Calc. Var. (2017) 56:49 DOI 10.1007/s00526-017-1138-4
Calculus of Variations
On the p-Laplacian with Robin boundary conditions and boundary trace theorems Hynek Kovaˇrík1 · Konstantin Pankrashkin2
Received: 28 September 2016 / Accepted: 20 February 2017 © Springer-Verlag Berlin Heidelberg 2017
Abstract Let ⊂ Rν , ν ≥ 2, be a C 1,1 domain whose boundary ∂ is either compact or behaves suitably at infinity. For p ∈ (1, ∞) and α > 0, define
|∇u| dx − α |u| p dσ ∂ , |u| p dx p
(, p, α) :=
inf
u∈W 1, p () u≡0
where dσ is the surface measure on ∂. We show the asymptotics p
(, p, α) = −( p − 1)α p−1 − (ν − 1)Hmax α + o(α), α → +∞, where Hmax is the maximum mean curvature of ∂. The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities. Mathematics Subject Classification 49R05 · 35P30 · 58C40
Communicated by L. Ambrosio.
B
Hynek Kovaˇrík
[email protected] http://dm.ing.unibs.it/~hynek.kovarik/ Konstantin Pankrashkin
[email protected] http://www.math.u-psud.fr/~pankrash/
1
DICATAM, Sezione di Matematica, Università degli studi di Brescia, Via Branze 38, 25123 Brescia, Italy
2
Laboratoire de Mathématiques d’Orsay, Univ.Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
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1 Introduction and main results 1.1 Problem setting Let ⊂ Rν , ν ≥ 2, be a domain with a sufficiently regular boundary. For α > 0 and p ∈ (1, ∞), consider the quantity |∇u| p dx − α |u| p dσ ∂ (, p, α) := inf , (1) u∈W 1, p () p |u| dx u≡0
where dσ is the surface measure on ∂. Standard variational arguments show that under suitable assumptions, e.g. if is bounded with a Lipschitz boundary, the problem (1) has a minimizer, see e.g. Proposition 6.1 below. The respective Euler-Lagrange equation takes the form of a non-linear eigenvalue problem ∂u (2) = α|u| p−2 u at ∂, ∂n where p is the p-Laplacian, p u = ∇ · |∇u| p−2 ∇u , n is the outer unit normal, and = (, p, α). Equation (2) should be understood in the weak sense, i.e. p−2 p−2 (, p, α) |u| u φ dx = |∇u| ∇u · ∇φ dx − α |u| p−2 u φ dσ (3) − p u = |u| p−2 u in ,
|∇u| p−2
∂
W 1, p ().
holds for all φ ∈ In the present paper we work with C 1,1 domains, either bounded or with a suitable behavior at infinity (see below), and we study the behavior of (1) as α tends to +∞. While the properties of (, p, α) for α < 0 are well understood for any p > 1, see e.g. [13] and references therein, the same problem for α → +∞ was previously studied for the linear case p = 2 only. It was shown in [27] that for bounded C 1 domains there holds (, 2, α) = −α 2 + o(α 2 ) as α → +∞. Under additional smoothness assumptions, a more detailed asymptotic expansion (, 2, α) = −α 2 − (ν − 1)Hmax () α + o(α) was obtained first in [15,29] for ν = 2 and then in [31,32] for the general case, where Hmax is the maximal mean curvature of the boundary. Further terms in the asymptotic expansion can be calculated under suitable geometric hypotheses, see e.g. [9,23,24,32]. Non-smooth domains were studied as well, see e.g. [25] and the recent paper [5]. The above mentioned papers used a number of techniques which are specific for the linear problems, such as the perturbation theory for self-adjoint operators or a separation of variables, which are not available for the p-Laplacian. In the present paper we are going to modify the existing approaches, which will allow us to feature the variational nature of the problem and to consider arbitrary values of p in a unified way. Furthermore, we work under weaker smoothness conditions when compared to the preceding works, and only C 1,1 regularity is assumed.
1.2 Main result Let us pass to the exact formulation of our main result. In the sequel a domain will be called admissible if the following conditions are satisfied:
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(i) the boundary ∂ is C 1,1 , i.e. is locally the graph of a function with a Lipschitz gradient, (ii) the principal curvatures of ∂ are essentially bounded, (iii) for some δ > 0 the map ∂ × (0, δ) (s, t) → s − tn(s) ∈ x ∈ : dist(x, ∂) < δ is bijective. The mean curvature H of ∂ is the arithmetic mean of the principal curvatures, and we set Hmax ≡ Hmax () := ess sup H.
(4)
We remark that we do not assume that this value is attained. An account of the differential geometry in the C 1,1 setting, including the precise definition of the curvatures, can be found e.g. in [20, Section 3]. In particular, the assumptions are satisfied for any domain with a compact C 1,1 boundary. Another obvious example of an admissible domain is given by any C 1,1 domain coinciding with a half-space outside a ball. Our main result reads as follows: Theorem 1.1 For any admissible domain ⊂ Rν and any p ∈ (1, ∞) there holds p
(, p, α) = −( p − 1)α p−1 − (ν − 1)Hmax () α + o(α) as α → +∞.
