Rigidity theorems for best Sobolev inequalities

For $n\geq 2$, $p\in(1,n)$, the"best $p$-Sobolev inequality"on an open set $\Omega\subset\mathbb{R}^n$ is identified with a family $\Phi_\Omega$ of variational problems with critical volume and trace constraints. When $\Omega$ is bounded we prove: (i) for every $n$ and $p$, the existence of generalized minimizers that have at most one boundary concentration point, and: (ii) for $n>2\,p$, the existence of (classical) minimizers. We then establish rigidity results for the comparison theorem"balls have the worst best Sobolev inequalities"by the first named author and Villani, thus giving the first affirmative answers to a question raised in [MV05].

1. Introduction 1.1.Overview.The goal of this paper is to answer some basic open questions concerning a "doubly critical" family {Φ Ω (T )} T ≥0 of minimization problems on Sobolev functions, which, in a precise sense to be clarified below, can be interpreted as collectively defining the best Sobolev inequality on an open set Ω ⊂ R n with C 1 -boundary.Given an integer n ≥ 2 and p ∈ (1, n), these problems are defined as and their minimizers, whenever they exist, satisfy the Euler-Lagrange equation for suitable Lagrange multipliers λ, σ ∈ R. We call ´Ω |u| p ⋆ = 1 and ´∂Ω |u| p # = T p # the "volume" and "trace" constraints of Φ Ω (T ).The critical Sobolev exponents associated to n and p, p ⋆ and p # , are defined by and their precise values guarantee the scale invariance of Φ Ω , i.e.
When Ω is bounded, the C 1 -regularity of ∂Ω guarantees that every u ∈ L 1 loc (Ω) with ∇u ∈ L p (Ω; R n ) lies in the competition class of Φ Ω (T ) for some T ≥ 0. In particular, Epi(Φ Ω ) = (T, G) ∈ R 2 : T ≥ 0 , G ≥ Φ Ω (T ) , (the epigraph of Φ Ω ) collects the best possible information on the range of values achievable by ∇u L p (Ω) when u L p ⋆ (Ω) is fixed: from this peculiar viewpoint, which is reminiscent of the one adopted in the study of Blaschke-Santaló diagrams, Epi(Φ Ω ) is "the best Sobolev inequality on Ω".The following list of results, summarized in Figure 1.1, aims to provide a hopefully complete state of the art on Φ Ω , and illustrates the wealth of information stored in this family of variational problems.As a disclaimer: here we are definitely not attempting to exhaustively frame the study of Φ Ω into the incredibly vast and layered context of the theory of Sobolev-type inequalities (see e.g.[Maz85]), as that would be a long and delicate exercise, lying well beyond the scope of this introduction.
This bound is sharp only if T = T E , and is nearly optimal only if T is close to T E ; but it largely suboptimal away from T E , see Figure 1.1.
(4) Balls have the worst best Sobolev inequalities: In [CL90, CL94, p = 2] (by symmetrization methods and conformal invariance) and in [MV05, p ∈ (1, n)] (via the mass transportation method pioneered in [Kno57, MS86, CENV04]) it is shown that if B is a ball, then for every T ∈ (0, ISO(B) 1/p # ) there is a unique α > 0 such that with U S defined in (1.4).Further elaborating on the proof of this partial characterization of Φ B , again in [MV05] the comparison theorem that balls have the worst best Sobolev inequalities (1.12) is proved.This sharp lower bound, combined with (1.11), allows one to infer some sharp and more traditional-looking Sobolev-type inequalities, like the following sharp interpolation between (1.3) and (1.10) and the following sharp Sobolev inequality, additive in the domain of the p-Dirichlet energy, which was first conjectured by Brezis and Lieb in [BL85].
1 The "only if" statement for p = 2 was left open in [Naz06], but was proven in [MN17, Theorem 2.3].
(5) Full characterization of Φ H : In [CL90, CL94, for p = 2] and, again by optimal mass transport arguments, in [MN17, for p for each T > T E (beyond Escobar range), there is a unique where Up to the natural dilation and translation invariances, these functions are the unique minimizers of Φ H (T ).Moreover, again by [MN17]: , where t T 0 = 0 and Φ H (T 0 ) = S(n, p)/2 Question 1: When does Φ Ω (T ) (T > 0) admit minimizers?
