Gradient stability for the Sobolev inequality: the case p≥2
aa r X i v : . [ m a t h . A P ] O c t GRADIENT STABILITY FOR THE SOBOLEV INEQUALITY:THE CASE p ≥ ALESSIO FIGALLI AND ROBIN NEUMAYER
Abstract.
We prove a strong form of the quantitative Sobolev inequality in R n for p ≥
2, wherethe deficit of a function u ∈ ˙ W ,p controls k∇ u − ∇ v k L p for an extremal function v in the Sobolevinequality. Introduction
Given n ≥ ≤ p < n , the Sobolev inequality provides a control of the L r norm of afunction in terms of a suitable L p norm of its gradient. More precisely, setting p ∗ := np/ ( n − p ),one defines the homogeneous Sobolev space ˙ W ,p as the space of functions in R n such that u ∈ L p ∗ and |∇ u | ∈ L p . Then the following holds: k∇ u k L p ≥ S p,n k u k L p ∗ ∀ u ∈ ˙ W ,p . (1.1)Throughout the paper, all the integrals and function spaces will be over R n , so we will omit thedomain of integration when no confusion arises.It is well known that the optimal constant in (1.1) is given by S p,n = √ πn /p (cid:18) n − pp − (cid:19) ( p − /p (cid:18) Γ( n/p )Γ(1 + n − n/p )Γ(1 + n/ n ) (cid:19) /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 ( λ ( x − y )) , c ∈ R , λ ∈ R + , y ∈ R n , where v ( x ) := κ (1 + | x | p ′ ) ( n − p ) /p , (1.2)see [31, 1] and [13] (here κ is chosen so that k v k L p ∗ = 1, therefore k cv λ,y k L p ∗ = c , and p ′ := p/ ( p −
1) denotes the H¨older conjugate of p ). In other words, M := { cv λ,y : c ∈ R , λ ∈ R + , y ∈ R n } (1.3)is the ( n + 2)-dimensional manifold of extremal functions in the Sobolev inequality (1.1).To quantify how close a function u ∈ ˙ W ,p is to achieving equality in (1.1), we define its deficit to be the p -homogeneous functional δ ( u ) := k∇ u k pL p − S pp,n k u k pL p ∗ . By (1.1), the deficit is nonnegative and equals zero if and only if v ∈ M . In [5], Brezis and Liebraised the question of stability for the Sobolev inequality, that is, whether the deficit controls anappropriate distance between a function u ∈ ˙ W ,p and the family of extremal functions.This question was first answered in the case p = 2 by Bianchi and Egnell in [3]: there, theyshowed that the deficit of a function u controls the L distance between the gradient of u and the gradient of closest extremal function v . The result is optimal both in the strength of thedistance and the exponent of decay. However, their proof is very specific to the case p = 2, asit strongly exploits the Hilbert structure of ˙ W , . Later on, in [10], Cianchi, Fusco, Maggi, andPratelli considered the case 1 < p < n and provided a stability result in which the deficit controlsthe L p ∗ distance between u and some v ∈ M . Their proof uses a combination of symmetrizationtechniques and tools from the theory of mass transportation. More recently, in [21], Figalli, Maggi,and Pratelli used rearrangement techniques and mass transportation theory to show that, in thecase p = 1, the deficit controls the appropriate notion of distance of u from M at the level ofgradients (see also [22, 8] for partial results when p = 1). As in [3], the distance considered in [21]is the strongest that one expects to control and the exponent of decay is sharp.In view of [3] and [21], one may expect that, for all 1 < p < n , the deficit controls the L p distancebetween ∇ u and ∇ v for some v ∈ M ; this would answer the question of Breizis and Lieb in theaffirmative with the deficit controlling the strongest possible notion of distance in this setting. Themain result of this paper shows that, in the case p ≥
2, this result is indeed true. More precisely,our main result states the following:
Theorem 1.1.
Let ≤ p < n . There exists a constant C > , depending only on p and n , suchthat for all u ∈ ˙ W ,p , k∇ u − ∇ v k pL p ≤ C δ ( u ) + C k u k p − L p ∗ k u − v k L p ∗ (1.4) for some v ∈ M . As a consequence of Theorem 1.1 and the main result of [10] (see Theorem 5.5 below), we deducethe following corollary, proving the desired stability at the level of gradients:
Corollary 1.2.
Let ≤ p < n . There exists a constant C > , depending only on p and n , suchthat for all u ∈ ˙ W ,p , (cid:18) k∇ u − ∇ v k L p k∇ u k L p (cid:19) ζ ≤ C δ ( u ) k∇ u k pL p (1.5) for some v ∈ M , where ζ = p ∗ p (cid:16) p − p +1 n (cid:17) . The topic of stability for functional and geometric inequalities has generated much interest inrecent years. In addition to the aforementioned papers, results of this type have been addressedfor the isoperimetric inequality [23, 20, 11], log-Sobolev inequality [26, 4, 17], the higher orderSobolev inequality [24, 2], the fractional Sobolev inequality [7], the Morrey-Sobolev inequality [9]and the Gagliardo-Nirenberg-Sobolev inequality [6, 29], as well as for numerous other geometricinequalities. Aside from their intrinsic interest, stability results have applications in the study ofgeometric problems (see [18, 19, 12]) and can be used to obtain quantitative rates of convergencefor diffusion equations (as in [6]).For the remainder of the paper, we will always assume that 2 ≤ p < n. Acknowledgments:
A. Figalli is partially supported by NSF Grants DMS-1262411 and DMS-1361122. R. Neumayer is supported by the NSF Graduate Research Fellowship under GrantDGE-1110007. Both authors warmly thank Francesco Maggi for useful discussions regarding thiswork.
RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Theorem 1.1: idea of the proof
As a starting point to prove stability of (1.1) at the level of gradients, one would like to followthe argument used to prove the analogous result in [3]. However, this approach turns out to besufficient only in certain cases, and additional ideas are needed to conclude the proof. Indeed,a Taylor expansion of the deficit δ ( u ) and a spectral gap for the linearized problem allow us toshow that the second variation is strictly positive, but in general we cannot absorb the higher orderterms. Let us provide a few more details to see to what extent this approach works, where it breaksdown, and how we get around it.2.1. The expansion approach.
The first idea of the proof of Theorem 1.1 is in the spirit of thestability result of Bianchi and Egnell in [3]. Ultimately, this approach will need modification, butlet us sketch how such an argument would go.In order to introduce a Hilbert space structure to our problem, we define a weighted L -typedistance of a function u ∈ ˙ W ,p to M at the level of gradients. To this end, for each v = cv λ,y ∈ M ,we define A v ( x ) := ( p − |∇ v | p − ˆ r ⊗ ˆ r + |∇ v | p − Id , ˆ r = x − y | x − y | , (2.1)where ( a ⊗ b ) c := ( a · c ) b . Then, with the notation A v [ a, a ] := a T A v a for a ∈ R n , we define theweighted L distance of u to M byd( u, M ) := inf (cid:26)(cid:16) Z A v [ ∇ u − ∇ v, ∇ u − ∇ v ] (cid:17) / : v ∈ M , k v k L p ∗ = k u k L p ∗ (cid:27) = inf (cid:26)(cid:16) Z A cv λ,y [ ∇ u − ∇ cv λ,y , ∇ u − ∇ cv λ,y ] (cid:17) / : λ ∈ R + , y ∈ R n , c = k u k L p ∗ (cid:27) . (2.2)Note that Z A v [ ∇ u − ∇ v, ∇ u − ∇ v ] = Z |∇ v | p − |∇ u − ∇ v | + ( p − Z |∇ v | p − | ∂ r u − ∂ r v | . A few remarks about this definition are in order.
Remark 2.1.
The motivation to define d( u, M ) in this way instead of, for instance,inf (cid:26)(cid:16) Z |∇ v | p − |∇ u − ∇ v | (cid:17) / : v ∈ M , k v k L p ∗ = k u k L p ∗ (cid:27) , will become apparent in Section 3. This choice, however, is only technical, as Z |∇ v | p − |∇ u − ∇ v | ≤ Z A v [ ∇ u − ∇ v, ∇ u − ∇ v ] ≤ ( p − Z |∇ v | p − |∇ u − ∇ v | . Remark 2.2.
One could alternatively define the distance in (2.2) without the constraint c = k u k L p ∗ , instead also taking the infimum over the parameter c . Up to adding a small positivityconstraint to ensure that the infimum is not attained at v = 0, this definition works, but ultimatelythe current presentation is more straightforward. Remark 2.3.
The distance d( u, M ) has homogeneity p/
2, that is, d( cu, M ) = c p/ d( u, M ) . In Proposition 4.1(1), we show that there exists δ = δ ( n, p ) > δ ( u ) ≤ δ k∇ u k pL p , (2.3) FIGALLI AND NEUMAYER then the infimum in d( u, M ) is attained. Given a function u ∈ ˙ W ,p satisfying (2.3), let v ∈ M attain the infimum in (2.2) and define ϕ := u − v k∇ ( u − v ) k L p , so that u = v + ǫϕ with ǫ = k∇ ( u − v ) k L p and R |∇ ϕ | p = 1. Since δ ≥ δ ( v ) = 0, theTaylor expansion of the deficit of u around v vanishes both at the zeroth and first order. Thus, theexpansion leaves us with δ ( u ) = ǫ p Z A v [ ∇ ϕ, ∇ ϕ ] − ǫ S pp,n p ( p ∗ − Z | v | p ∗ − | ϕ | + o ( ǫ ) . (2.4)Since v ∈ M minimizes the distance between u and M , ǫϕ = u − v is orthogonal (in someappropriate sense) to the tangent space of M at v , which we shall see coincides with the spanthe first two eigenspaces of an appropriate weighted linearized p -Laplacian. Then, a gap in thespectrum in this operator allows us to show that c d( u, M ) = c ǫ Z A v [ ∇ ϕ, ∇ ϕ ] ≤ ǫ p Z A v [ ∇ ϕ, ∇ ϕ ] − ǫ S pp,n p ( p ∗ − Z | v | p ∗ − | ϕ | for a positive constant c = c ( n, p ). Together with (2.4), this impliesd( u, M ) + o ( ǫ ) ≤ Cδ ( u ) . Now, if the term o ( ǫ ) could be absorbed into d( u, M ) , then we could use the estimate (2.6) belowto obtain Z |∇ u − ∇ v | p ≤ Cδ ( u ) , which would conclude the proof.2.2. Where this approach falls short.
The problem arises exactly when trying to absorb theterm o ( ǫ ). Indeed, recalling that ǫ = k∇ ( u − v ) k L p , we are asking whether o ( k∇ u − ∇ v k L p ) ≪ d( u, M ) ≈ Z |∇ v | p − |∇ u − ∇ v | (recall Remark 2.1), and unfortunately this is false in general. Notice that this problem never arisesin [3] for the case p = 2, as the above inequality reduces to o ( k∇ u − ∇ v k L ) ≪ k∇ u − ∇ v k L , which is clearly true.2.3. The solution.
A Taylor expansion of the deficit will not suffice to prove Theorem 1.1 aswe cannot hope to absorb the higher order terms. Instead, for a function u ∈ ˙ W ,p , we givetwo different expansions, each of which gives a lower bound on the deficit, by splitting the termsbetween the second order term and the p th order term using elementary inequalities (Lemma 3.2).Pairing this with an analysis of the second variation, we obtain the following: RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Proposition 2.4.