(5)
Remark 1.2 The asymptotics can be different for domains with a weaker regularity, see e.g. Proposition 6.8 below for Lipschitz domains. Remark 1.3 A significant feature of our method of proving (5) is that it does not assume existence of minimizers. This is important since there exist admissible domains for which problem (1) does not have a minimizer; for example if p = 2 and is an infinite cylinder with sufficiently smooth boundary, then it is easily seen that the infimum in (1) is not attained. The proof of Theorem 1.1 is presented in Sects. 3–5 and is organized as follows. In Sect. 3 we estimate the eigenvalue (, p, α) using some auxiliary operators in a tubular neighborhood of ∂. In Sect. 4 we obtain an upper bound by a suitable choice of test functions. The lower bound is obtained in Sect. 5 using an analysis of an auxiliary one-dimensional operator. In turns out that Theorem 1.1 has various applications to Sobolev boundary trace theorems, extension operators and isoperimetric inequalities. These are described in Sect. 2. Apart from the asymptotic behavior of (, p, α) it is natural to address the question of the behavior of the associated eigenfunction, in other words the minimizer of (1). Indeed, in [9] it was shown, for p = 2, that as α → ∞, the eigenfunctions concentrate at the boundary of . In Sect. 6 we carry this analysis, for a general p > 1, a bit further. In particular, we prove an exponential localization near the boundary, Theorem 6.4, and a localization near the part of the boundary at which the mean curvature attains its maximum, Theorem 6.6. Some explicitly solvable cases are discussed in “Appendix 1”. In “Appendix 2” we prove an auxiliary elementary inequality used in the proofs, and in “Appendix 3” we show how the remainder estimate in Theorem 1.1 can be improved under stronger regularity assumptions on ∂.
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2 Applications 2.1 Best constants for boundary trace theorems The asymptotic expansion (5) provides a number of consequences for maps between various Sobolev spaces. Note first that a simple scaling argument gives (, p, μ p−1 α) = μ p (μ, p, α), μ > 0. In particular, as the half-space Rν−1 ×R+ is invariant under dilations, the first term on the right-hand side of (5) can be represented as p
(Rν−1 ×R+ , p, α) ≡ −( p − 1)α p−1 , ν ≥ 1,
(6)
see also “Appendix 1”. Theorem 1.1 thus admits the following version for expanding domains: Corollary 2.1 For any admissible domain ⊂ Rν , any p ∈ (1, ∞) and α > 0 one has, as μ tends to +∞, (μ, p, α) = (Rν−1 ×R+ , p, α) − (ν − 1)Hmax ()αμ−1 + o(μ−1 ). Furthermore, one checks easily that the function R+ α → (, p, α) is strictly decreasing and continuous with (, p, 0) = 0 and limα→+∞ (, p, α) = −∞ and, hence, defines a bijection between R+ and R− . Moreover, the function α → (, p, α) is concave being an infimum of affine functions. Denote by S(, p, q) the best constant in the trace embedding W 1, p () → L q (∂), which is defined through p
|u|q dσ
q
1 = sup ∂ , S(, p, q) u∈W 1, p () |∇u| p + |u| p dx u≡0
with
1 < q < p∗ ,
⎧ ⎨ pν − p , ν−p p∗ := ⎩∞,
if p ∈ (1, ν), if p ∈ [ν, ∞).
The value of S(, p, p) is then uniquely determined by the implicit equation , p, S(, p, p) = −1.
(7)
In particular, in view of (6) it holds S(Rν−1 ×R+ , p, p) = ( p − 1)
1− p p
, ν ≥ 1,
see also [12, Lemmas 3.1 and 3.3]. Various estimates for S(, p, q) were extensively studied in the literature, see e.g. the review [33]. In particular, it was shown in [10] that for any q > 2 there exists a constant γ = γ (q, ν) > 0 independent of with S(μ, 2, q) = S(Rν−1 ×R+ , 2, q) − γ Hmax () μ−1 + o(μ−1 ) for μ → +∞.
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Furthermore, by [16, Theorem 1.3], for each there exist positive constants c1 and c2 such that for μ → +∞ there holds ⎧ ⎨ (q − p)(ν − 1) , if 1 < q < p, κ κ q c1 μ ≤ S(μ, p, q) ≤ c2 μ , κ := ⎩ 0, if p ≤ q < p∗ . In particular, for any p ∈ (1, ∞) the constant S(μ, p, p) remains uniformly bounded and separated from 0 as μ → +∞. The substitution of Corollary 2.1 into (7) gives the following improvement in the spirit of (8): Corollary 2.2 For any admissible domain ⊂ Rν and any p ∈ (1, ∞) there holds ν−1
S(μ, p, p) = S(R
( p − 1)
×R+ , p, p) −
2− p p
(ν − 1)
p
Hmax () μ−1 + o(μ−1 ) (9)
as μ tends to +∞.
2.2 Extension operators Recall that a bounded linear operator E from W 1, p () to W 1, p (Rν ) is called an extension operator if E f coincides with f in for any f . The existence and various estimates for the extension operators in terms of are of interest, see e.g. [7]. We will be concerned with the lower bound for the norms E (, p) = inf E : E : W 1, p () → W 1, p (Rν ) is an extension operator . It is known, in particular, that E (, p) ≥
1+
S(c , p, p) S(, p, p)
1
p
, c := Rν \ .
(10)
see [26, Theorem 3.1]. Note that the work [26] deals formally with the case p = 2 only, but the proof holds literally for any p ∈ (1, ∞). Remark also that 1
E (Rν−1 ×R+ , p) = 2 p .
In fact, the lower bound follows from by the operator of extension by (10), and it is attained parity (E f )(x1 , . . . , xν−1 , xν ) := f x1 , . . . , xν−1 , |xν | . We have the following result: Corollary 2.3 Assume that both and c are admissible domains in Rν and that p ∈ (1, ∞), then for μ → +∞ there holds 1
ν−1
E (μ, p) ≥ E (R
×R+ , p) +
( p − 1) p 2
p−1 p
(ν − 1) Hmax () + Hmin () μ−1 + o(μ−1 ),
p2
where Hmin () := ess inf H . Proof We have Hmax (c ) = −Hmin (), and the substitution into (9) and then into (10) gives the result.