Question 2: Does rigidity hold in the comparison theorem (1.12)?
The main idea of this paper is attacking these two closely related questions by systematically exploiting the complete characterization of Φ H obtained in [MN17].
Concerning Question 1, a classical concentration-compactness argument characterizes the limit behavior of minimizing sequences of Φ Ω (T ) as the superposition of a standard weak limit plus at most countably many concentration points, located either in the interior of Ω, or on its boundary.By exploiting properties of Φ H we are able to (i): exclude all interior concentrations and all but at most one boundary concentration, thus proving existence of minimizers for a suitable "relaxed problem" Φ * Ω (T ); and (ii): completely exclude concentrations, and thus establish the existence of minimizers of Φ Ω (T ), as soon as ∂Ω is of class C 2 and n > 2p .To give precise statements, it is convenient to let X Ω (T ) denote the competition class of Φ Ω (T ), and let Y Ω (T ) denote the set of triples (u, v, t) with either u ∈ X Ω (T ) and v = t = 0, or u ∈ W 1,p (Ω), v ∈ (0, 1], t ∈ (0, T ], and (1.20) The relaxed problem associated to Φ Ω (T ) is then given by and Ω is a bounded open set with C 1 -boundary in R n , then: (i): for every T > 0, there is a minimizer (u, v, t) of Φ * Ω (T ), and (ii): if Ω has boundary of class C 2 , n > 2p, T > 0, and (u, v, t) is a minimizer of Φ * Ω (T ), then v = t = 0, and thus u is a minimizer of Φ Ω (T ).
Question 2 is motivated by the various rigidity statements associated to comparison theorems in Riemannian geometry (see, e.g.[CE08]).In that setting, a certain model space provides a universal bound on a certain global geometric quantity (comparison theorem), which is then shown to be saturated by the model space alone (rigidity statement).With this analogy in mind, we can reformulate more precisely Question 2 as follows: Question 2, weak form: Does Φ Ω = Φ B on (0, ISO(B) 1/p # ) imply that Ω is a ball?Question 2, strong form: Does Φ Ω (T ) = Φ B (T ) at just one value of T ∈ (0, ISO(B) 1/p # ) imply that Ω is a ball?Concerning the weak form of Question 2, through a careful use of the properties of Φ H we answer affirmatively whenever Ω is bounded and connected.These conditions are optimal, as shown by unbounded or disconnected non-rigidity examples presented in [MV05].In fact, the argument we propose gives rigidity under the mere assumption that Φ Ω = Φ B holds on an open neighborhood of T = 0. Concerning the strong form of Question 2, which was originally formulated in [MV05, Section 1.9], we can answer in the affirmative as a direct by-product of our existence result for minimizers of Φ Ω (T ) (thus, when Ω has C 2 -boundary and n > 2p) thanks to the following "conditional rigidity" statement, which is proved in [MV05] as a direct by-product of the proof of (1.12): if Ω is connected (possibly unbounded) , if Φ Ω (T ) = Φ B (T ) for a value of T ∈ (0, ISO(B) 1/p # ) , (1.23) and if Φ Ω (T ) is known to admit minimizers (possibly just for that T ) , then Ω is a ball .
(We notice for future use an important consequence of (1.23), namely, we have indeed, by [MN17], Φ H (T ) admits minimizers for every T > 0.) With these premises, we now state our main results concerning Question 2.
Theorem 1.3 (Rigidity of "Balls have the worst best Sobolev inequalities").
, Ω an open, bounded, connected set with C 1 -boundary in R n , and assume that one of the following two conditions holds: (i): there is T * > 0 such that Φ Ω (T ) = Φ B (T ) for every T ∈ (0, T * ); or (ii): n > 2p, the boundary of Ω is of class C 2 , and there is T ∈ (0, ISO(B) Then, Ω is a ball.
1.3.Strategy of proof.Concentration-compactness arguments and the use of sharp Sobolev-type inequalities (like the Sobolev and Escobar inequalities (1.3) and (1.6)) are the standard tools of the trade in the analysis of variational problems with critical growth.As seen, if interpreted as assertions about Φ H , (1.3) and (1.6) contain only very partial information (respectively, "Φ H (0) = S(n, p)" and "Φ H (T ) ≥ E(n, p) T for every T > 0").From this viewpoint, our arguments provide an interesting example of the potentialities of using, in the familiar context of concentration-compactness, the full characterization of Φ H obtained in [CL94,MN17].We now explain how this characterization is used in this paper.