There exist constants c , C , and C , depending only on p and n , such that thefollowing holds. Let u ∈ ˙ W ,p be a function satisfying (2.3) and let v ∈ M be a function where theinfimum of the distance (2.2) is attained. Then c d( u, M ) − C Z |∇ u − ∇ v | p ≤ δ ( u ) , (2.5) − C d( u, M ) + 14 Z |∇ u − ∇ v | p ≤ δ ( u ) . (2.6)Individually, both inequalities are quite weak. However, as shown in Corollary 4.3, they allow usto prove Theorem 1.1 (in fact, the stronger statement R |∇ u − ∇ v | p ≤ δ ( u )) for the set of functions u such that d( u, M ) = Z A v [ ∇ u − ∇ v, ∇ u − ∇ v ] ≪ Z |∇ u − ∇ v | p ord( u, M ) = Z A v [ ∇ u − ∇ v, ∇ u − ∇ v ] ≫ Z |∇ u − ∇ v | p . (2.7)We are then left to consider the middle regime, where Z A v [ ∇ u − ∇ v, ∇ u − ∇ v ] ≈ Z |∇ u − ∇ v | p . We handle this case as follows. Let u t := (1 − t ) u + tv be the linear interpolation between u and v . Choosing t ∗ small enough, u t ∗ falls in the second regime in (2.7), so Theorem 1.1 holds for u t ∗ .We then must relate the deficit and distance of u t ∗ to those of u . While relating the distances isstraightforward, it is not clear for the deficits whether the estimate δ ( u t ∗ ) ≤ Cδ ( u ) holds. Still, wecan show that δ ( u t ∗ ) ≤ Cδ ( u ) + C k v k p − L p ∗ k u − v k L p ∗ , which allows us to conclude the proof. It is this point in the proof that introduces that term k u − v k L p ∗ in Theorem 1.1, and for this reason we rely on the main theorem of [10] to proveCorollary 1.2. We note that the application of [10] is not straightforward, since the function v which attains the minimum in our setting is a priori different from the one considered there (seeSection 5 for more details).2.4. Outline of the paper.
The paper is structured as follows.In Section 3, we introduce the operator L v that will be important in our analysis of the secondvariation of the deficit and prove some facts about the spectrum of this operator. We also provesome elementary but crucial inequalities in Lemma 3.2 and provide orthogonality constraints thatarise from taking the infimum in (2.2).In Section 4, we prove Proposition 2.4 by exploiting a gap in the spectrum of L v and using theinequalities of Lemma 3.2.In Section 5, we combine Proposition 2.4 with an interpolation argument to obtain Theorem 1.1.We then apply the main result of [10] in order to prove Corollary 1.2. FIGALLI AND NEUMAYER
In Section 6, we prove the compact embedding that shows that L v has a discrete spectrum andjustify the use of Sturm-Liouville theory in the proof of Proposition 3.1.Section 7 is an appendix in which we prove a technical claim.3. Preliminaries
In this section, we state a few necessary facts and tools.3.1.
The tangent space of M and the operator L v . The set M of extremal functions defined in(1.3) is an ( n + 2)-dimensional smooth manifold except at 0 ∈ M . For a nonzero v = c v λ ,y ∈ M ,the tangent space is computed to be T v M = span { v, ∂ λ v, ∂ y v, . . . , ∂ y n v } , where y i denotes the i th component of y and ∂ λ v = ∂ λ | λ = λ v , ∂ y i v = ∂ y i | y i = y i v .Since the functions v = v λ ,y minimize u δ ( u ) and have k v λ ,y k L p ∗ = 1, by computing theEuler-Lagrange equation one discovers that − ∆ p v = S pp,n v p ∗ − , (3.1)where the p -Laplacian ∆ p is defined by ∆ p w := div ( |∇ w | p − ∇ w ). Hence, differentiating (3.1) withrespect to y i or λ , we see that − div ( A v ( x ) ∇ w ) = ( p ∗ − S pp,n v p ∗ − w, w ∈ span { ∂ λ v, ∂ y v, . . . , ∂ y n v } , (3.2)where A v ( x ) is as defined in (2.1). This motivates us to consider the weighted operator L v w := − div ( A v ( x ) ∇ w ) v − p ∗ (3.3)on the space L ( v p ∗ − ), where, for a measurable weight ω : R n → R , we let k w k L ( ω ) = (cid:16) Z R n | w | ω (cid:17) / , L ( ω ) = { w : R n → R : k w k L ( ω ) < ∞} . Proposition 3.1.
The operator L v has a discrete spectrum { α i } ∞ i =1 , with < α i < α i +1 for all i ,and α = ( p − S pp,n , H = span { v } , (3.4) α = ( p ∗ − S pp,n , H = span { ∂ λ v, ∂ y v, . . . , ∂ y n v } , (3.5) where H i denotes the eigenspace corresponding to α i . In particular, Proposition 3.1 implies that T v M = span { H ∪ H } . (3.6)The Rayleigh quotient characterization of eigenvalues implies that α = inf (cid:26) hL v w, w ih w, w i = R A v [ ∇ w, ∇ w ] R v p ∗ − w : w ⊥ span { H ∪ H } (cid:27) , (3.7)where orthogonality is with respect to the inner product defined by h w , w i := Z v p ∗ − w w . (3.8)Note that the eigenvalues of L v are invariant under changes in λ and y . RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Proof of Proposition 3.1.
The discrete spectrum of L v follows in the usual way after establishingthe right compact embedding theorem; we show the compact embedding in Corollary 6.2 and givedetails confirming the discrete spectrum in Corollary 6.3. Since a scaling argument shows that theeigenvalues of L v are invariant under changes of λ and y , it suffices to consider the operator L = L v for v = v , , letting A = A v .One easily verifies that v is an eigenfunction of L with eigenvalue ( p − S pp,n and that ∂ λ v and ∂ y i v are eigenfunctions with eigenvalue ( p ∗ − S pp,n , using (3.1) and (3.2) repectively. Furthermore,since v >
0, it follows that α = ( p − S pp,n is the first eigenvalue, which is simple, so (3.4) holds.To prove (3.5), we must show that α = ( p ∗ − S pp,n is the second eigenvalue and verify thatthere are no other eigenfunctions in H . Both of these facts follow from separation of variables andSturm-Liouville theory. Indeed, an eigenfunction ϕ of L satisfiesdiv ( A ( x ) ∇ ϕ ) + αv p ∗ − ϕ = 0 . (3.9)Assume that ϕ takes the form ϕ ( x ) = Y ( θ ) f ( r ), where Y : S n − → R and f : R → R . In polarcoordinates, div( A ( x ) ∇ ϕ ) = ( p − |∇ v | p − ∂ rr ϕ + ( p − n − r |∇ v | p − ∂ r ϕ + 1 r |∇ v | p − n − X j =1 ∂ θ j θ j ϕ + ( p − p − |∇ v | p − ∂ r v ∂ rr v ∂ r ϕ (3.10)(this computation is given in the appendix for the convenience of the reader). As v is radiallysymmetric, that is, v ( x ) = w ( | x | ), we introduce the slight abuse of notation by letting v ( r ) alsodenote the radial component: v ( r ) = w ( r ), so v ′ ( r ) = ∂ r v and v ′′ ( r ) = ∂ rr v. From (3.10), we seethat (3.9) takes form0 = ( p − | v ′ | p − f ′′ ( r ) Y ( θ ) + ( p − n − r | v ′ | p − f ′ ( r ) Y ( θ )+ 1 r | v ′ | p − f ( r )∆ S n − Y ( θ ) + ( p − p − | v ′ | p − v ′ v ′′ f ′ ( r ) Y ( θ ) + αv p ∗ − f ( r ) Y ( θ ) , which yields the system0 = ∆ S n − Y ( θ ) + µY ( θ ) on S n − , (3.11)0 = ( p − | v ′ | p − f ′′ + ( p − n − r | v ′ | p − f ′ − µr | v ′ | p − f +( p − p − | v ′ | p − v ′ v ′′ f ′ + αv p ∗ − f on [0 , ∞ ) . (3.12)The eigenvalues and eigenfunctions of (3.11) are explicitly known; these are the spherical harmon-ics. The first two eigenvalues are µ = 0 and µ = n − µ = µ = 0 in (3.12), we claim that:- α = ( p − S pp,n and the corresponding eigenspace is span { v } ;- α = ( p ∗ − S pp,n with the corresponding eigenspace span { ∂ λ v } .Indeed, Sturm-Liouville theory ensures that each eigenspace is one-dimensional, and that the i th eigenfunction has i − v (resp. ∂ λ v ) solves (3.12) with µ = 0 and α = ( p − S pp,n (resp. α = ( p ∗ − S pp,n ), having no zeros (resp. one zero) it must be the first (resp. FIGALLI AND NEUMAYER second) eigenfunction.For µ = n −
1, the eigenspace for (3.11) is n dimensional with n eigenfunctions giving the spher-ical components of ∂ y i v, i = 1 , . . . , n . The corresponding equation in (3.12) gives α = ( p ∗ − S pp,n .As the first eigenvalue of (3.12) with µ = µ , α is simple.The eigenvalues are strictly increasing, so this shows that α > ( p ∗ − S pp,n and α > ( p ∗ − S pp,n ,concluding the proof. (cid:3) The application of Sturm-Liouville theory in the proof above is not immediately justified becauseours is a singular
Sturm-Liouville problem. The proof of Sturm-Liouville theory in our setting,that is, that each eigenspace is one-dimensional and that the i th eigenfunction has i − Some useful inequalities.
The following lemma contains four elementary inequalities forvectors and numbers. This lemma is a key tool for getting around the issues presented in theintroduction; in lieu of a Taylor expansion, these inequalities yield bounds on the deficit by splittingthe higher order terms between the second order terms and the p th or p ∗ th order terms. Lemma 3.2.
Let x, y ∈ R n and a, b ∈ R . The following inequalities hold.For all κ > , there exists a constant C = C ( p, n, κ ) such that | x + y | p ≥ | x | p + p | x | p − x · y + (1 − κ ) (cid:16) p | x | p − | y | + p ( p − | x | p − ( x · y ) (cid:17) − C | y | p . (3.13) For all κ > , there exists C = C ( p, κ ) such that | a + b | p ∗ ≤ | a | p ∗ + p ∗ | a | p ∗ − ab + (cid:16) p ∗ ( p ∗ − κ (cid:17) | a | p ∗ − | b | + C | b | p ∗ . (3.14) There 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) There exists C = C ( p ) such that | a + b | p ∗ ≤ | a | p ∗ + p ∗ | a | p ∗ − ab + C | a | p ∗ − | b | + 2 | b | p ∗ . (3.16) Proof of Lemma 3.2.