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2.3 Isoperimetric inequalities Numerous works studied isoperimetric inequalities for the quantities (, 2, α) and S(, 2, 2). In particular, in [2] it was conjectured that the balls maximize (, 2, α) among all fixed volume domains for any α > 0. An analogous question for S(, 2, 2) was asked e.g. in [34]. The conjecture was supported e.g. by the consideration of the first and second variations of the respective functionals and by showing that the balls are at least local extrema, see e.g. [17,34]. It was shown only recently in [19] that the conjecture in the general form is wrong by comparing the eigenvalues of the balls with those for the spherical shells for large α, while it remains true at least in two dimensions for a restricted range of positive α. (It is worth noting that the case α < 0 is well understood for any p, see [6,8].) In fact, the conjecture appears to be closely related to some estimates for the maximum mean curvature Hmax as discussed in [31], and the asymptotics (5) and (9) allow us to include into consideration all possible values of p. More precisely, let us recall the following known results: (A) The balls do not minimize the quantity Hmax among the bounded C 1,1 domains having the same volume. In particular, consider the domains Bρ := x ∈ Rν : |x| < ρ , Ur,R := x ∈ Rν : r < |x| < R , 0 < r < ρ < R,
Rν − r ν = ρν ,
then vol Bρ = vol Ur,R ,
Hmax (Bρ ) =
1 1 > = Hmax (Ur,R ). ρ R
(B) For ν = 2, the balls are the strict minimizers of Hmax among all bounded simply connected C 2 domains of a fixed area, see e.g. the discussion in [30]. (C) For ν ≥ 3, the balls are the strict minimizers of Hmax among the bounded star-shaped C 2 domains of the same volume, see e.g. [31, Theorem 2]. (D) For ν = 3, the balls do not minimize the quantity Hmax among the bounded C 1,1 domains having the same volume and homeomorphic to a ball. Moreover, there is no strictly positive lower bound for Hmax in terms of the volume. The respective examples were constructed recently in [18]. The combination of (A–C) with Theorem 1.1 gives the following observations, with an arbitrary p ∈ (1, ∞): • The balls do not maximize (, p, α) among the domains of a fixed volume. In particular, for sufficiently large α > 0 there holds (Bρ , p, α) < (Ur,R , p, α). • Let B ⊂ Rν be a ball and ⊂ Rν be a simply connected bounded C 2 domain of the same volume, and for ν ≥ 3 assume additionally that is star-shaped, then there exists α > 0 such that (B, p, α) ≥ (, p, α) for α > α , with an equality iff is a ball. • At least for ν = 3, the balls do not maximize (, p, α) among the domains homeomorphic to balls and having a fixed volume. In a similar way, Corollary 2.2 combined with (A–C) gives the following assertions valid for any p ∈ (1, ∞): • The balls do not maximize S(, p, p) among the domains of a fixed volume. In particular, for sufficiently large μ > 0 there holds S(μBρ , p, p) < S(μUr,R , p, p). • Let B ⊂ Rν be a ball and ⊂ Rν be a simply connected bounded C 2 domain of the same volume, and for ν ≥ 3, assume additionally that is star-shaped, then there exists μ > 0 such that S(μB, p, p) ≥ S(μ, p, p) for μ > μ , with an equality iff is a ball.
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• At least for ν = 3, the balls do not maximize S(, p, p) among the domains homeomorphic to balls and having a fixed volume. In view of (8), the same considerations hold for S(, 2, q) with any q > 2.
3 Proof of theorem 1.1: bracketing and a change of variables The construction of this section is quite standard and represents a suitable adaptation of [31, Sections 2.2 and 2.3]. For s ∈ S := ∂, let n(s) denote the outer unit normal and Ts S the tangent space to S at s, and let L s := dn(s) : Ts S → Ts S be the shape operator, which is defined for almost all s ∈ S, see [20, Section 3]. Recall that the eigenvalues κ1 (s) ≤ · · · ≤ κν−1 (s) of L s are the so-called principal curvatures at s, and the mean curvature H (s) at s is defined by H (s) =
κ1 (s) + · · · + κν−1 (s) 1 ≡ tr L s . ν−1 ν−1
By assumptions, all κ j are essentially bounded, and the maximal mean curvature Hmax ≡ Hmax () := ess sup H is well-defined. In what follows, it will be convenient to use the quantities M(s) := (ν − 1)H (s),
Mmax := (ν − 1)Hmax .
For δ > 0 denote δ := x ∈ : inf |x − s| < δ}, δ := \ δ , s∈S |∇u| p dx − α |u| p dσ δ ∪δ S inf , N (, p, α) := u∈W 1, p (δ ∪δ ) p |u| dx u ≡0 δ ∪δ p |∇u| dx − α |u| p dσ S inf , D (, p, α) := u∈W 1, p () p |u| dx u=0 on ∂δ u≡0
(11)
then one clearly has N (, p, α) ≤ (, p, α) ≤ D (, p, α). Furthermore, denote p |∇u| dx − α |u| p dσ δ S N ,δ inf (, p, α) := u∈W 1, p (δ ) p |u| dx u≡0 δ |∇u| p dx δ N ,δ , inf 0 (, p) := u∈W 1, p (δ ) p |u| dx u≡0 δ
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|∇u| p dx − α |u| p dσ δ S D,δ (, p, α) := inf 1, p (δ ) u∈W p |u| dx u≡0 δ |∇u| p dx δ inf 0D,δ (, p) := 1, p u∈W0 (δ ) |u| p dx
u ≡0
δ
where 1, p (δ ) := u ∈ W 1, p (δ ) : u = 0 on ∂δ \ S . W It follows that
N (, p, α) = min N ,δ (, p, α), 0N ,δ (, p) D (, p, α) = min D,δ (, p, α), 0D,δ (, p) .