We have already mentioned how the mere knowledge of the existence of minimizers in Φ H (T ) for every T > 0 allows one to reduce the analysis of concentrations to the simplest possible case of a single boundary concentration (thus leading to Theorem 1.1-(i)).Finer properties of Φ H are exploited in the proof of Theorem 1.1-(ii), which goes as follows.We consider the existence of a minimizer (u, v, t) of Φ * Ω (T ) with v > 0, and, keeping in mind that Φ Ω (T ) = Φ * Ω (T ), we aim to obtain a contradiction to v > 0 by constructing a competitor v of Φ Ω (T ) with We seek v in the form v = u ε , for the Ansatz given by Here x 0 ∈ ∂Ω is a boundary point of Ω with positive mean curvature 2 , i.e.H ∂Ω (x 0 ) > 0; ϕ ε is a cut-off function between B ε β (x 0 ) and B 2 ε β (x 0 ) for β = β(n, p) ∈ (0, 1) to be suitably chosen (the condition n > 2 p enters in this choice); g is a boundary flattening diffeomorphism near x 0 ; and, finally, U = U τ + b ε V τ for τ = t/v, V τ a standard perturbation of U τ , and b a constant suitably chosen depending on n, p, H ∂Ω (x 0 ) and τ .The energy, volume and trace expansions for u ε as ε → 0 + are computed to be where λ H (T ) is the volume Lagrange multiplier of U T (see (3.7) below), and where L and M are functionals defined on U : H → R by Modulo o(ε)-perturbations of u ε aimed at correcting the volume and trace constraints to the exact values needed for inclusion in X Ω (T ), we have constructed the required competitors, and proved Theorem 1.1-(ii), if we can show the existence of c(n, p, T ) > 0 such that Of course, the full characterization of Φ H plays a crucial role in our proof of (1.30), see Lemma 3.1 below.
While Theorem 1.3-(ii) is immediate from Theorem 1.1-(ii) thanks to the rigidity criterion (1.23), the proof of Theorem 1.3-(i) requires an additional argument, which once more exploits several fine properties of Φ B and Φ H : these include (1.23), (1.24), and information on the signs of the Lagrange multipliers λ H (T ) and σ H (T ) for minimizers U T of Φ H (T ) (see (3.12) and (3.11) below).
2 Our convention is that the scalar mean curvature of ∂Ω is computed with respect to the outer unit normal to Ω, so that every bounded open set with C 2 -boundary has at least one boundary point of positive mean curvature.
1.4.Organization of the paper.After collecting a few preliminary results in section 2, in section 3 we study in detail various properties of Φ H and of its minimizers: in particular, we prove the key inequality (1.30) (see Lemma 3.1), and discuss in detail the Ansatz (1.25) (see Lemma 3.4).Sections 4 and 5 contain, respectively, the proofs of Theorem 1.1 and Theorem 1.3.Finally, we collect some auxiliary, routine proofs in an appendix.

Notation and preparations
Some basic notation is presented in section 2.1.We then discuss, in separate subsections, four useful technical lemmas: a concentration-compactness lemma with boundary terms (Lemma 2.1); a second order expansion for the boundary flattening diffeomorphisms used in the Ansatz (1.25) (Lemma 2.3); some basic regularity information on minimizers of Φ Ω (T ) (Lemma 2.5); and the basic technique of "volume/trace correcting variations" (Lemma 2.6).Some proofs are postponed to the appendix.
2.1.Notation.Throughout the paper we always assume that n ≥ 2 and p ∈ (1, n).We denote by L n and H k the Lebesgue measure and the k-dimensional Hausdorff measure of R n , although we simply set |E| in place of L n (E).We denote by B r (x) the open ball of center x ∈ R n and radius r > 0, and set B r = B r (0), while B denotes a ball of unspecified center and radius.
Following a standard shorthand notation, by "f a rate that is uniform with respect to the parameters a and b.
In general, we will use capital letters (e.g.U, V, Ψ) to denote functions defined on the half space H and lowercase letters (e.g.u, v, ϕ) to denote functions defined on an open bounded domain Ω.