We only give the proof of (3.13), as the proofs of (3.14)-(3.16) are analogous.Observe that if p is an even integer or p ∗ is an integer, these inequalities follow (with explicitconstants) from a binomial expansion and splitting the intermediate terms between the secondorder and p th or p ∗ th order terms using Young’s inequality.Suppose (3.13) fails. Then there exists κ > { 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 ) (cid:18) p | x j | p − | y j | + p ( p − | x j | p − ( x j · y j ) (cid:19) − C j | y j | p . If x j = 0 , we immediately get a contradiction. Otherwise, we divide by | x j | p to obtain | x j + y j | p | x j | p − < p x j · y j | x j | + (1 − κ ) p (cid:18) | y j | | x j | + ( p −
2) ( x j · y j ) | x j | (cid:19) − C j | y j | p | x j | p . (3.17) RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ The left-hand side is bounded below by −
1, so in order for (3.17) to hold, | y j | / | x j | must convergeto 0 at a sufficiently fast rate. In this case, | y j | is much smaller that | x j | , so a Taylor expansionreveals that the left-hand side behaves like p x j · y j | x j | + p | y j | | x j | + p ( p − x j · y j ) | x j | + o (cid:16) | y j | | x j | (cid:17) , (3.18)which is larger than the right-hand side, contradicting (3.17). (cid:3) With the same proof, one can show (3.14) with the opposite sign: For all κ >
0, there exists C = C ( p, κ ) such that | a + b | p ∗ ≥ | a | p ∗ + p ∗ | a | p ∗ − ab − (cid:16) p ∗ ( p ∗ − κ (cid:17) | a | p ∗ − | b | − C | b | p ∗ . Therefore, applying this and (3.14) to functions v and v + ϕ with R | v | p ∗ = R | v + ϕ | p ∗ , one obtains (cid:12)(cid:12)(cid:12)(cid:12)Z | v | p ∗ − vϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) p ∗ ( p ∗ − κ (cid:17) Z | v | p ∗ − | ϕ | + C Z | ϕ | p ∗ . (3.19)3.3. Orthogonality constraints for u − v . Given a function u ∈ ˙ W ,p satisfying (2.3), supposethat v = c v λ ,y is a function at which the infimum is attained in (2.2). Then Z | u | p ∗ = Z | v | p ∗ = c p ∗ , (3.20)and the energy E ( v ) = E ( λ, y ) = Z A c v λ,y [ ∇ u − c ∇ v λ,y , ∇ u − c ∇ v λ,y ] , (3.21)arising from (2.2) when u is fixed, has a critical point at ( λ , y ) in the n + 1 parameters λ and y i , i = 1 , . . . , n . In other words,0 = ∂ λ | λ = λ Z A c v λ,y [ ∇ u − c ∇ v λ,y , ∇ u − c ∇ v λ,y ] , ∂ y i | y i = y i Z A c v λ,y [ ∇ u − c ∇ v λ,y , ∇ u − c ∇ v λ,y ] . (3.22)We express u as u = v + ǫϕ , with ϕ scaled such that R |∇ ϕ | p = 1 . Computing the derivatives in(3.22) gives ǫ Z A v [ ∇ ∂ λ v, ∇ ϕ ] = ǫ ( p − Z |∇ ϕ | |∇ v | p − ∇ v · ∇ ∂ λ v + ǫ ( p − Z |∇ ϕ | |∇ v | p − ∂ r v ∂ rλ v,ǫ Z A v [ ∇ ∂ y i v, ∇ ϕ ] = ǫ ( p − Z |∇ ϕ | |∇ v | p − ∇ v · ∇ ∂ y i v + ǫ ( p − Z |∇ ϕ | |∇ v | p − ∂ r v ∂ ry i v + ǫ ( p − Z |∇ v | p − ∂ r ϕ ∇ ϕ · ∂ y i ˆ r, (3.23) where ˆ r is as in (2.1). Furthermore, multiplying (3.2) by ǫϕ and integrating by parts implies S pp,n ( p ∗ − ǫ Z | v | p ∗ − ∂ λ v ϕ = ǫ Z A v [ ∇ ∂ λ v, ∇ ϕ ] ,S pp,n ( p ∗ − ǫ Z | v | p ∗ − ∂ y i v ϕ = ǫ Z A v [ ∇ ∂ y i v, ∇ ϕ ] , so (3.23) becomes ǫ Z | v | p ∗ − ∂ λ v ϕ = ǫ C (cid:20)Z |∇ ϕ | |∇ v | p − ∇ v · ∇ ∂ λ v + ( p − Z |∇ ϕ | |∇ v | p − ∂ r v ∂ rλ v (cid:21) , (3.24) ǫ Z | v | p ∗ − ∂ y i v ϕ = ǫ C (cid:20)Z |∇ ϕ | |∇ v | p − ∇ v · ∇ ∂ y i v + ( p − Z |∇ ϕ | |∇ v | p − ∂ r v ∂ ry i v (3.25)+ 2 Z |∇ v | p − ∂ r ϕ ∇ ϕ · ∂ y i ˆ r (cid:21) , where C = ( p − p ∗ − S pp,n .A Taylor expansion of the constraint (3.20) implies − ǫ Z | v | p ∗ − vϕ = ǫ Z | v | p ∗ − | ϕ | + o ( ǫ ) . However, in view of the comments in the introduction, we cannot generally absorb the term o ( ǫ ),so this is not quite the form of the orthogonality constraint that we need. In its place, using (3.20)and (3.19), we have (cid:12)(cid:12)(cid:12)(cid:12) ǫ Z | v | p ∗ − vϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ p ∗ − κ Z | v | p ∗ − | ϕ | + C ǫ p ∗ Z | ϕ | p ∗ (3.26)for any κ >
0, with C = C ( p, n, κ ).The conditions (3.24), (3.25), and (3.26) show that ϕ is “almost orthogonal” to T v M withrespect to the inner product given in (3.8). Indeed, dividing through by ǫ , the inner product of ϕ with each basis element of T v M appears on the left-hand side of (3.24), (3.25), and (3.26), whilethe right-hand side is O ( ǫ ). As a result of (3.6) and ϕ being “almost orthogonal” to T v M , it isshown that ϕ satisfies a Poincar´e-type inequality (4.14), which is an essential point in the proof ofProposition 2.4. Remark 3.3.
In [3], the analogous constraints give orthogonality rather than almost orthogonality;this is easily seen here, as taking p = 2 makes the right-hand sides of (3.24) and (3.25) vanish.4. Proof of Proposition 2.4 and its consequences
We prove Proposition 2.4 combining an analysis of the second variation and the inequalities ofLemma 3.2. As a consequence (Corollary 4.3), we show that, up to removing the assumption (2.3),Theorem 1.1 holds for the two regimes described in (2.7).To prove Proposition 2.4, we will need two facts. First, we want to know that the infimum in(2.2) is attained, so that we can express u as u = v + ǫϕ where R |∇ ϕ | p = 1, and ϕ satisfies (3.24),(3.25), and (3.26). Second, it will be important to know that if δ in (2.3) is small enough, then ǫ is small as well. For this reason we first prove the following: Proposition 4.1.
The following two claims hold.
RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ (1) There exists δ = δ ( n, p ) > such that if δ ( u ) ≤ δ k∇ u k pL p , (4.1) then the infimum in (2.2) is attained. In other words, there exists some v ∈ M with R | v | p ∗ = R | u | p ∗ such that Z A v [ ∇ u − ∇ v, ∇ u − ∇ v ] = d( u, M ) . (2) For all ǫ > , there exists δ = δ ( n, p, ǫ ) > such that if u ∈ ˙ W ,p satisfies (4.1) , then ǫ := k∇ u − ∇ v k L p < ǫ where v ∈ M is a function that attains the infimum in (2.2) .Proof. We begin by showing the following fact, which will be used in the proofs of both parts ofthe proposition: for all γ >
0, there exists δ = δ ( n, p, γ ) > δ ( u ) ≤ δ k∇ u k pL p , theninf {k∇ u − ∇ v k L p : v ∈ M} ≤ γ k∇ u k L p . (4.2)Otherwise, for some γ >
0, there exists a sequence { u k } ⊂ ˙ W ,p such that k∇ u k k L p = 1 and δ ( u k ) → {k∇ u k − ∇ v k L p : v ∈ M} > γ. A concentration compactness argument as in [27, 30] ensures that there exist sequences { λ k } and { y k } such that, up to a subsequence, λ n/p ∗ k u k ( λ k ( x − y k )) converges strongly in ˙ W ,p to some¯ v ∈ M . Since γ < (cid:13)(cid:13)(cid:13)(cid:13) ∇ u k − ∇ h λ − n/p ∗ k ¯ v (cid:16) · λ k + y k (cid:17)i(cid:13)(cid:13)(cid:13)(cid:13) L p = (cid:13)(cid:13)(cid:13) ∇ h λ n/p ∗ k u k ( λ k ( · − y k )) i − ∇ ¯ v (cid:13)(cid:13)(cid:13) L p → k sufficiently large, hence (4.2) holds. Proof of (1).
Suppose u satisfies (4.1), with δ to be determined in the proof. Up to multiplicationby a constant, we may assume that k u k L p ∗ = 1 . By the claim above, we may take δ small enoughso that (4.2) holds for γ as small as needed.The infimum on the left-hand side of (4.2) is attained. Indeed, let { v k } be a minimizing sequencewith v k = c k v λ k ,y k . The sequences { c k } , { λ k } , { /λ k } , and { y k } are bounded: if λ k → ∞ or λ k → k large enough there will be little cancellation in the term |∇ u − ∇ v k | p , so that Z |∇ u − ∇ v k | p ≥ Z |∇ u | p , contradicting (4.2). The analogous argument holds if | y k | → ∞ or | c k | → ∞ . Thus { c k } , { λ k } , { /λ k } , and { y k } are bounded and so, up to a subsequence, ( c k , λ k , y k ) → ( c , λ , y ) for some( c , λ , y ) ∈ R × R + × R n . Since the functions cv λ,y are smooth, decay nicely, and depend smoothlyon the parameters, we deduce that v k → c v λ ,y = ˜ v in ˙ W ,p (actually, they also converge in C k for any k ), hence ˜ v attains the infimum.To show that the infimum is attained in (2.2), we obtain an upper bound on the distance by using¯ v = ˜ v/ k ˜ v k L p ∗ as a competitor. Indeed, recalling Remark 2.1, it follows from H¨older’s inequalitythat d( u, M ) ≤ ( p − Z |∇ ¯ v | p − |∇ u − ∇ ¯ v | ≤ ( p − S ( p − /pp,n k∇ u − ∇ ¯ v k /pL p . Notice that, since k u k L p ∗ = 1, it follows by (4.1) that k∇ u k L p ≤ S pp,n provided δ ≤ /
2. Hence,since (cid:12)(cid:12) k ¯ v k L p ∗ − (cid:12)(cid:12) ≤ k ¯ v − u k L p ∗ ≤ S − pn,p k∇ ¯ v − ∇ u k L p , it follows by (4.2) and the triangle inequality that k∇ u − ∇ ¯ v k L p ≤ C ( n, p ) γ , therefored( u, M ) ≤ C ( n, p ) γ /p . (4.3)Hence, if { v k } is a minimizing sequence for (2.2) with v k = v λ k ,y k (so that R | v k | p ∗ = R | u | p ∗ = 1),the analogous argument as above shows that if either of the sequences { λ k } , { /λ k } , or { y k } areunbounded, then d( u, M ) ≥ , contradicting (4.3) for γ sufficiently small. This implies that v k → v λ ,y in ˙ W ,p , and by continuity v λ ,y attains the infimum in (2.2). Proof of (2).