Note also that 0D,δ (, p) ≥ 0N ,δ (, p) ≥ 0 and D,δ (, p, α) ≥ N ,δ (, p, α). Hence if D,δ (, p, α) ≤ 0, then N (, p, α) = N ,δ (, p, α) and D (, p, α) = D,δ (, p, α), so that N ,δ (, p, α) ≤ (, p, α) ≤ D,δ (, p, α). We will study the quantities N ,δ (, p, α) and D,δ (, p, α) using a change of variables. By assumption we can choose δ > 0 sufficiently small such that the map := S × (0, δ) (s, t) → (s, t) := s − tn(s) ∈ δ
(12)
is bijective and uniformly locally bi-Lipschitz. The metric G on induced by this embedding is G = g ◦ (Is − t L s )2 + dt 2 ,
(13)
where Is : Ts S → Ts S is the identity map, and g is the metric on S induced by the embedding in Rν . The associated volume form d on is 1 1 d(s, t) = det G(s, t) 2 ds dt = ϕ(s, t) det g(s) 2 ds dt = ϕ(s, t) dσ (s) dt, 1 where dσ (s) = det g(s) 2 ds is the induced (ν − 1)-dimensional volume form on S, and the weight ϕ is given by ν−1 ϕ(s, t) := det(Is − t L s ) = 1 − κ j (s)t = 1 − M(s)t + P(s, t)t 2 ,
(14)
j=1
where P is a polynomial in t with coefficients which are essentially bounded functions of s, and we assume in addition that δ > 0 is sufficiently small to have 1/2 ≤ ϕ ≤ 2 almost everywhere in . In particular,
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(∇u) ◦ (s, t) p d(s, t) − α u ◦ (s, 0) p dσ (s) S , N ,δ (, p, α) = inf u∈W 1, p (δ ) u ◦ (s, t) p d(s, t) u ≡0 (∇u) ◦ (s, t) p d(s, t) − α u ◦ (s, 0) p dσ (s) S D,δ (, p, α) = inf , p 1, p (δ ) u∈W u ◦ (s, t) d(s, t) u ≡0
1, p and the map u → u ◦ defines a bijection between W 1, p (δ ) and W () as well as 1, p 1, p 1, p between W (δ ) and W () := v ∈ W () : v(·, δ) = 0 . Furthermore, for 2 v = u ◦ we have (∇u) ◦ = G jk ∂ j v∂k v with (G jk ) := G −1 , and due to (13) we can estimate, with some 0 < C− < C+ ,
C− g −1 + dt 2 ≤ G −1 ≤ C+ g −1 + dt 2 , (g ρμ ) := g −1 , 2 which gives C− |∇s v|2 + |∂t v|2 ≤ (∇u) ◦ ≤ C+ |∇s v|2 + |∂t v|2 with |∇s v|2 := g ρμ ∂ρ v∂μ v. Hence, with the notation
− ( p, α) :=
inf
u∈W 1, p () u≡0
p 2 p 2 C− ∇s u(s, t) + u t (s, t)2 ϕ(s, t)dσ (s)dt − α u(s, 0) dσ (s) S u(s, t) p ϕ(s, t)dσ (s)dt
and
+ ( p, α) :=
inf
1, p () u∈W u≡0
p 2 p 2 C+ ∇s u(s, t) + u t (s, t)2 ϕ(s, t)dσ (s)dt − α u(s, 0) dσ (s) S , u(s, t) p ϕ(s, t)dσ (s)dt
we conclude that − ( p, α) ≤ (, p, α) ≤ + ( p, α) holds true provided + ( p, α)
(15)
0. Now we obtain separately an upper bound for + ( p, α),
≤ see Sect. 4, and a lower bound for − ( p, α), see Sect. 5.
4 Proof of theorem 1.1: upper bound For an upper bound for + ( p, α) we will test on functions of a special form. To have shorter expressions we introduce the parameter 1
β := α p−1 .
(16)
Pick a C ∞ function χ : (0, δ) → [0, 1] which equals one in a neighborhood of 0 and zero in a neighborhood of δ, and define ψ(t) := e−βt χ(t). As β tends to +∞, δ 1 M(s) 1 − 2 2 +O 3 , ψ(t) p ϕ(s, t)dt = pβ p β β 0 (17) δ p ψ (t) ϕ(s, t)dt = β p 1 − M(s) + O 1 , pβ p2 β 2 β3 0
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where the remainder estimates are uniform in s ∈ S due to the essential boundedness of the coefficients of ϕ. Now we are going to consider two cases separately.
Case I: p ∈ (1, 2] Using the inequality (a + b)q ≤ a q + bq valid for a, b ≥ 0 and q ∈ (0, 1] we estimate, with some C > 0, p p C ∇s u(s, t) + u t (s, t) ϕ(s, t)dσ (s)dt inf + ( p, α) ≤ 1, p () u∈W u ≡0
u(s, 0) p dσ (s)
− β p−1 ×
S
u(s, t) p ϕ(s, t)dσ (s)dt
−1
.
Considering the functions u of the form u(s, t) = v(s)ψ(t) with v ∈ W 1, p (S) and using the estimates (17) we arrive, as β → +∞, at 1 p 1 M(s) + ( p, α) ≤ C ∇s v(s) − 2 2 + O 3 dσ (s) inf pβ p β β v∈W 1, p (S) S v ≡0
v(s) p 1 − M(s) + O 1 dσ (s) − β p−1 v(s) p dσ (s) pβ p2 β 2 β3 S S −1
p 1 1 M(s) v(s) , (18) × − 2 2 + O 3 dσ (s) pβ p β β S
+βp
where the O-terms are uniform in s ∈ S and do not depend on v. To construct a suitable function v, we adapt the procedure appearing e.g. in [3, Lemma 3.2] for Schrödinger operators with strong potentials. Take an arbitrary ε > 0. By assumption, the set Sε := s ∈ S : Mmax − ε ≤ M(s) ≤ Mmax has a non-zero measure, and almost any point s of Sε has density one with respect to σ , i.e. σ Bρ (s) ∩ Sε /σ Bρ (s) → 1 for ρ → 0, where Bρ (s) is the geodesic ball in S centered at s of radius ρ, see e.g. [14, Section 1.7, Corollary 3]. Let us choose any s ∈ Sε with this property. In what follows, we denote by Br the ball of radius r center at 0 in Rν−1 and consider the exponential map E at s, Br y → E(y) := exps (y) ∈ S, where r > 0 is sufficiently small to ensure that E is a diffeomorphism between Br and Br (s). For a function v defined on Br (s) we denote in what follows v∗ := v ◦ E, which is a function on Br , and σ∗ the measure on Br given by σ∗ (U ) := σ E(U ) , and we recall that Bρ (s) = E(Bρ ) for 1, p ρ ≤ r . Then for any v ∈ W0 Br (s) , v ≡ 0, due to (18) one has, with some c > 0, c + ( p, α) ≤
123
Br
∇v∗ (y) p dσ∗ (y) + β p v∗ (y) p (1 − p) − M(y) + O 1 dσ∗ (y) pβ β2 Br .