Concentration-compactness.
The following lemma is a version of Lions' celebrated concentration-compactness lemma and provides a natural starting point to study minimizing sequences of Φ Ω (T ).
Lemma 2.1 (Concentration-compactness).Let n ≥ 2, p ∈ (1, n), and let Ω ⊂ R n be open and bounded with C 1 -boundary.If {u j } j is a sequence in L 1 loc (Ω), {∇u j } j is bounded in L p (Ω; R n ) and u j ⇀ u as distributions in Ω, then the Radon measures on R n defined by have subsequential weak-star limits µ, ν and τ which satisfy where {x i } i∈I ⊂ Ω is at most countable set, v i > 0 and t i ≥ 0 for every i ∈ I, t i > 0 only if x i ∈ ∂Ω, and In particular, g i ≥ S v i whenever x i ∈ Ω.
Proof.See appendix A.
2.3.Near-boundary coordinates.In this section, we introduce two types of coordinates for a neighborhood of a boundary point of a domain Ω: one that requires minimal regularity of the boundary of Ω and will suffice in the proofs of Theorem 1.1(i) and Theorem 1.3(i), and a second that requires C 2 regularity of the boundary of Ω and will be used in the proof of Theorem 1.1(ii) and Theorem 1.3(ii).
Given an open set Ω with C 1 -boundary, we denote by ν Ω its outer unit normal and by T x (∂Ω) the tangent space to x ∈ ∂Ω.When Ω has C 2 -boundary, we denote by A Ω and H Ω the second fundamental form and the scalar mean curvature of ∂Ω defined by ν Ω .To define coordinates near boundary points of Ω, for then we can find r 0 > 0 and ℓ : D r 0 → (−r 0 , r 0 ) such that ℓ(0) = 0, ∇ℓ(0) = 0, and We then define the maps (2.9) In this way, for every y ∈ (∂Ω) ∩ C r 0 , if we set y = F (px), then Notice that the map f need not be of class C 1 if the boundary of Ω is only of class C 1 , while the map f will be as regular as the boundary of Ω.The following lemma summarizes basic properties about the map f .Lemma 2.2 (Near-boundary coordinates, one).If H = {x n > 0}, Ω is an open set with C 1 -boundary and (2.6) holds, then there exist r 0 and C 0 positive such that the map f in (2.8) defines a C 1 -diffeomorphism from C r 0 to its image, taking D r 0 into ∂Ω and with (2.10) (2.12) The orders in (2.11) and (2.12) depend on ∂Ω and on 0 ∈ ∂Ω.
Proof.See appendix B.
The map f defined in (2.9) has the advantage that, when the boundary of Ω is at least of class C 2 , curvature quantities appear in expansions of the metric coefficients and the volume form in these coordinates.These properties are the content of the following lemma.
Lemma 2.3 (Near-boundary coordinates, two).If H = {x n > 0}, Ω is an open set with C 2 -boundary and (2.6) holds, then there exist r 0 and C 0 positive such that the map f in (2.9) defines a C 1 -diffeomorphism from C r 0 to its image, taking D r 0 into ∂Ω and with (2.13) Moreover, for x ∈ C r 0 and x ∈ D r 0 respectively, we have denote the corresponding eigenvalues (so that, by (2.6), they are the principal curvatures of ∂Ω with respect to ν Ω computed at 0 ∈ ∂Ω, and in particular H Ω (0) = n−1 i=1 κ i ), then, letting g = f −1 denote the inverse of f , we have (2.15) The orders in (2.14) and (2.15) depend on ∂Ω and on 0 ∈ ∂Ω.
Proof.See appendix B.
Remark 2.4.Given x 0 ∈ ∂Ω, we denote by π x 0 the rigid motion of R n that maps x 0 to 0 such that (2.6) holds with π x 0 (Ω) in place of Ω.Then we set, for f and f defined as in (2.8) and (2.9) respectively but with Clearly these maps are diffeomorphisms on C r 0 , mapping H ∩ C r 0 into a neighborhood of x 0 in Ω and D r 0 = (∂H) ∩ C r 0 into a neighborhood of x 0 in ∂Ω, and satisfies proper reformulations of the estimates in Lemmas 2.2 and 2.3.Here r 0 and C 0 depend also on the choice of x 0 , and can of course be assumed uniform across x 0 ∈ ∂Ω if ∂Ω is bounded.