We have shown that (4.2) holds for δ sufficiently small. Therefore, we need only toshow that, up to further decreasing δ , there exists C = C ( p, n ) such that k∇ u − ∇ v k L p ≤ C inf {k∇ u − ∇ v k L p : v ∈ M} , where v ∈ M is the function where the infimum is attained in (2.2).Suppose for the sake of contradiction that there exists a sequence { u j } such that δ ( u j ) → k∇ u j k L p = 1 but Z |∇ u j − ∇ v j | p ≥ j Z |∇ u j − ∇ ¯ v j | p , (4.4)where v j , ¯ v j ∈ M are such that Z A v j [ ∇ u j − ∇ v j , ∇ u j − ∇ v j ] = d( u j , M ) and Z |∇ u j − ∇ ¯ v j | p = inf n Z |∇ u j − ∇ v j | p : v ∈ M o . Since δ ( u j ) →
0, the same concentration compactness argument as above implies that there existsequences { λ j } and { y j } such that, up to a subsequence, λ n/p ∗ j u j ( λ j ( x − y j )) converges in ˙ W ,p tosome v ∈ M with k∇ v k L p = 1. By an argument analogous to that in part (1), we determine that v j → v in C k and ¯ v j → v in C k for any k . Let φ j = u j − v j k∇ u j − ∇ v j k L p and ¯ φ j = u j − ¯ v j k∇ u j − ∇ v j k L p . Then (4.4) implies that 1 = Z |∇ φ j | p ≥ j Z |∇ ¯ φ j | p . (4.5)In particular, ∇ ¯ φ j → L p . Now define ψ j = φ j − ¯ φ j = ¯ v j − v j k∇ u j − ∇ v j k L p . For any η >
0, (4.5) implies that 1 − η ≤ k∇ ψ j k L p ≤ η for j large enough. In particular, {∇ ψ j } is bounded in L p and so ∇ ψ j ⇀ ∇ ψ in L p for some ψ ∈ ˙ W ,p . RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ We now consider the finite dimensional manifold ¯ M := { v − ¯ v : v, ¯ v ∈ M} . Since v j , ¯ v j → v, thesequences { λ j } , { /λ j } , { y j } , { ¯ λ j } , { / ¯ λ j } and { ¯ y j } are contained in some compact set, and thusall norms of ¯ v j − v j are equivalent: for any norm |||·||| on ¯ M there exists µ > µ k∇ ¯ v j − ∇ v j k L p ≤ |||∇ ¯ v j − ∇ v j ||| ≤ µ k∇ ¯ v j − ∇ v j k L p . (4.6)Dividing (4.6) by k∇ u j − ∇ v j k L p gives µ (1 − η ) ≤ µ k∇ ψ j k L p ≤ |||∇ ψ j ||| ≤ µ k∇ ψ i k L p ≤ ηµ . (4.7)Taking the norm |||·||| = k · k C k , the upper bound in (4.7) and the Arzel`a-Ascoli theorem imply that ψ j converges, up to a subsequence, to ψ in C k . The lower bound in (4.7) implies that k ψ k C k = 0.To get a contradiction, we use the minimality of v j for d( u j , M ) to obtain Z |∇ ¯ v j | p − |∇ ¯ φ j | + ( p − Z |∇ ¯ v j | p − | ∂ r ¯ φ j | ≥ Z |∇ v j | p − |∇ φ j | + ( p − Z |∇ v j | p − | ∂ r φ j | = Z |∇ v j | p − |∇ ¯ φ j | + 2 Z |∇ v j | p − ∇ ¯ φ j · ∇ ψ j + Z |∇ v j | p − |∇ ψ j | + ( p − (cid:18)Z |∇ v j | p − | ∂ r ¯ φ j | + 2 Z |∇ v j | p − ∂ r ¯ φ j ∂ r ψ j + Z |∇ v j | p − | ∂ψ j | (cid:19) . (4.8)Since Z |∇ ¯ v j | p − |∇ ¯ φ j | − Z |∇ v j | p − |∇ ¯ φ j | → Z |∇ ¯ v j | p − | ∂ r ¯ φ j | − Z |∇ v j | p − | ∂ r ¯ φ j | → , (4.8) implies that0 ≥ j →∞ Z |∇ v j | p − ∇ ¯ φ j · ∇ ψ j + lim j →∞ Z |∇ v j | p − |∇ ψ j | + ( p − (cid:18) j →∞ Z |∇ v j | p − ∂ r ¯ φ j ∂ r ψ j + lim j →∞ Z |∇ v j | p − | ∂ψ j | (cid:19) . (4.9)However, since ∇ ¯ φ j → L p ,lim j →∞ Z |∇ v j | p − ∇ ¯ φ j · ∇ ψ j = 0 and lim j →∞ Z |∇ v j | p − ∂ r ¯ φ j ∂ r ψ j = 0 . In addition, the terms Z |∇ v j | p − |∇ ψ j | and Z |∇ v j | p − | ∂ r ψ j | converge to something strictly positive, as ψ j → ψ v j → v with ∇ v ( x ) = 0 for all x = 0.This contradicts (4.9) and concludes the proof. (cid:3) The following Poincar´e inequality will be used in the proof of Proposition 2.4:
Lemma 4.2.
There exists a constant
C > such that Z | v | p ∗ − | ϕ | ≤ C Z |∇ v | p − |∇ ϕ | (4.10) for all ϕ ∈ ˙ W ,p .Proof. Let v ∈ M and ϕ ∈ C ∞ . As v is a local minimum of the functional δ ,0 ≤ d dǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 δ ( v + ǫϕ ) = p Z |∇ v | p − |∇ ϕ | + p ( p − Z |∇ v | p − | ∂ r ϕ | − S pp,n (cid:18) p (cid:16) pp ∗ − (cid:17)(cid:16) Z | v | p ∗ (cid:17) p/p ∗ − (cid:16) Z v p ∗ − v ϕ (cid:17) + p ( p ∗ − (cid:0) Z | v | p ∗ (cid:1) p ∗ /p − Z | v | p ∗ − ϕ (cid:19) . Noting that Z |∇ v | p − | ∂ r ϕ | ≤ Z |∇ v | p − |∇ ϕ | and (cid:16) Z | v | p ∗ (cid:17) p/p ∗ − (cid:16) Z v p ∗ − v ϕ (cid:17) ≥ , this implies that0 ≤ p ( p − Z |∇ v | p − |∇ ϕ | − S pp,n p ( p ∗ − (cid:16) Z | v | p ∗ (cid:17) p ∗ /p − Z | v | p ∗ − ϕ . Thus (4.10) holds for ϕ ∈ C ∞ , and for ϕ ∈ ˙ W ,p by approximation. (cid:3) We now prove Proposition 2.4.
Proof of Proposition 2.4.
First of all, thanks to (2.3), we can apply Proposition 4.1(1) to ensurethat some v = c v λ ,y ∈ M attains the infimum in (2.2). Also, expressing u as u = v + ǫϕ where R |∇ ϕ | p = 1, it follows from Proposition 4.1(2) and the discussion in Section 3.3 that ǫ can beassumed to be as small as desired (provided δ is chosen small enough) and that ϕ satisfies (3.24),(3.25), and (3.26). Note that, since all terms in (2.5) and (2.6) are p -homogeneous, without loss ofgenerality we may take c = 1 . Proof of (2.5) . The inequalities (3.13) and (3.14) are used to expand the gradient term and thefunction term in δ ( u ) respectively, splitting higher order terms between the second order and the p th or p ∗ th order terms.From (3.13) and for κ = κ ( p, n ) > Z |∇ u | p ≥ Z |∇ v | p + ǫp Z |∇ v | p − ∇ v · ∇ ϕ + ǫ p (1 − κ )2 (cid:16)Z |∇ v | p − |∇ ϕ | + ( p − Z |∇ v | p − | ∂ r ϕ | (cid:17) − ǫ p C Z |∇ ϕ | p . (4.11)Note that the second order term is precisely ǫ p (1 − κ ) R A v [ ∇ ϕ, ∇ ϕ ] . Similarly, (3.14) gives Z | u | p ∗ ≤ ǫp ∗ Z v p ∗ − ϕ + ǫ (cid:16) p ∗ ( p ∗ − p ∗ κ S pp,n (cid:17) Z v p ∗ − ϕ + C ǫ p ∗ Z | ϕ | p ∗ . (4.12)From the identity (3.1), the first order term in (4.12) is equal to ǫp ∗ Z v p ∗ − ϕ = ǫp ∗ S − pp,n Z |∇ v | p − ∇ v · ∇ ϕ. (4.13) RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Using (4.13) and recalling that ( p ∗ − S pp,n = α (see (3.5)), (4.12) becomes Z | u | p ∗ ≤ ǫp ∗ S pp,n Z |∇ v | p − ∇ v · ∇ ϕ + ǫ p ∗ ( α + κ )2 S pp,n Z v p ∗ − ϕ + C ǫ p ∗ , The following estimate holds, and is shown below: ǫ Z v p ∗ − ϕ ≤ (1 + 2 κ ) ǫ α Z A v [ ∇ ϕ, ∇ ϕ ] + C ǫ p , (4.14)Philosophically, (4.14) follows from a spectral gap analysis, using (3.7) and the fact that (3.24),(3.25), and (3.26) imply that ϕ is “almost orthogonal” to H and H .As ǫ may be taken as small as needed, using (4.14) we have Z | u | p ∗ ≤ p ∗ S pp,n (cid:18) ǫ Z |∇ v | p − ∇ v · ∇ ϕ + ǫ ( α + κ )(1 + 2 κ )2 α Z A v [ ∇ ϕ, ∇ ϕ ] + C ǫ p (cid:19) . The function z
7→ | z | p/p ∗ is concave, so k u k pL p ∗ ≤ pp ∗ ( R | u | p ∗ − S pp,n k u k pL p ∗ ≤ S pp,n + p (cid:18) ǫ Z |∇ v | p − ∇ v · ∇ ϕ + ǫ ( α + κ )(1 + 2 κ )2 α Z A v [ ∇ ϕ, ∇ ϕ ] + C ǫ p (cid:19) . (4.15)Subtracting (4.15) from (4.11) gives δ ( u ) ≥ ǫ p (cid:18) − κ − ( α + κ )(1 + 2 κ ) α (cid:19) Z A [ ∇ ϕ, ∇ ϕ ] − C ǫ p . Since 1 − α α >