v∗ (y) p 1 − M(y) + O 1 dσ∗ (y) 2 pβ β Br (19)
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Assuming that r is sufficiently small we have 2−1 dy ≤dσ∗ (y) ≤ 2dy in Br . Furthermore, due to the choice of s we have meas E −1 (Bρ (s) ∩ Sε ) / meas(Bρ ) → 1 as ρ → 0, where ν−1 meas stands forthe Lebesgue measure in R . Let μ ∈ (0, 1/2), then one can pick ρ ∈ (0, r ) −1 such that meas E (Bρ (s) ∩ Sε ) ≥ (1 − μ) meas(Bρ ). Denote θ ≡ θ (ε, ρ) := meas E −1 (Bρ (s) \ Sε ) / meas(Bρ ), so that 0 ≤ θ ≤ μ. For a ball ω ⊂ Br , let ω denote a non-negative minimizer of λω := inf
1, p
ω
|∇ f | p dy : f ∈ W0 (ω),
ω
| f | p dy = 1 .
(20)
By [4], the function ω is uniquely defined, and it is radially decreasing. Furthermore, set ω = cω ω with cω > 0 chosen such that ω
ω (y) p dσ∗ (y) = 1.
Due to the above computations one has cω ∈ (2 1, p function v ∈ W0 Br (s) such that
− 1p
1
, 2 p ). We are going to test in (19) on the
v∗ = Bρ extended by zero to Br .
(21)
1 Set η := θ ν−1 , then meas E −1 (Bρ (s) \ Sε ) = meas(Bηρ ). As Bρ is radially decreasing, we have p p p Bρ (y)dy. Bρ (y) dσ∗ (y) ≤ 2 Bρ (y)dy ≤ 2 E −1 Bρ (s)\Sε
E −1 Bρ (s)\Sε
Bηρ
1
Furthermore, using the inequality η ≤ μ ν−1 and the equality Bρ (y) = ρ have p p Bρ (y)dy ≤ Bρ (y)dy Bηρ
1−ν p
B1 (y/ρ), we
B
1 μ ν−1 ρ
p
= ρ 1−ν B
1 μ ν−1 ρ
B1 (y/ρ)dy =
p
B
B1 (y)dy =: γ (μ).
1 μ ν−1
Putting all together, for an arbitrarily chosen μ ∈ (0, 1/2) we can take ρ sufficiently small and make the choice (21), which gives
E −1 Bρ (s)\Sε
p v∗ (y) dσ∗ (y) ≤ γ (μ) = o(1),
μ → 0.
Furthermore, using the fact that ρ was chosen small but fixed and that p ≤ 2, we have, with suitable C j > 0 and β large enough
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1 M(y) v∗ (y) p (1 − p) − + O 2 dσ∗ (y) pβ β Br Br p−1 M β max p−1 Mmax − M(y) v∗ (y) p dσ∗ (y) + C1 β p−2 + ≤ 2cλ Bρ + (1 − p)β p − β p p Br Mmax p−1 β p−1 p = 2cλ Bρ + (1 − p)β p − + β Mmax − M(y) v∗ (y) dσ∗ (y) p p E −1 Bρ (s)∩Sε β p−1 p p−2 + Mmax − M(y) v∗ (y) dσ∗ (y) + C1 β p E −1 Bρ (s)\Sε Mmax p−1 ε p ≤ (1 − p)β p − + β p−1 β v∗ (y) dσ∗ (y) p p E −1 Bρ (s)∩Sε Mmax − M∞ p−1 p + β v∗ (y) dσ∗ (y) + C2 p E −1 Bρ (s)\Sε
A := c
∇v∗ (y) p dσ∗ (y) + β p
Mmax p−1 ε Mmax − M∞ γ (μ) p−1 + β p−1 + + C2 β β p p p
1 Mmax − ε − Mmax − M∞ γ (μ) β p−1 + C2 . = (1 − p)β p − p ≤ (1 − p)β p −
In the same way we obtain 1 M(y) + O 2 dσ∗ (y) v∗ (y) p 1 − B := pβ β Br Mmax −1 1 −1 =1− β + β Mmax − M(y) v∗ (y) p dσ∗ (y) + O(β −2 ) p p Br
1 Mmax − ε − Mmax − M∞ γ (μ) β −1 + C2 β −2 . ≤1− p For large β one has A < 0, and by (19)
1 Mmax − ε − Mmax − M∞ γ (μ) β p−1 + C1 (1 − p)β p − A p + ( p, α) ≤ ≤
1 B Mmax − ε − Mmax − M∞ γ (μ) β −1 + C2 β −2 1− p
≤ (1 − p)β p − Mmax − ε − Mmax − M∞ γ (μ) β p−1 + C3
= (1 − p)β p − Mmax β p−1 + ε + Mmax − M∞ γ (μ) β p−1 + C3 . In particular + ( p, α) ≤ 0 for α large enough, and from (15) it follows that p
p
(, p, α) − (1 − p)α p−1 + Mmax α + ( p, α) − (1 − p)α p−1 + Mmax α ≤ lim sup lim sup α α α→+∞ α→+∞ + ( p, α) − (1 − p)β p + Mmax β p−1 ≤ ε + Mmax − M∞ γ (μ). β p−1 β→+∞
= lim sup
As ε > 0 is arbitrary and γ (μ) can be made arbitrary small by taking μ arbitrary small, we p have the sought estimate (, p, α) ≤ (1 − p)α p−1 − Mmax α + o(α) for large α.