Properties of minimizers.
The following lemma gathers some fundamental properties of minimizers of Φ Ω that will be needed in the sequel.Proof.By a standard argument, based on similar considerations to the one presented in Lemma 2.6 below, one sees that a minimizer u of Φ Ω (T ) is a W 1,p (Ω)-distributional solution of the Euler-Lagrange equation (1.2) for some λ, σ ∈ R. As soon as Ω is bounded and has Lipschitz boundary, one can exploit (1.2) in conjunction with a Moser iteration argument to prove that u ∈ L ∞ (Ω) (see, e.g.[MW19, Theorem 3.1]; their result applies to (1.2) by taking, in the notation of their paper, A(x, u, ∇u) = |∇u| p−2 ∇u, B(x, u, ∇u) = λu p ⋆ −1 , and C(x, u) = σu p # −1 ).On further assuming that ∂Ω is of class C 2 , then the classical result [Lie92, Theorem 1.7] can be applied to deduce that u ∈ C 1,β (Ω) for a suitable β = β(n, p) ∈ (0, 1) (for more details, see [MW19, Theorem 3.9]).In particular, u is bounded and Lipschitz continuous on Ω, as claimed.
2.5.Volume/trace correcting variations.At various stages in our arguments we will need to slightly modify certain competitors so to restore the volume and trace constraints defining X Ω (T ).The following lemma describes the basic mechanism used to this end.
and if x 0 ∈ R n and r > 0 are such that ´Ω\Br(x 0 ) v p ⋆ and ´(∂Ω)\Br(x 0 ) v p # are positive and finite , (2.17) then there exist positive constants η and C, and functions )), all depending on n, p, v and M only, and with the following property.

Boundary concentrations
3.1.Properties of Φ H -minimizers.We recall some facts proved in [MN17] about Φ H and its minimizers.Recall that we denote by T 0 the minimum point of Φ H , so that where T E is the "Escobar trace" defined in (1.8).If we set H = {x n > 0}, the minimizers of U T of Φ H (T ) for T > 0 are characterized (modulo the obvious scaling and translation invariance of Φ H ) as where the constants c T , t T and s T are chosen in such a way that It is convenient to keep in mind that the various formulas for U T listed in (3.2) all share the same decay behavior at infinity, that is (see (3.20) below), we have (where the rate depends on the specific value of T under consideration).The constants t T and s T have the following properties: T ∈ (0, T E ) → t T is continuous and strictly decreasing, with t T > 0 if and only if T ∈ (0, T 0 ), and lim while T ∈ (T E , ∞) → s T is continuous, negative, strictly increasing, with lim Denoting by ∆ p v = div (|∇v| p−2 ∇v) the p -Laplace operator, we have where λ H , σ H : (0, ∞) → R are continuous and satisfy the relations (see [MN17, Lemma 3.3]3 ) as well as lim The signs of σ H and λ H can be easily deduced from (3.2), and satisfy (3.12)

3.2.
A key inequality and further properties of U T .In this section, we prove the key inequality (3.13) for the functions L and M introduced in (1.28) and (1.29), namely Whenever U satisfies the decay properties (3.4) (e.g., when U is a compactly supported perturbation of some U T ), we have that M(U ) < ∞; however, L(U ) < ∞ under (3.4) if and only if n > 2 p − 1; see (3.24) and (3.25) below.
Lemma 3.1 (Key inequality).If n ≥ 2, p ∈ (1, n), n > 2p − 1, and T > 0, then there is a positive constant c(n, p, T ) such that The following lemma will be useful in proving Lemma 3.1.
We now prove Lemma 3.1.
Proof of Lemma 3.1.Testing (3.7) with x n U T we find Here the integration by parts is justified since x n = 0 on ∂H and since by (3.4), Now, since |∇U T | is symmetric by reflection with respect to the hyperplanes {x i = 0}, i = 1, ..., n − 1, we see that so that continuing from above we have In particular, the lemma is proved by showing that where the first inequality, (3.15), is immediate from Lemma 3.2 (recall that n > 2p − 1 and U T is radially symmetric with respect to te n for some t ∈ R).