0, we may choose κ sufficiently small so that 1 − κ − ( α + κ )(1+2 κ ) α >
0. To concludethe proof of (2.5), we need only to prove (4.14).
Proof of (4.14) . If ϕ were orthogonal to T v M instead of almost orthogonal, that is, if the right-hand sides of (3.24), (3.25), and (3.26) were equal to zero, then (4.14) would be an immediateconsequence of (3.7). Therefore, the proof involves showing that the error in the orthogonalityrelations is truly higher order, in the sense that it can be absorbed in the other terms.Up to rescaling u and v , we may assume that λ = 1 and y = 0. We recall the inner product h w, y i defined in (3.8) which gives rise to the norm k w k = (cid:16) Z | v | p ∗ − w (cid:17) / . As in Section 3, we let H i denote the eigenspace of L v in L ( v p ∗ − ) corresponding to eigenvalue α i , so H i = span { Y i,j } N ( i ) j =1 , where Y i,j is an eigenfunction with eigenvalue α i with k Y i,j k = 1 . Weexpress ǫϕ in the basis of eigenfunctions: ǫϕ = ∞ X i =1 N ( i ) X j =1 β i,j Y i,j where β i,j := ǫ Z | v | p ∗ − ϕY i,j . We let ǫ ˜ ϕ be the truncation of ǫϕ : ǫ ˜ ϕ = ǫϕ − X i =1 N ( i ) X j =1 β i,j Y i,j , so that ˜ ϕ is orthogonal to span { H ∪ H } and, introducing the shorthand β i := P N ( i ) j =1 β i,j , Z | v | p ∗ − ( ǫϕ ) = Z | v | p ∗ − ( ǫ ˜ ϕ ) + β + β . (4.16)Applying (3.7) to ˜ ϕ implies that Z | v | p ∗ − ( ǫ ˜ ϕ ) ≤ ǫ α hL v ˜ ϕ, ˜ ϕ i , which combined with (4.16) gives Z | v | p ∗ − ( ǫϕ ) ≤ ǫ α hL v ˜ ϕ, ˜ ϕ i + β + β = 1 α ∞ X i =3 α i β i + β + β ≤ ǫ α hL v ϕ, ϕ i + (cid:16) − α α (cid:17) ( β + β ) . (4.17)We thus need to estimate β + β . The constraint (3.26) implies β ≤ (cid:16) ǫ p ∗ − κ Z | v | p ∗ − | ϕ | + C ǫ p ∗ Z | ϕ | p ∗ (cid:17) ≤ C ǫ (cid:16) Z | v | p ∗ − | ϕ | (cid:17) + C ǫ p ∗ (cid:16) Z | ϕ | p ∗ (cid:17) . By (4.10), R | v | p ∗ − | ϕ | ≤ R ∇ v | p − |∇ ϕ | . Furthermore, both R |∇ v | p − |∇ ϕ | and R | ϕ | p ∗ are uni-versally bounded, so for ǫ sufficiently small depending only on p and n and κ , β ≤ κǫ α (cid:16) Z |∇ v | p − |∇ ϕ | + ( p − Z |∇ v | p − | ∂ r ϕ | (cid:17) + C ǫ p . (4.18)For β , , we notice that H¨older’s inequality and (3.24) imply β , ≤ (cid:16) C p,n ǫ Z |∇ v | p − |∇ ϕ | |∇ ∂ λ v |k ∂ λ v k (cid:17) ≤ C p,n R |∇ v | p − |∇ ∂ λ v | k ∂ λ v k Z |∇ v | p − | ǫ ∇ ϕ | = C p,n ǫ Z |∇ v | p − |∇ ϕ | , (4.19)where the final equality follows because the term R |∇ v | p − |∇ ∂ λ v | / k ∂ λ v k is bounded (in fact, itis bounded by α ). Then, using Young’s inequality, we get β , ≤ ǫ κ ( n + 1) α (cid:16) Z |∇ v | p − |∇ ϕ | + ( p − Z |∇ v | p − | ∂ r ϕ | (cid:17) + C κ,p ǫ p Z |∇ ϕ | p . The analogous argument using (3.25) implies that β ,j ≤ C p,n ǫ Z |∇ v | p − |∇ ϕ | + C p,n ǫ (cid:18)Z |∇ v | p − ∂ r ϕ ∇ ϕ · ∂ y i ˆ r k ∂ y v k (cid:19) . (4.20)for j = 2 , . . . , n + 1. For the second term in (4.20), H¨older’s inequality implies that (cid:18)Z |∇ v | p − ∂ r ϕ ∇ ϕ · ∂ y i ˆ r k ∂ y v k (cid:19) ≤ Z |∇ v | p − |∇ ϕ | Z |∇ v | p | ∂ y i ˆ r | k ∂ y i v k . RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Since ∂ y i ˆ r = x i x | x | , | ∂ y i ˆ r | ≤ | x | , we find that R |∇ v | p | ∂ yi ˆ r | k ∂ yi v k converges, so (4.20) implies that β ,j ≤ C p,n ǫ Z |∇ v | p − |∇ ϕ | . Then using Young’s inequality just as in (4.19), we find that β ,j ≤ ǫ κ ( n + 1) α (cid:16) Z |∇ v | p − |∇ ϕ | + ( p − Z |∇ v | p − | ∂ r ϕ | (cid:17) + C κ,p ǫ p Z |∇ ϕ | p , and thus β ≤ ǫ κα (cid:16) Z |∇ v | p − |∇ ϕ | + ( p − Z |∇ v | p − | ∂ r ϕ | (cid:17) + C κ,p ǫ p . (4.21)Together (4.17), (4.18), and (4.21) imply (4.14), as desired. Proof of (2.6) . The proof of (2.6) is similar to, but simpler than, the proof of (2.5), as no spectralgap or analysis of the second variation is needed. The principle of the expansion is the same, butnow we use (3.15) and (3.16) for the expansion, putting most of the weight of the higher orderterms on the second order term and preserving the positivity of the p th order term.From (3.15), we have Z |∇ u | p ≥ Z |∇ v | p + pǫ Z |∇ v | p − ∇ v · ∇ ϕ − C ǫ Z |∇ v | p − |∇ ϕ | + ǫ p Z |∇ ϕ | p . (4.22)Similarly, (3.16) implies Z | u | p ∗ ≤ ǫp ∗ Z v p ∗ − ϕ + C ǫ Z v p ∗ − ϕ + 2 ǫ p ∗ Z | ϕ | p ∗ . (4.23)As before, the identity (3.1) implies (4.13), so (4.23) becomes Z | u | p ∗ ≤ ǫp ∗ S − pp,n Z |∇ v | p − ∇ v · ∇ ϕ + C ǫ Z v p ∗ − ϕ + 2 ǫ p ∗ Z | ϕ | p ∗ . By the Poincar´e inequality (4.10), Z | u | p ∗ ≤ ǫp ∗ S − pp,n Z |∇ v | p − ∇ v · ∇ ϕ + C ǫ Z |∇ v | p − |∇ ϕ | + 2 ǫ p ∗ . As in (4.15), the concavity of z
7→ | z | p/p ∗ yields S pp,n k u k pL p ∗ ≤ S pp,n + ǫp Z |∇ v | p − ∇ v · ∇ ϕ + C ǫ Z |∇ v | p − |∇ ϕ | + C ǫ p ∗ . (4.24)Subtracting (4.24) from (4.22) gives δ ( u ) ≥ − C ǫ Z |∇ v | p − |∇ ϕ | + ǫ p − C ǫ p ∗ ≥ − C d( u, M ) + ǫ p . The final inequality follows from Remark 2.1 and once more taking ǫ is as small as needed. Thisconcludes the proof of (2.6). (cid:3) Corollary 4.3.
Suppose u ∈ ˙ W ,p is a function satisfying (2.3) and v ∈ M is a function wherethe infimum in (2.2) is attained. There exist constants C ∗ , c ∗ and c , depending on n and p only,such that if C ∗ ≤ R A v [ ∇ u − ∇ v, ∇ u − ∇ v ] R |∇ u − ∇ v | p or c ∗ ≥ R A v [ ∇ u − ∇ v, ∇ u − ∇ v ] R |∇ u − ∇ v | p , (4.25) then c Z |∇ u − ∇ v | p ≤ δ ( u ) . Proof.
Let C ∗ = C c and let c ∗ = C where c , C and C are as defined in Proposition 2.4.First suppose that u satisfies the first condition in (4.25). Then in (2.5), we may absorb the term C R |∇ u − ∇ v | p into the term c d( u, M ) , giving us c u, M ) ≤ δ ( u ) . Given this control, we may bootstrap using (2.6) to gain control of the stronger distance:14 Z |∇ u − ∇ v | p ≤ δ ( u ) + C d( u, M ) ≤ C δ ( u ) . Similarly, if u satisfies the second condition in (4.25), then we may absorb the term C d( u, M ) into the term R |∇ u − ∇ v | p in (2.6), giving us18 Z |∇ u − ∇ v | p ≤ δ ( u ) . (cid:3) Proofs of Theorem 1.1 and Corollary 1.2
Corollary 4.3 implies Theorem 1.1 for the functions u ∈ ˙ W ,p that satisfy (2.3) and that lie in oneof the two regimes described in (2.7). Therefore, to prove Theorem 1.1, it remains to understandthe case when the terms R A v [ ∇ u − ∇ v, ∇ u − ∇ v ] and R |∇ u − ∇ v | p are comparable and to removethe assumption (2.3). The following proposition accomplishes the first. Proposition 5.1.
Let u ∈ ˙ W ,p be a function satisfying (2.3) , and let v ∈ M be a function wherethe infimum in (2.2) is attained. If c ∗ ≤ R A v [ ∇ u − ∇ v, ∇ u − ∇ v ] R |∇ u − ∇ v | p ≤ C ∗ , (5.1) where c ∗ and C ∗ are the constants from the Corollary 4.3, then Z |∇ u − ∇ v | p ≤ Cδ ( u ) + C k v k p − L p ∗ k u − v k L p ∗ (5.2) for a constant C depending only on p and n . RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Proof.
Suppose u lies in the regime (5.1). Then we consider the linear interpolation u t := tu +(1 − t ) v and notice that R A v [ ∇ u t − ∇ v, ∇ u t − ∇ v ] R |∇ u t − ∇ v | p = t R A v [ ∇ u − ∇ v, ∇ u − ∇ v ] t p R |∇ u − ∇ v | p ≥ t − p c ∗ . Hence, there exists t ∗ sufficiently small, depending only on p and n , such that t − p ∗ c ∗ > C ∗ .We claim that we may apply Corollary 4.3 to u t ∗ . This is not immediate because v may notattain the infimum in (2.2) for u t ∗ . However, each step of the proof holds if we expand u t ∗ around v . Indeed, keeping the previous notation of u − v = ǫϕ with R |∇ ϕ | p = 1, we have u t ∗ − v = t ∗ ǫϕ .so the orthogonality constraints in (3.24), (3.25), and (3.26) still hold for u t ∗ and v by simplymultiplying through by t ∗ (this changes the constants by a factor of t ∗ but this does not affect theproof). Furthermore, (2.3) is used in the proofs of Proposition 2.4 and (4.14) to ensure that ǫ is asmall as needed to absorb terms. Since t ∗ <
1, if ǫ is sufficiently small then so is t ∗ ǫ . With thesetwo things in mind, every step in the proof of Proposition 2.4, and therefore Corollary 4.3 goesthrough for u t ∗ .Corollary 4.3 then implies that t p ∗ Z |∇ u − ∇ v | p = Z |∇ u t ∗ − ∇ v | p ≤ Cδ ( u t ∗ ) . Therefore, (5.2) follows if we can show δ ( u t ∗ ) ≤ Cδ ( u ) + C k v k p − L p ∗ k u − v k L p ∗ . (5.3)In the direction of (5.3), by convexity and recalling that k∇ v k L p = S p,n k v k L p ∗ = S p,n k u k L p ∗ , wehave δ ( u t ∗ ) = Z | t ∗ ∇ u + (1 − t ∗ ) ∇ v | p − S pp,n k t ∗ u + (1 − t ∗ ) v k pL p ∗ ≤ t ∗ Z |∇ u | p + (1 − t ∗ ) Z |∇ v | p − S pp,n k t ∗ u + (1 − t ∗ ) v k pL p ∗ = t ∗ δ ( u ) + S pp,n (cid:16) k v k pL p ∗ − k t ∗ u + (1 − t ∗ ) v k pL p ∗ (cid:17) . (5.4)Also, by the triangle inequality, k t ∗ ( u − v ) + v k pL p ∗ ≥ ( k v k L p ∗ − k t ∗ ( u − v ) k L p ∗ ) p , and by the convexity of the function f ( z ) = | z | p , f ( z + y ) ≥ f ( z ) + f ′ ( z ) y , and so( k v k L p ∗ − k t ∗ ( u − v ) k L p ∗ ) p ≥ k v k L p ∗ − p k v k p − L p ∗ k u − v k L p ∗ . These two inequalities imply that k v k pL p ∗ − k t ∗ u + (1 − t ∗ ) v k pL p ∗ ≤ p k v k p − L p ∗ k u − v k L p ∗ . Combining this with (5.4) yields (5.3), concluding the proof. (cid:3)
From here, the proof of Theorem 1.1 follows easily:
Proof of Theorem 1.1.