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Case II: p ∈ (2, ∞) Let q > 1, then one can find a constant c > 0 such that for all ε0 ∈ (0, 1) and all a, b ≥ 0 there holds c (a + b)q ≤ (1 + ε0 )a q + q−1 bq , (22) ε0 see e.g. “Appendix 2”. Therefore, with a suitable C > 0 and any ε ∈ (0, 1) one can estimate 2− p p p + ( p, α) ≤ Cε0 2 ∇s u(s, t) + (1 + ε0 )u t (s, t) ϕ(s, t)dσ (s)dt inf 1, p () u∈W u ≡0
−1 u(s, 0) p dσ (s) × u(s, t) p ϕ(s, t)dσ (s)dt .
− β p−1
S
Considering the functions u of the form u(s, t) = v(s)φ(t) with v ∈ W 1, p (S) and using the estimates (17) we arrive, as β → +∞, at + ( p, α) ≤
2− p ∇s v(s) p 1 − M(s) + O 1 dσ (s) Cε0 2 2 2 1, p pβ p β β3 v∈W (S) S inf
v≡0
v(s) p 1 − M(s) + O 1 dσ (s) − β p−1 v(s) p dσ (s) pβ p2 β 2 β3 S S −1
v(s) p 1 − M(s) + O 1 dσ (s) × , (23) pβ p2 β 2 β3 S
+ (1 + ε0 )β p
where the O-terms are uniform in s ∈ S and do not depend on v and ε0 , and by taking 3 ε0 := β − 2 and choosing suitable C j > 0 we arrive at C1 β + ( p, α) ≤
3( p−2) 4
S
1 p M(s) + O 3 v(s) dσ (s) pβ S β2 . 1 p M(s) v(s) 1 − + O 2 dσ (s) pβ β S
∇s v(s) p dσ (s) + β p
(1 − p) −
Now using the same notation and the same test function as in the case I we arrive at
1 3 Mmax − ε − Mmax − M∞ γ (μ) β p−1 + C1 β p− 2 (1 − p)β p − p + ( p, α) ≤
1 Mmax − ε − Mmax − M∞ γ (μ) β −1 + C2 β −2 1− p
3 p ≤ (1 − p)β − Mmax β p−1 + ε + Mmax − M∞ γ (μ) β p−1 + C3 β p− 2 , where C j > 0 are suitable constants, and p
(, p, α) − (1 − p)α p−1 + Mmax α lim sup ≤ ε + Mmax − M∞ γ (μ), α α→+∞ while ε and γ (μ) can be chosen arbitrarily small, which gives the result.
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5 Proof of theorem 1.1: lower bound 2 The minoration C− ∇s u(s, t) + u t (s, t)2 ≥ u t (s, t)2 gives δ u t (s, t) p ϕ(s, t)dt − α u(s, 0) p dσ (s) S 0 − ( p, α) ≥ inf . u∈W 1, p () u(s, t) p ϕ(s, t)dσ (s)dt u ≡0
(24)
Denote
λ(α, p, s) :=
δ
0
inf
u∈W 1, p (0,δ)
p u (t) ϕ(s, t) dt − α u(0) p , δ u(t) p ϕ(s, t) dt
(25)
0
then for a.e. s ∈ S one has δ u t (s, t) p ϕ(s, t)dt − α u(s, 0) p ≥ λ(α, p, s) 0
u(s, t) p ϕ(s, t)dt,
0
and (24) implies − ( p, α) ≥
δ
inf
u∈W 1, p () u ≡0
p λ(α, p, s)u(s, t) ϕ(s, t)dσ (s)dt ≥ ess inf λ(α, p, s). s∈S u(s, t) p ϕ(s, t)dσ (s)dt
Hence the result follows from the following lemma: Lemma 5.1 There holds p
p−2
λ ≡ λ(α, p, s) = (1 − p)α p−1 − M(s)α + O(α p−1 log α) as α → +∞, where the remainder estimate is uniform in s outside a zero-measure set. Proof Introducing β as in (16) and testing on u(t) = e−βt we obtain by a direct computation the upper bound p
p−2
λ ≤ (1 − p)β p − M(s)β p−1 + O(β p−2 ) ≡ (1 − p)α p−1 − M(s)α + O(α p−1 ), where the remainder depends on κ j ∞ only, see (14), and, hence, is uniform in s outside a zero-measure set. In particular, for sufficiently large α we have, λ≤
1− p p β , 2
(26)
uniformly in s ∈ S. It follows by standard arguments that problem (25) admits a minimizer v, see e.g. Proposition 6.1 below. Without loss of generality we may assume that δ 1 p v(t) p ϕ(s, t) dt = vϕ p p = 1. (27) v ≥ 0, 0
The Euler-Lagrange equation for v reads p−2 v ϕ = −λv p−1 ϕ, |v |
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(28)
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49
where the prime means the derivative in t, with the boundary conditions p−2 v (0) v (0) = −αv(0) p−1 , v (δ) = 0.