We are thus left to prove (3.16).This is immediate in the case when T ≥ T 0 , because in that case, by (3.5) and (3.6), U T has center of symmetry at t e n for some t ≤ 0, and thus ∂ n U T < 0 on H.By (3.5), if T ∈ (0, T 0 ), then U T has center of symmetry at t e n for some t > 0. Correspondingly, U T ∂ n U T is odd with respect to {x n = t}, with U T ∂ n U T < 0 on {x n > t} and U T ∂ n U T > 0 on {0 < x n < t}: in particular, if p t denotes the reflection with respect to {x n = t}, then and the latter integral is positive because ∂ n U T < 0 on {x n > t}.

3.3.
Standard variations of Φ H -minimizers.We now introduce a "class of standard variations" of minimizers of Φ H .With Given T > 0 we denote by U T (3.18) the family of functions U : H → R of the form The following lemma contains some basic properties of functions in U T .We notice that every U ∈ U T is symmetric by reflection with respect to the coordinates x 1 , ..., x n−1 .
(3.19) Lemma 3.3 (Standard variations of U T ).If n ≥ 2, p ∈ (1, n), and T > 0, then there are positive constants R 0 and C 0 depending on n, p, T , and V T such that the following properties hold: for dilations introduced in (1.5).
Step one: We start by noticing the following estimates for the energy, volume and trace of v ε in transition region for the cut-off function ϕ ε .The estimates in this step hold in identical form with the same proofs for v ε defined from f as in (3.29) and for v ε defined from f as in (3.33); we write the proof for (3.33).First, with v ∈ W 1,p (Ω), and, second, under the additional assumption that v ∈ Lip(Ω), By (2.13), and thanks to |∇g|, Jf ≤ 2 on C r 0 , we find where in the last inequality we have used (3.22).Concerning b ε , we notice that if we only know that v ∈ W 1,p (Ω) then by v ∈ L p ⋆ (Ω) and ∇v ∈ L p (Ω) we find that where the latter quantity converges to 0 at an non-quantified rate as ε → 0 + (as stated in (3.37)); while, if v ∈ Lip(Ω), then Step two: We prove statement (i).By (3.37), This proves (3.30).Entirely analogous arguments prove (3.31) and (3.32).
Step three: We now start the proof of statement (ii); in particular, from now on, Ω has C 2 -boundary, n > 2p, and v ε is defined as in (3.33); moreover, for the sake of brevity, we set h = H ∂Ω (0).In this step, we discuss the choice of β = β(n, p) ∈ (0, 1), which is determined by the rates in (3.38), (3.39) and (3.40), and by the fact that in (3.34), (3.35) and (3.36) we want errors of size o(ε): therefore, by we are led to choose Step four: We prove (3.34).We first notice that by (3.43) and (3.41) we have Now, by (2.15) we have and thus Then, by (2.13), Now, by the reflection symmetries of U with respect to {x i = 0}, i = 1, ..., n − 1 (recall (3.19)), we have and therefore we can thus rewrite (3.45) as At the same time, ε 2 |x| 2 ≤ C ε r 1 |x| for any x ∈ B C r 1 /ε , so where we have used the facts that V T is compactly supported and that n > 2p − 1 to guarantee the convergence of the integrals in the final line.Hence, where the o(ε) term depends on n, p, and T , and in the second line we have applied (3.25).By (3.44) we deduce (3.34).
Step five: We prove (3.35).We first notice that by (3.43), v ∈ Lip(Ω), r n 2 = ε β n = o(ε) and our choice of β we have Then keeping in mind (2.13) and (2.14), By (3.21) and (3.24), along with our choice of β, we see that Moreover, since U = U T + tV T with V T compactly supported in H and |t| ≤ 1, we have with o(ε) depending on n, p, and T .So, the entire term in brackets above can be written as o(ε).Combining this estimate with (3.47), we deduce (3.35).
Step six: We finally prove (3.36).Notice that, by (3.43 Now, by J ∂H f ≥ 1, (3.23) and our choice of β we have At the same time, by thanks to n > 2p − 1.This completes the proof.