Together, Corollary 4.3 and Proposition 5.1 imply the following: there existssome constant C such that if u ∈ ˙ W ,p satisfies (2.3), then there is some v ∈ M such that Z |∇ u − ∇ v | p ≤ Cδ ( u ) + C k v k p ∗ − L p ∗ k u − v k L p ∗ . Therefore, we need only to remove the assumption (2.3) in order to complete the proof of Theo-rem 1.1. However, in the case where (2.3) fails, then trivially,inf {k∇ u − ∇ v k pL p : v ∈ M} ≤ k∇ u k pL p ≤ δ δ ( u ) . Therefore, by choosing the constant to be sufficiently large, Theorem 1.1 is proven. (cid:3)
We now prove Corollary 1.2 using the main result from [10], which we recall here:
Theorem 5.2 (Cianchi, Fusco, Maggi, Pratelli, [10]) . There exists C such that λ ( u ) ζ ′ k u k L p ∗ ≤ C ( k∇ u k L p − S p,n k u k L p ∗ ) , (5.5) where λ ( u ) = inf (cid:8) k u − v k p ∗ L p ∗ / k u k p ∗ L p ∗ : v ∈ M , R | v | p ∗ = R | u | p ∗ (cid:9) and ζ ′ = p ∗ (cid:16) p − p +1 n (cid:17) .Proof of Corollary 1.2. As before, if (2.3) does not hold, then Corollary 1.2 holds trivially by simplychoosing the constant to be sufficiently large. Now suppose u ∈ ˙ W ,p satisfies (2.3). There are twoobstructions to an immediate application of Theorem 5.2. The first is the fact that the deficit in(5.5) is defined as k∇ u k L p − S p,n k u k L p ∗ , while in our setting it is defined as k∇ u k pL p − S pp,n k u k pL p ∗ .However, this is easy to fix. Indeed, using the elementary inequality a p − b p ≥ a − b ∀ a ≥ b ≥ , we let a = k∇ u k L p /S p,n k u k L p ∗ and b = 1 to get k∇ u k L p − S p,n k u k L p ∗ S p,n k u k L p ∗ ≤ k∇ u k pL p − S pp,n k u k pL p ∗ S pp,n k u k pL p ∗ ≤ − δ k∇ u k pL p − S pp,n k u k pL p ∗ k∇ u k pL p , where the last inequality follows from (2.3). Therefore, up to increasing the constant, (5.5) impliesthat λ ( u ) ζ ′ ≤ C δ ( u ) k∇ u k pL p . (5.6)The second obstruction to applying Theorem 5.2 is the fact that (5.5) holds for the infimum in λ ( u ), while we must control k u − v k L p ∗ for v attaining the infimum in (2.2). To solve this issue itis sufficient to show that there exists some constant C = C ( n, p ) such that Z | ¯ v − u | p ∗ ≤ C inf n k u − v k p ∗ L p ∗ : v ∈ M , Z | v | p ∗ = Z | u | p ∗ o where ¯ v attains the infimum in (2.2). The proof of this fact is nearly identical (with the obviousadaptations) to that of part (2) of Proposition 4.1, with the only nontrivial difference being thatone must integrate by parts to show that the analogue of first term in (4.9) goes to zero.Therefore, (5.5) implies (cid:18) k u − v k L p ∗ k u k L p ∗ (cid:19) ζ ′ ≤ C δ ( u ) k∇ u k L p where v ∈ M attains the infimum in (2.2). Paired with Theorem 1.1, this proves Corollary 1.2with ζ = ζ ′ p . (cid:3) RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Spectral Properties of L v In this section, we give the proofs of the compact embedding theorem and Sturm-Liouville theorythat were postponed in the proof of Proposition 3.1. As in Proposition 3.1, by scaling, it sufficesto consider the operator L = L v where v = v , . The discrete spectrum of L . Given two measurable functions ω , ω : Ω → R , let W , (Ω , ω , ω ) := { g : k g k W , (Ω ,ω ,ω ) < ∞} , where k · k W , (Ω ,ω ,ω ) is the norm defined by k g k W , (Ω ,ω ,ω ) = (cid:18)Z Ω g ω + Z Ω |∇ g | ω (cid:19) / . (6.1)The space W , (Ω , ω , ω ) is defined as the completion of the space C ∞ (Ω) with respect to thenorm k · k W , (Ω ,ω ,ω ) . The following compact embedding result was shown in [28]: Theorem 6.1 (Opic, [28]) . Let Z = W , ( R n , ω , ω ) and suppose ω i ∈ L and ω − / i ∈ L ∗ loc , (6.2) i = 0 , . If there are local compact embeddings W , ( B k , ω , ω ) ⊂⊂ L ( B k , ω ) , k ∈ N , (6.3) where B k = { x : | x | < k } , and if lim k →∞ sup (cid:8) k u k L ( R n \ B k ,ω ) : u ∈ Z, k u k Z ≤ (cid:9) = 0 , (6.4) then Z embeds compactly in L ( R n , ω ) . We apply Theorem 6.1 to show that the space X = W , ( R n , v p ∗ − , |∇ v | p − ) , (6.5)embeds compactly into L ( R n , v p ∗ − ). Corollary 6.2.
The compact embedding X ⊂⊂ L ( R n , v p ∗ − ) holds, with X as in (6.5) .Proof. Let us verify that Theorem 6.1 may be applied in our setting, taking ω = v p ∗ − , ω = |∇ v | p − . In other words, we must show that (6.2), (6.3) and (6.4) are satisfied. A simple computation verifies(6.2). To show (6.3), we fix δ > n and p ) and show thethree inclusions below: W , ( B r , ω , ω ) (1) ⊂ W , n + δ ) / ( n +2) ( B r ) (2) ⊂⊂ L ( B r ) (3) ⊂ L ( B r , ω ) . Since (2 n/ (2 + n )) ∗ = 2, the Rellich-Kondrachov compact embedding theorem implies (2), whilethe inclusion (3) holds simply because v p ∗ − ≥ c n,p,r for x ∈ B r . In the direction of showing (1),we use this fact and H¨older’s inequality to obtain (cid:16) Z B r | u | n + δ ) / ( n +2) (cid:17) ( n +2) / ( n + δ ) ≤ | B r | (2 − δ ) / ( n + δ ) Z B r | u | ≤ C n,p,r Z B r | v | p ∗ − | u | . (6.6) Furthermore, since |∇ v | p − = C (1 + | x | p ′ ) − n ( p − /p | x | ( p − / ( p − ≥ c n,p,r | x | ( p − / ( p − for x ∈ B r , H¨older’s inequality implies that (cid:16) Z B r |∇ u | n + δ ) / ( n +2) (cid:17) ( n +2) / ( n + δ ) ≤ (cid:16) Z B r | x | ( p − / ( p − |∇ u | (cid:17)(cid:16) Z B r | x | − β (cid:17) (2 − δ ) / ( n + δ ) ≤ C n,p,r Z B r |∇ v | p − |∇ u | , (6.7)where β = (cid:0) p − p − (cid:1)(cid:0) n + δn +2 (cid:1)(cid:0) n +22 − δ (cid:1) . Then the inclusion (1) follows from (6.6) and (6.7), and thus (6.3)is verified.To show (6.4), let u k be a function almost attaining the supremum in (6.4), in other words, fora fixed η >
0, let u k be such that u k ∈ X, k u k k X ≤
1, andsup (cid:8) k u k L ( R n \ B k ,ω ) : u ∈ X, k u k X ≤ (cid:9) ≤ k u k k L ( R n \ B k ,ω ) + η. By mollifying u and multiplying by a smooth cutoff η ∈ C ∞ ( R n \ B k ), we may assume without lossof generality that u k ∈ C ∞ ( R n \ B k ). Recalling that v = v with v as in (1.2), we have Z R n \ B k v p ∗ − u k = Z R n \ B k κ (1 + | x | p ′ ) − ( p ∗ − n − p ) /p u k ≤ κ Z R n \ B k | x | − ( p ∗ − n − p ) / ( p − u k (6.8)for k ≥
2. We use Hardy’s inequality in the form Z R n | x | s u ≤ C Z R n | x | s +2 |∇ u | (6.9)for u ∈ C ∞ ( R n ) (see, for instance, [32]). Applying (6.9) to the right-hand side of (6.8) implies Z R n \ B k | x | − ( p ∗ − n − p ) / ( p − u k ≤ C Z R n \ B k | x | − ( p ∗ − n − p ) / ( p − |∇ u k | (6.10)and (6.8) and (6.10) combined give Z R n \ B k v p ∗ − u k ≤ C Z R n \ B k | x | − ( p ∗ − n − p ) / ( p − |∇ u k | = C Z R n \ B k | x | − p ′ | x | − ( p − n − / ( p − |∇ u k | ≤ Ck − p ′ Z R n \ B k |∇ v | p − |∇ u k | , where the final inequality follows because |∇ v | p − ≥ C | x | − ( p − n − / ( p − for x ∈ R n \ B . Thus Z R n \ B k v p ∗ − u k ≤ Ck − p ′ k u k k X , and (6.4) is proved. (cid:3) Thanks to the compact embedding X ⊂⊂ L ( R n , ω ), we can now prove the following importantfact: RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Corollary 6.3.
The operator L has a discrete spectrum { α i } ∞ i =1 .Proof. We show that the operator L − : L ( v p ∗ − ) → L ( v p ∗ − ) is bounded, compact, and self-adjoint. From there, one applies the spectral theorem (see for instance [15]) to deduce that L − has a discrete spectrum, hence so does L .Approximating by functions in C ∞ ( R n ) , the Poincar´e inequality (4.10) holds for all functions ϕ ∈ X , with X as defined in (6.5). Thanks to this fact, the existence and uniqueness of solutionsto L u = f for f ∈ L ( v p ∗ − ) follow from the Direct Method, so the operator L − is well defined.Self-adjointness is immediate. From (4.10) and H¨older’s inequality, we have c k u k X ≤ Z |∇ v | p − |∇ u | ≤ Z A [ ∇ u, ∇ u ] ≤ k u k X kL u k L ( v p ∗− ) . This proves that L − is bounded from L ( v p ∗ − ) to L ( v p ∗ − ), and by Corollary 6.2 we see that L − is a compact operator. (cid:3) Sturm-Liouville theory.
