, since f 1 ( 1 ) = f 1 ( ) = 0 we cannot have f 1 ( 1) = 0 o f 1 ( ) = 0, as othewise f 1 would vanish identically. Futhemoe, f is nonnegative on [ 1, ], so we conclude that the ight-hand side is nonnegative, giving us a contadiction. Case : Suppose that =. Again integating the identity (6.16) fom 1 to, we obtain 0 > (α 1 α ) 1 wf 1 f = lim (P W )() (P W )( 1 ) = lim [P ()(f 1 ()f () f 1()f ())] P ( 1 )(f 1 ( 1 )f ( 1 ) f 1( 1 )f ( 1 )). We notice that Lemma 6.6 implies that Indeed, taking n p (p 1) < β < n p p 1, and, ecalling Remak 6.5, implying that lim [P ()(f 1()f () f 1()f ())] = 0. f 1f f 1 f f 1 f + f 1 f C β 1, P () C (n 1)/(p 1), P f 1f f 1 f C γ 0, (6.17)