Existence of minimizers
We first establish the existence of generalized minimizers.Recall that Φ * Ω (T ) was defined in (1.21).(i): for every T > 0, Φ Ω (T ) = Φ * Ω (T ); (ii): there is a minimizer (u, v, t) of Φ * Ω (T ); and there exists λ, σ ∈ R such that In particular, u ∈ Lip(Ω).If, in addition, v > 0, then λ and σ are given by and, in particular, Step one: Since (u, 0, 0) ∈ Y Ω (T ) if u ∈ X Ω (T ), we have Φ * Ω (T ) ≤ Φ Ω (T ).To prove the converse inequality it is enough to show that for every (u, v, t) ∈ Y Ω (T ), Looking back at the definition of Y Ω (T ) in the paragraph preceding the statement of Theorem 1.1, we can assume without loss of generality that v > 0 and t > 0.Moreover, given (u, v, t) ∈ Y Ω (T ) with v and t positive we can easily find (u j , v j , t j ) ∈ Y Ω (T ) with v j , t j , ´Ω u p ⋆ j , and ´∂Ω u p # j positive and such that E(u j , v j , t j ) → E(u, v, t).By a diagonal argument, it is thus sufficient proving (4.5) under the assumption that ´Ω u p ⋆ and ´∂Ω u p # are positive.This said, we apply Lemma 3.4(i) with to find functions v j with v j = v on Ω \ B 2 ε j (x 0 ) for some x 0 ∈ ∂Ω and ε j → 0 + , and with where G j , V j , T j → 0 as j → ∞ at a rate depending on n, p, Ω, t/v and u only.By Lemma 2.6, there exist η and C depending on n, p, Ω, t/v and u, but independent from j, such that for any (a j , b j ) with |a j | + |b j | < η, we have functions w j such that For j large enough we can apply this statement with a j = −T j and b j = −V j to find a sequence {w j } j with Setting u j = v w j , we obtain a sequence in X Ω (T ) that satisfies (4.5).
Step two: We prove that there is a minimizer for the generalized problem Φ * Ω (T ).By the argument in step one we can find a sequence {u j } j in X Ω (T ) such that ´Ω |∇u j | p → Φ * Ω (T ) p .By Lemma 2.1, the measures µ j , ν j and τ j defined in (2.1) have subsequential weak-star limits µ, ν and τ satisfying (2.2), (2.3) and (2.4) and (2.5).In particular, there is an at most countable set {x i } i∈I ⊂ Ω and corresponding v i > 0 and t i ≥ 0 for every i ∈ I, such that where u is the subsequential weak limit of u j , and By an immediate adaptation of the proof of Lemma 3.4 we can easily construct a sequence {W j } j in X H (t c /v c ) with the property that while (4.11) gives (u, v c , t c ) ∈ Y Ω (T ).This proves that (u, v c , t c ) is a minimizer of Φ * Ω (T ).
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1.Statement (i) is an immediate consequence of Theorem 4.1.We thus focus on statement (ii), and assume that Ω is of class C 2 and that n > 2p.We want to prove that if (u, v, t) is a minimizer of Φ * Ω (T ), then v = t = 0. We assume by way of contradiction that either v > 0 or t > 0; recalling the definition of Y Ω (T ), this implies that v > 0 and t > 0. We apply Lemma 3.4 (ii) with the choice (v, T ) = (u/v, τ ) at a point x 0 ∈ ∂Ω of positive mean curvature, noting that if v = 1 then v ≡ 0 and that v is Lipschitz continuous if v ∈ (0, 1) thanks to (4.1) and Lemma 2.5.Then, for every U ∈ U τ , we have where L(U ) and M(U ) are defined in (3.46) and (1.29).We apply this with U ∈ U τ given by The reason for the choice of b will become apparent in a moment.Indeed, thanks to (3.26), (3.27) and (3.28), we have which, combined with (4.13), (4.14), (4.15) and where in (4.16) we have used the choice of b to deduce In the same spirit, by lim where in the second line we apply Lemma 3.1.We thus conclude that It remains to modify the functions w ε to obtain w * ε ∈ X Ω (T ) also satisfying (4.20), allowing us to conclude the proof of the theorem by choosing ε sufficiently small.We will distinguish between two cases, applying Lemma 2.6 in different ways in the two cases.
Case one: Suppose first that v < 1 and thus ´Ω u p ⋆ > 0. This also implies that ´∂Ω u p # > 0 by Theorem 4.1-(iii).Taking into account (4.17) and (4.18), we can thus apply Lemma 2.6 in an analogous way to step one of the proof of Theorem 4.1 in order to slightly modify thus reaching a contradiction.