Multiplying by the integrating factor r n − , the ordinary differen-tial equation (3.12) takes the form of the Sturm-Liouville eigenvalue problem Lf + αf = 0 on [0 , ∞ ) , (6.11)where Lf = 1 w [( P f ′ ) ′ − Qf ]with P ( r ) = ( p − | v ′ | p − r n − ,Q ( r ) = µr n − | v ′ | p − ,w ( r ) = v p ∗ − r n − . (6.12)This is a singular Sturm-Liouville problem; first of all, our domain is unbounded, and secondof all, the equation is degenerate because v ′ (0) = 0. Nonetheless, we show that Sturm-Liouvilletheory holds for this singular problem. Lemma 6.4 (Sturm-Liouville Theory) . The following properties hold for the singular Sturm-Liouville eigenvalue problem (6.11) : (1) If f and f are two eigenfunctions corresponding to the eigenvalue α , then f = cf . Inother words, each eigenspace of L is one-dimensional. (2) The i th eigenfunction of L has i − interior zeros. Note that L has a discrete spectrum because L does (Corollary 6.3), and that eigenfunctions f of L live in the space Y = W , (cid:0) [0 , ∞ ) , v p ∗ − r n − , | v ′ | p − r n − (cid:1) , using the notation introduced at the beginning of Section 6.1. In any ball B R around zero, theoperator L is degenerate elliptic with the matrix A bounded by an A -Muckenhoupt weight, soeigenfunctions of L are H¨older continuous; see [16, 25]. Therefore, eigenfunctions of L are H¨oldercontinuous on [0 , ∞ ). Remark 6.5.
The function P ( r ) as defined in (6.12) has the following behavior: P ( r ) ≈ r ( p − p − n − in [0 , ,P ( r ) ≈ r ( n − / ( p − as r → ∞ . In particular, the weight | v ′ | p − r n − ≈ r ( n − / ( p − goes to infinity as r → ∞ , which implies that R ∞ | f ′ | dr < ∞ for any f ∈ Y .In order to prove Lemma 6.4, we first prove the following lemma, which describes the asymptoticdecay of solutions of (6.11). Lemma 6.6.
Suppose f ∈ Y is a solution of (6.11) . Then, for any < β < n − pp − , there exist C and r such that | f ( r ) | ≤ Cr − β and | f ′ ( r ) | ≤ Cr − β − for r ≥ r .Proof. Step 1: Qualitative Decay of f . For any function f ∈ Y , f ( r ) → r → ∞ . Indeed, nearinfinity, | v ′ | p − r p − behaves like Cr γ where γ := n − p − >
1. Then for any r, s large enough with r < s , | f ( r ) − f ( s ) | ≤ Z ∞ r | f ′ ( t ) | dt ≤ (cid:16) Z ∞ r f ′ ( t ) t γ dt (cid:17) / (cid:16) Z ∞ r t − γ dt (cid:17) / (6.13)by H¨older’s inequality. As both integrals on the right-hand side of (6.13) converge, for any ǫ > r large enough such that the right-hand side is bounded by ǫ , so the limit of f ( r ) as r → ∞ exists.We claim that this limit must be equal to zero. Indeed, since Y is obtained as a completion of C ∞ , if we apply (6.13) to a sequence f k ∈ C ∞ ([0 , ∞ )) converging in Y to f and we let s → ∞ , weget | f k ( r ) | ≤ (cid:16) Z ∞ r f ′ k ( t ) t γ dt (cid:17) / (cid:16) Z ∞ r t − γ dt (cid:17) / , thus, by letting k → ∞ , | f ( r ) | ≤ (cid:16) Z ∞ r f ′ ( t ) t γ dt (cid:17) / (cid:16) Z ∞ r t − γ dt (cid:17) / . Since the right-hand side tends to zero as r → ∞ , this proves the claim. Step 2: Qualitative Decay of f ′ . For r >
0, (6.11) can be written as L ′ f := f ′′ + af ′ + bf = 0 (6.14)where a = P ′ P and b = − Q + wαP . Fixing ǫ >
0, an explicit computation shows that there exists r large enough such that(1 − ǫ )( n − p − r ≤ a ≤ (1 + ǫ )( n − p − r and − µp − r + (1 − ǫ ) c p,n αr (3 p − / ( p − ≤ b ≤ − µp − r + (1 + ǫ ) c p,n αr (3 p − / ( p − p ≥ for r ≥ r , where c n,p is a positive constant depending only on n and p . Asymptotically, therefore,our equation behaves like f ′′ + n − p − f ′ r + (cid:16) c p,n αr p ′ − µp − (cid:17) fr = 0 . If f is a solution of (6.11), then squaring (6.14) on [ r , ∞ ), we obtain | f ′′ | ≤ (cid:18)(cid:16) n − p − ǫ (cid:17) f ′ r (cid:19) + 2 (cid:18)(cid:16) (1 + ǫ ) c p,n αr p ′ + µp − (cid:17) fr (cid:19) ≤ C ( | f | + | f ′ | ) . Integrating on [
R, R + 1] for R ≥ r implies Z R +1 R | f ′′ | ≤ C Z R +1 R | f ′ | + C Z R +1 R | f | . Step 1 and Remark 6.5 ensure that both terms on the right-hand side go to zero. Applying Mor-rey’s embedding to f ′ η R , where η R is a smooth cutoff equal to 1 in [ R, R + 1], we determine that k f ′ k L ∞ ([ R,R +1]) → R → ∞ , proving that f ′ ( r ) → r → ∞ . Step 3: Quantitative Decay of f and f ′ . Standard arguments (see for instance [14, VI.6]) showthat, also in our case, the i th eigenfunction f of L has at most i − f ( r )does not change sign for r sufficiently large. Without loss of generality, we assume that eventually f ≥ r as in Step 2 and applying the operator L ′ defined in (6.14) to the function g = Cr − β + c , c >
0, for r ≥ r gives L ′ g ≤ Cβ ( β + 1) r − β − − (1 − ǫ )( n − p − Cβr − β − + (cid:16) (1 + ǫ ) c p,n αr (3 p − / ( p − − µp − (cid:17) ( Cr − β − + c ) ≤ Cr − β − (cid:16) β ( β + 1) − (1 − ǫ )( n − p − β + (1 + ǫ ) c p,n αr p ′ (cid:17) + (1 + ǫ ) c p,n αr (3 p − / ( p − c. For any 0 < β < ( n − p ) / ( p − r may be taken large enough (and therefore ǫ small enough)such that L ′ g < r , ∞ ) , so g is a supersolution of the equation on this interval.Choosing C = f ( r ) r β and c >
0, then ( g − f )( r ) > g − f )( r ) → c > r → ∞ . Since L ′ ( g − f ) < , we claim that g − f > r , ∞ ). Indeed, otherwise, g − f would have a negativeminimum at some r ∈ ( r , ∞ ), implying that( g − f )( r ) ≤ , ( g − f ) ′ ( r ) = 0 , and ( g − f ) ′′ ( r ) ≥ , forcing L ′ ( g − f ) ≥ , a contradiction. This proves that 0 ≤ f ≤ g on [ r , ∞ ), and since c > f ≤ Cr − β on [ r , ∞ ).We now derive bounds on f ′ : by the fundamental theorem of calculus and using (6.14) and thebound on f for r ≥ r , we get | f ′ ( r ) | = (cid:12)(cid:12)(cid:12) Z ∞ r f ′′ (cid:12)(cid:12)(cid:12) ≤ Cr (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r f ′ (cid:12)(cid:12)(cid:12)(cid:12) + C (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r t − β − (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cr | f ( r ) | + Cβ + 2 r − β − ≤ Cr − β − . (cid:3) With these asymptotic decay estimates in hand, we are ready to prove Lemma 6.4.
Proof of Lemma 6.4.
We begin with the following remark about uniqueness of solutions. If f and f are two solutions of (6.11) and f ( r ) = f ( r ) , f ′ ( r ) = f ′ ( r )for some r >
0, then f = f on [0 , ∞ ). Indeed, for r >
0, we may express our equation as in(6.14). As a and b are continuous on (0 , ∞ ) , the standard proof of uniqueness for (non-degenerate)second order ODE holds. Once f = f on (0 , ∞ ), they are also equal at r = 0 by continuity. Proof of (1).
Suppose α is an eigenvalue of L with f and f satisfying (6.11). In view of theuniqueness remark, if there exists r > f of f and f such that f ( r ) = f ′ ( r ) = 0 , then f is constantly zero and f and f are linearly dependent. Let W ( r ) = W ( f , f )( r ) := det (cid:20) f f f ′ f ′ (cid:21) ( r )denote the Wronskian of f and f . This is well defined for r > f and f are C there)and a standard computation shows that ( P W ) ′ = 0 on (0 , ∞ ): indeed, since W ′ = f f ′′ − f f ′′ , weget ( P W ) ′ = P W ′ + P ′ W = P ( f f ′′ − f f ′′ ) + P ′ ( f f ′ − f f ′ ) , and by adding and subtracting the term ( αw − Q ) f f it follows that( P W ) ′ = f (cid:0) P f ′′ + P ′ f ′ + ( αw − Q ) f (cid:1) − f (cid:0) P f ′′ + P ′ f ′ + ( αw − Q ) f (cid:1) = 0 . Thus
P W is constant on (0 , ∞ ). We now show that that P W is continuous up to r = 0 and that( P W )(0) = 0 . Indeed, (6.11) implies that(
P f ′ i ) ′ = ( Q − αw ) f i for i = 1 ,
2. The right-hand side is continuous, so (
P f ′ i ) ′ is continuous, from which it follows easilythat P W is also continuous on [0 , ∞ ).To show that ( P W )(0) = 0, we first prove that (
P f ′ i )(0) = 0. Indeed, let c i := ( P f ′ i )(0). If c i = 0, then keeping in mind Remark 6.5, f ′ i ( r ) ≈ c i P ( r ) ≈ c i r ( p − / ( p − n − for r ≪ , (6.15)therefore Z R | v ′ | p − | f ′ | r n − dr & Z R r ( p − / ( p − n − | f ′ | dr & Z R drr ( p − / ( p − n − = + ∞ , contradicting the fact that f ∈ Y . Hence, we conclude that lim r → ( P f ′ i )( r ) = 0 , and using this factwe obtain ( P W )(0) = lim r → ( P f ′ f − P f ′ f ) = lim r → ( P f ′ ) lim r → f − lim r → ( P f ′ ) lim r → f = 0 . Therefore (
P W )( r ) = 0 for all r ∈ [0 , ∞ ). Since P ( r ) > r >
0, we determine that W ( r ) = 0for all r >
0. In particular, given r ∈ (0 , ∞ ), there exist c , c such that c + c = 0 and c f ( r ) + c f ( r ) = 0 ,c f ′ ( r ) + c f ′ ( r ) = 0 . RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ Then f := c f + c f solves (6.11) and f ( r ) = f ′ ( r ) = 0. By uniqueness, f ≡ t ∈ (0 , ∞ ),and so f = cf . Proof of (2).