(29)
In order to establish suitable decay properties of v in the spirit of Agmon [1], let us take f ∈ C 1 [0, δ] with f ≥ 0 and f (0) = 0. Using the weak formulation of (28) with the test function f p v then gives λ
δ
δ
f (t) p v(t) p ϕ(s, t) dt = −
0
|v | p−2 v ϕ (t) f (t) p v(t) dt
0
=
δ
p−2 v (t) v (t)( f p v) (t)ϕ(s, t) dt
δ
p v (t) f (t) p ϕ(s, t) dt
0
= 0
+p
δ
p−2 v (t) v (t) f (t) p−1 f (t)v(t)ϕ(s, t) dt.
(30)
0
An application of the Young inequality |AB| ≤ ε|A|q + ε
1 − q−1
q
|B| q−1 ,
A, B ∈ R, ε > 0, q > 1,
(31)
p−2 v (t) f (t) p−1 , B = f (t)v(t) and q = p/( p − 1) to the second term on with A = v (t) the right-hand side of (30) gives
δ
−λ
f (t) p v(t) p ϕ(s, t) dt ≤ ( pε − 1)
0
δ
0
+ pε
−( p−1)
p v (t) f (t) p ϕ(s, t) dt
δ
p f (t) v(t) p ϕ(s, t) dt.
0
Taking ε = 1/(2 p) and using (26) we arrive at
δ
p v (t) f (t) p ϕ(s, t) dt + ( p − 1)β p
0
≤ (2 p) p
δ
v(t) p f (t) p ϕ(s, t) dt
0 δ
p v(t) p f (t) ϕ(s, t) dt.
(32)
0
Choose f in the form f (t) = χ(t)eωt , where χ ∈ C 1 [0, δ]) with χ(0) = 0, 0 ≤ χ ≤ 1, χ(t) = 1 for t ≥ θ, χ ∞ ≤
2 , θ
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and ω > 0 and θ ∈ (0, δ) will be chosen later. Using (22) with q = p and ε0 = 1 we obtain δ δ p p v(t) p f (t) ϕ(s, t) dt = v(t) p e pωt χ (t) + ωχ(t) ϕ(s, t) dt 0
0
p v(t) p χ (t) e pωt ϕ(s, t)dt + 2ω p
δ
≤c 0
p v(t) χ (t) e pωt ϕ(s, t)dt + 2ω p p
0 − p pωθ
≤ c1 θ
δ
v(t) p χ(t) p e pωt ϕ(s, t)dt
0
θ
≤c
e
θ
v(t) ϕ(s, t)dt + 2ω
0
≤ c1 θ − p e pωθ + 2ω p
δ
v(t) p χ(t) p e pωt ϕ(s, t)dt
0
p
δ
p
(33)
v(t) χ(t) e p
p pωt
ϕ(s, t)dt
0
δ
v(t) p χ(t) p e pωt ϕ(s, t)dt,
0
where c1 := 2 p c, and on the last step we used the normalization (27) for v. The substitution into (32) gives δ δ p v (t) f (t) p ϕ(s, t) dt + ( p − 1)β p v(t) p f (t) p ϕ(s, t) dt ≤ C1 θ − p e pωθ 0
0
δ
+ C2 ω p
v(t) p f (t) p ϕ(s, t)dt
0
with C1 := (2 p) p c1 and C2 := 2(2 p) p . Let us set ω := κβ so that
δ
0
with
κ :=
p − 1 1
p
2C2
p v (t) f (t) p ϕ(s, t) dt + p − 1 β p 2
δ
,
v(t) p f (t) p ϕ(s, t) dt
0
≤ C1 θ − p e pκβθ . Finally, we put θ :=
1 . β
Then, with a suitable C2 > 0, δ v(t) p f (t) p ϕ(s, t) dt 0
≤ C2 ,
δ
p v (t) f (t) p ϕ(s, t) dt ≤ C2 β p .
0
Further, as ϕ0 := inf (s,t)∈S×(0,δ) ϕ(s, t) > 0, we have, with C3 := C2 /ϕ0 > 0, δ δ p p p v (t) f (t) p dt ≤ C3 β p , v(t) f (t) dt ≤ C3 , 0
0
and (33) gives
δ 0
123
p v(t) p f (t) dt ≤ C4 β p
(34)
On the p-Laplacian with Robin boundary conditions...
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49
for some C4 > 0. Using again (22) with q = p and ε = 1 we conclude with
δ
(v f ) (t) p dt ≤ 2
0
δ
p v(t) p f (t) dt + c
0
δ
p v (t) f (t) p dt ≤ C5 β p .
0
The integral bounds obtained allow us to estimate the values of v(δ) and v(0) as follows. First, v(δ) p f (δ) p = p
δ
p−1
(v f ) (t)(v f ) p−1 (t) dt ≤ pv f p
0
(v f ) p ≤ C6 β,
implying v(δ) p ≤ C6 βe− pκδβ .
(35)
Furthermore, v(0) p = ϕ(s, 0)v(0) p
= ϕ(s, δ)v(δ) − p p
δ
v(t)
p−1
ϕ(s, t)
p−1 p
0 1
p−1
≤ ϕ(s, δ)v(δ) p + pvϕ p p
1
v ϕ p p −
1 p
δ
v (t)ϕ(s, t) dt −
v(t) p ∂t ϕ(s, t)dt
0
δ
v(t) p ∂t ϕ(s, t)dt.
0
Using the normalization of v and the estimate (35) we arrive at 1
v(0) p ≤ pv ϕ p p −
δ
v(t) p ∂t ϕ(s, t)dt + O(β −2 ).
(36)
0
In order to estimate the integral on the right-hand side we remark that, for any b > 0 and as α is sufficiently large,
δ
bβ −1 log β
v(t) p ϕ(s, t) dt ≤ f (bβ −1 log β)− p ≤
1
β pκb
δ
δ
cβ −1 log β
v(t) p f (t) p ϕ(s, t) dt
v(t) p f (t) p ϕ(s, t) dt ≤
0
C3 , β pκb
where we have used (34). Hence for b = 2/( pκ) we obtain
δ
bβ −1 log β
v(t) p ϕ(s, t) dt = O(β −2 ) .