In this case, we will pull the relevant quantities back to the half space H and apply Lemma 2.6 there to correct the volume and trace constraints.More specifically, for ε < ε 0 , the support of ϕ ε is entirely contained in the domain of the diffeomorphism f x 0 , and so we can define Ψ ε : In this way, we can rewrite Thanks to (2.13), Ψ ε is identically equal to one in B ε β−1 /C ∩ H and vanishes outside of Using the area formula, we rewrite (4.17), (4.18), and (4.20) as where we have set m ε (x) = Jf (ε x) , mε (x) = J ∂H f (ε x) , A ε (x) = (∇g • f (ε x)) * .
We can thus apply the inverse function theorem uniformly in ε, to obtain functions W * ε : giving us a contradiction in this case as well.This completes the proof of the theorem.

Rigidity theorems for best Sobolev inequalities
In this section, we prove Theorem 1.3.
By Theorem 4.1, for every T > 0 there is (u T , v T , t T ) a minimizer of Φ * Ω (T ).The basic idea of the proof will be to show that the trace-to-volume ratio of u T must be, on one hand, uniformly positive and, on the other hand, tending to zero as T → 0, giving us a contradiction.Since Ω is connected and is not a ball by assumption, the rigidity criterion (1.23) together with (1.24) and (5.1) tell us that a classical minimizer for Φ Ω (T ) cannot exist for T ∈ (0, T * ), and so we immediately deduce that v T < 1 for all such T .In other words, if we set ν T = 1 − v p ⋆ T ) 1/p ⋆ = u T L p ⋆ (Ω) , τ T = (T p # − t p # ) 1/p # = u L p # (∂Ω) , then ν T > 0 for all T ∈ (0, T * ).So, Theorem 4.1-(iii) implies that u T /ν T is a minimizer of Φ Ω (τ T /ν T ) . (5.2) In particular, this means that τ T ν T ≥ T * for all T ∈ (0, T * ) , (5.3) since as we noted above, no minimizer of Φ Ω ( T ) can exist for T = τ T /ν T < T * .Since T ≥ τ T , (5.3) tells us that T /ν T ≥ T * ; rearranging this inequality gives us the following lower bound on v T : v p ⋆ T ≥ 1 − (T /T * ) p ⋆ ∀T ∈ (0, T * ) . (5.4) From this, an upper bound on the ratio t T /v T follows immediately: t T v T ≤ 1 − (T /T * ) p ⋆ −1/p ⋆ T ∀T ∈ (0, T * ) . (5.5) In particular t T /v T → 0 as T → 0 + .and from (B.5) and (B.6) we directly compute that A j i = g ik A kj is given by Step 2: Next, we use the previous step to compute geometric quantities associated to the coordinates defined by f .For x ∈ C r 0 and i = 1, . . ., n − 1, we note that ∂ i p(x) = e i and thus from (B.7),In what follows we will suppress the composition with p in our notation, writing for instance τ i in place of τ i • p(x).For i = 1, . . ., n − 1, we have In particular, from (B.2) we see that the volume form is given by Jf = det g ij = 1 − x n H Ω (0) + O(|x| 2 ) , giving us the first estimate in (2.14).We see directly from the definition that f is a C 1 map, and since we see from the expression for ∇f above that ∇f (0) = Id R n , we may apply the inverse function theorem to see that, up to decreasing r 0 , f defines a C 1 diffeomorphism onto its image.Letting g = f −1 and using the expansion of the inverse (B.1), we find Finally, (2.13) follows from these expressions for ∇f and ∇g, along with the assumptions that ∇ℓ(0) = 0.This completes the proof of the lemma.
2) Escobar inequality: The Escobar inequality ([Esc88, for p = 2], [Naz06, for p ∈ (1, n)]) states that if H is an (open) half-space in R n with outer unit normal ν H , then The Ansatz for boundary concentrations.We next use the standard variations of minimizers of Φ H described in Lemma 3.3 to define certain competitors for Φ Ω that provide us with a notion of "standard boundary concentration."Recall the notation U(ε) .25) (ii): for every U ∈ U T we haveˆH |∇U | p = Φ H (T ) p + p λ H (T ) t + o(t) ,