Thanks to our preliminary estimates on the behavior of f i at infinity, the followingis an adaptation of the standard argument in, for example, [14, VI.6].Suppose that f and f are eigenfunctions of L corresponding to eigenvalues α and α respec-tively, with α < α , that is, ( P f ′ i ) ′ − Qf i + α i wf i = 0 . Our first claim is that between any two consecutive zeros of f is a zero of f , including zeros atinfinity. Note that( P W ) ′ = P [ f f ′′ − f f ′′ ] + P ′ [ f f ′ − f f ′ ]= f [( P f ′ ) ′ + ( α − Q ) f ] − f [( P f ′ ) ′ + ( α w − Q ) f ] + ( α − α ) wf f = ( α − α ) wf f . (6.16)Suppose that f has consecutive zeros at r and r , and suppose for the sake of contradictionthat f has no zeros in the interval ( r , r ). With no loss of generality, we may assume that f and f are both nonnegative in [ r , r ]. Case 1:
Suppose that r < ∞ . Then integrating (6.16) from r to r implies0 > ( α − α ) Z r r wf f = ( P W )( r ) − ( P W )( r )= P ( r )[ f ( r ) f ′ ( r ) − f ′ ( r ) f ( r )] − P ( r )[ f ( r ) f ′ ( r ) − f ′ ( r ) f ( r )]= − P ( r ) f ′ ( r ) f ( r ) + P ( r ) f ′ ( r ) f ( r ) . The function f is positive on ( r , r ), so f ′ ( r ) ≥ f ′ ( r ) ≤
0. Also, since f ( r ) = f ( r ) = 0we cannot have f ′ ( r ) = 0 or f ′ ( r ) = 0, as otherwise f would vanish identically. Furthermore, f is nonnegative on [ r , r ], so we conclude that the right-hand side is nonnegative, giving us acontradiction. Case 2:
Suppose that r = ∞ . Again integrating the identity (6.16) from r to ∞ , we obtain0 > ( α − α ) Z ∞ r wf f = lim r →∞ ( P W )( r ) − ( P W )( r )= lim r →∞ [ P ( r )( f ( r ) f ′ ( r ) − f ′ ( r ) f ( r ))] − P ( r )( f ( r ) f ′ ( r ) − f ′ ( r ) f ( r )) . (6.17)We notice that Lemma 6.6 implies thatlim r →∞ [ P ( r )( f ( r ) f ′ ( r ) − f ′ ( r ) f ( r ))] = 0 . Indeed, taking n − p p − < β < n − pp − , | f ′ f − f f ′ | ≤ | f ′ || f | + | f || f ′ | ≤ Cr − β − , and, recalling Remark 6.5, P ( r ) ≤ Cr ( n − / ( p − , implying that P | f ′ f − f f ′ | ≤ Cr γ → , where γ = − β − n − p − < . Then (6.17) becomes0 > − P ( r ) f ′ ( r ) f ( r ) . Since f ′ ( r ) > f ( r ) ≥ f has a zero in the interval [0 , r ), where r is the first zero of f . Again,we assume for the sake of contradiction that f has no zero in this interval and that, without lossof generality, f and f are nonnegative in [0 , r ]. Integrating (6.16) implies0 > ( α − α ) Z r wf f = P W ( r ) − P W (0) . (6.18)The same computation as in the proof of Part (1) of this lemma implies that ( P W )(0) = 0, so(6.18) becomes 0 > − P ( r ) f ′ ( r ) f ( r ) , once more giving us a contradiction.The first eigenfunction of an operator is always positive in the interior of the domain, so thesecond eigenfunction of L must have at least one interior zero by orthogonality. Thus the claimsabove imply that the i th eigenfunction has at least i − i th eigenfunctionhas at most i − (cid:3) Appendix
In this section we give the proofs of Lemma 3.2 and of the polar coordinates form of the operatordiv ( A ( x ) ∇ ϕ ) given in (3.10). Proof of (3.10) . We will use the following classical relations: ∂ r ˆ r = 0 ∂ r ˆ θ i = 0 , ∂ θ i ˆ r = ˆ θ i , ∂ θ i ˆ θ i = − ˆ r, ∂ θ j ˆ θ i = 0 for i = j. The chain rule implies thatdiv( A ( x ) ∇ ϕ ) = tr( A ( x ) ∇ ϕ ) + tr( ∇ A ( x ) ∇ ϕ ) . (7.1)We compute the two terms on the right-hand side of (7.1) separately. For the first, we begin bycomputing the Hessian of ϕ in polar coordinates, starting from ∇ ϕ = ∂ r ϕ ˆ r + 1 r n − X j =1 ∂ θ j ϕ ˆ θ j , (7.2)We have ∇ ϕ = ∂ r (cid:16) ∂ r ϕ ˆ r + 1 r n − X j =1 ∂ θ j ϕ ˆ θ j (cid:17) ˆ r + 1 r n − X i =1 ∂ θ i (cid:16) ∂ r ϕ ˆ r + 1 r n − X j =1 ∂ θ j ϕ ˆ θ j (cid:17) ˆ θ i = ∂ rr ϕ ˆ r ⊗ ˆ r − r n − X j =1 ∂ θ j ϕ ˆ θ j ⊗ ˆ r + 1 r n − X j =1 ∂ θ j r ϕ ˆ θ j ⊗ ˆ r + 1 r n − X i =1 ∂ θ i r ϕ ˆ r ⊗ ˆ θ i + 1 r n − X i =1 ∂ r ϕ θ i ⊗ θ i + 1 r n − X i =1 n − X j =1 ∂ θ i θ j ϕ ˆ θ j ⊗ ˆ θ i − r n − X i =1 ∂ θ i ϕ ˆ r ⊗ ˆ θ i . RADIENT STABILITY FOR THE SOBOLEV INEQUALITY: THE CASE p ≥ In order to compute A ( x ) ∇ ϕ , we note that(ˆ r ⊗ ˆ r )(ˆ r ⊗ ˆ r ) = ˆ r ⊗ ˆ r, (ˆ r ⊗ ˆ r )(ˆ θ j ⊗ ˆ θ i ) = 0 , (ˆ r ⊗ ˆ r )(ˆ r ⊗ ˆ θ i ) = 0 , (ˆ r ⊗ ˆ r )(ˆ θ i ⊗ ˆ r ) = ˆ θ i ⊗ ˆ r. Thus we have A ( x ) ∇ ϕ = ( p − |∇ v | p − ˆ r ⊗ ˆ r ( ∇ ϕ ) + |∇ v | p − Id( ∇ ϕ )= ( p − |∇ v | p − h ∂ rr ϕ ˆ r ⊗ ˆ r − r n − X j =1 ∂ θ j ϕ ˆ θ j ⊗ ˆ r + 1 r n − X j =1 ∂ θ j r ϕ ˆ θ j ⊗ ˆ r i + |∇ v | p − h ∂ rr ϕ ˆ r ⊗ ˆ r − r n − X j =1 ∂ θ j ϕ ˆ θ j ⊗ ˆ r + 1 r n − X j =1 ∂ θ j r ϕ ˆ θ j ⊗ ˆ r + 1 r n − X i =1 ∂ θ i r ϕ ˆ r ⊗ ˆ θ i + 1 r n − X i =1 ∂ r ϕ θ i ⊗ θ i + 1 r n − X i =1 n − X j =1 ∂ θ i θ j ϕ ˆ θ j ⊗ ˆ θ i − r n − X i =1 ∂ θ i ϕ ˆ r ⊗ ˆ θ i i , and the first term in (7.1) istr( A ( x ) ∇ ϕ ) = ( p − |∇ v | p − ∂ rr ϕ + n − r |∇ v | p − ∂ r ϕ + 1 r |∇ v | p − n − X i =1 ∂ θ i θ i ϕ. (7.3)Now we compute the second term in (7.1), starting by computing ∇ A ( x ). We reintroduce theslight abuse of notation by letting v ( r ) = v ( x ), so v ′ = ∂ r v , v ′′ = ∂ rr v . Note that ∂ θ Id = ∂ r Id = 0,thus ∇ A ( x ) = ∂ r A ( x ) ⊗ ˆ r + 1 r n − X j =1 ∂ θ j A ( x ) ⊗ ˆ θ j = ( p − | v ′ | p − v ′ v ′′ ˆ r ⊗ ˆ r ⊗ ˆ r + ( p − | v ′ | p − v ′ v ′′ Id ⊗ ˆ r + p − r n − X j =1 h | v ′ | p − ˆ θ j ⊗ ˆ r ⊗ ˆ θ j + | v ′ | p − ˆ r ⊗ ˆ θ j ⊗ ˆ θ j i . Recalling (7.2), we then have ∇ A ( x ) ∇ ϕ = ( p − | v ′ | p − v ′ v ′′ ∂ r ϕ (ˆ r ⊗ ˆ r ⊗ ˆ r )ˆ r + ( p − | v ′ | p − v ′ v ′′ ∂ r ϕ (Id ⊗ ˆ r )ˆ r + p − r n − X j =1 h | v ′ | p − ∂ r ϕ (ˆ θ j ⊗ ˆ r ⊗ ˆ θ j )ˆ r + | v ′ | p − ∂ r ϕ (ˆ r ⊗ ˆ θ j ⊗ ˆ θ j )ˆ r i + 1 r n − X i =1 h ( p − | v ′ | p − v ′ v ′′ ∂ θ i ϕ (ˆ r ⊗ ˆ r ⊗ ˆ r )ˆ θ i + ( p − | v ′ | p − v ′ v ′′ ∂ θ i ϕ (Id ⊗ ˆ r )ˆ θ i i + p − r n − X i =1 n − X j =1 h | v ′ | p − ∂ θ i ϕ (ˆ θ j ⊗ ˆ r ⊗ ˆ θ j )ˆ θ i + | v ′ | p − ∂ θ i ϕ (ˆ r ⊗ ˆ θ j ⊗ ˆ θ j )ˆ θ i i , where we used that ( a ⊗ b ⊗ c ) d = ( a · d ) b ⊗ c. Writing out these terms gives ∇ A ( x ) ∇ ϕ = ( p − p − | v ′ | p − v ′ v ′′ ∂ r ϕ ˆ r ⊗ ˆ r + p − r | v ′ | p − n − X j =1 ∂ r ϕ ˆ θ j ⊗ ˆ θ j + p − r | v ′ | p − v ′ v ′′ n − X j =1 ∂ θ j ϕ ˆ θ j ⊗ ˆ r + p − r | v ′ | p − n − X j =1 ∂ θ j ϕ ˆ r ⊗ ˆ θ j , thus the second term in (7.1) istr( ∇ A ( x ) ∇ ϕ ) = ( p − p − |∇ v | p − ∂ r v ∂ rr v ∂ r ϕ + ( n − p − r |∇ v | p − ∂ r ϕ. (7.4)Combining (7.3) and (7.4), (7.1) implies thatdiv( A ( x ) ∇ ϕ ) = ( p − |∇ v | p − ∂ rr ϕ + ( p − n − r |∇ v | p − ∂ r ϕ + 1 r |∇ v | p − n − X j =1 ∂ θ j θ j ϕ + ( p − p − |∇ v | p − ∂ r v ∂ rr v ∂ r ϕ, as desired. (cid:3) References [1] T. Aubin. Probl`emes isop´erim´etriques et espaces de Sobolev.
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Department of Mathematics, ETH Z¨urich,HG G 63.2, R¨amistrasse 101, CH-8092 Z¨urich, Switzerland
E-mail address : [email protected] (Robin Neumayer) Department of Mathematics, The University of Texas at Austin,2515 Speedway Stop C1200, Austin, TX 78712, USA
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