On the other hand the normalization (27) implies 0
bβ −1 log β
v(t) p ϕ(s, t) dt = 1 −
δ
bβ −1 log β
v(t) p ϕ(s, t) dt = 1 + O(β −2 ).
Now, as ∂t ϕ/ϕ and its derivative in t are uniformy bounded in S × (0, δ), we have
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H. Kovaˇ
δ
v(t) p ∂t ϕ(s, t)dt =
0
0
rík, K. Pankrashkin
∂t ϕ(s, t) v(t) p ϕ(s, t)dt ϕ(s, t)
δ ∂t ϕ(s, t) ∂t ϕ(s, t) v(t) p ϕ(s, t)dt + v(t) p ϕ(s, t)dt ϕ(s, t) 0 bβ −1 log β ϕ(s, t) bβ −1 log β
∂t ϕ(s, 0) = + O(β −1 log β) v(t) p ϕ(s, t)dt + O(β −2 ) ϕ(s, 0) 0 bβ −1 log β
= − M(s) + O(β −1 log β) v(t) p ϕ(s, t)dt + O(β −2 )
bβ −1 log β
=
0
= −M(s) + O(β −1 log β), 1
and the substitution into (36) gives v(0) p ≤ pv ϕ p p + M(s) + O(β −1 log β). Finally, using the definition of λ we infer that 1
1
p−2
1
λ = v ϕ p p − αv(0) p ≥ v ϕ p p − pαv ϕ p p − α M(s) + O(α p−1 log α) p
p
p−2
p
≥ inf (x p − pαx) − α M(s) + O(α p−1 log α) = (1 − p)α p−1 − α M(s) x∈R+
p−2
+O(α p−1 log α), where the remainder estimate depends again on κ j ∞ only and is uniform for s outside a zero-measure set.
6 Behaviour of minimizers: concentration effects So far we have been dealing only with the asymptotic behavior of the eigenvalue (, p, α). In this section we will discuss some properties of the minimizers, as soon as they exist. In contrast to the most of the paper, for a part of the results we only require that ∂ be Lipschitz. For the sake of completeness, we include the proof of the existence for bounded Lipschitz domains. Proposition 6.1 If ⊂ Rν is a bounded Lipschitz domain, then the variational problem (1) has a minimizer for every α ∈ R. Proof Fix α ∈ R and let {u j } j∈N be a minimizing sequence for (, p, α) normalized to one in L p ();
lim |∇u j | p dx − α |u j | p dσ = (, p, α), u j L p () = 1 ∀ j ∈ N.(37) j→∞
∂
By [22, Thm. 1.5.1.10] for any ε ∈ (0, 1) there exits a constant K ε > 0 such that the upper bound p p |u| p dσ ≤ ε ∇u L p () + K ε u L p () (38) ∂
holds true for all u ∈ W 1, p (). Applying this inequality with u = u j and ε sufficiently small, depending on α, we deduce from (37) that sup ∇u j L p () < ∞. j∈N
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It follows that the sequence {u j } j∈N is bounded in W 1, p () and therefore admits a weakly converging subsequence, which we still denote by {u j } j∈N . Let u α be its weak limit in W 1, p (). The compactness of the embeddings W 1, p () → L p () and W 1, p () → L p (∂) implies that there exists another subsequence {u jk }k∈N of {u j } j∈N such that u jk L p () → u α L p () and u jk L p (∂) → u α L p (∂) as k → ∞. Henceu α L p () = 1 and using the normalization of u j and the weak lower semi-continuity of |∇ · | p in (37) we arrive at (, p, α) = lim inf
k→∞
|∇u jk | p dx − α
∂
|u jk | p dσ ≥ |∇u α | p dx − α
∂
|u α | p dσ.
This shows that u α is a minimizer.
We mention the paper [28] discussing further properties of the minimizers such as the uniqueness and the strict positivity. These properties are not used in our estimates below. The following simple estimate for the eigenvalue is an adaption of a result from [21]. Proposition 6.2 For any bounded Lipschitz domain one has the inequality p
(, p, α) ≤ (1 − p)α p−1 for all α ≥ 0 and p ∈ (1, ∞). 1
Proof Set β := α p−1 . Without loss of generality one may assume that is contained in the half-space x1 > 0. Let us test on the function u(x) = eβx1 . Consider the vector field F(x) = (e pβx1 , 0, . . . , 0), then the divergence theorem gives
∂
u p dσ =
∂
e pβx1 dσ ≥
and, subsequently,
(, p, α) ≤
∂
F · n dσ =
∇ · F dx = pβ
|∇u| p dx − β p−1 u p dx
∂
= (1 − p)β = (1 − p)α
u p dσ
βp ≤
p
e pβx1 dx = pβ
u p dx,
u p dx − pβ p u p dx
u p dx
p p−1
.
Remark 6.3 Note that the inequality established in Proposition 6.2 is strict. This follows from the fact the equality in e pβx1 dσ ≥ F · n dσ ∂
∂
is achieved only if ∂ is a half-space which is excluded by assumption. In fact the result of Proposition 6.2 can be somewhat strengthened by using the estimate 1 n · e1 ≤ 1 − (1 − n · e1 )2 , 2
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where e1 = (1, 0, . . . , 0). This leads to the lower bound 1 e pβx1 dσ ≥ F · n dσ + e pβx1 (1 − n · e1 )2 dσ 2 ∂ ∂ ∂ and subsequently to
(, p, α) ≤ (1 − p)α
p p−1
−
α
1 p−1
∂
2
e pβx1 (1 − n · e1 )2 dσ . e pβx1 dx
Since the existence minimizers is not always guaranteed, see Remark 1.3, in the following statements we will include it as an assumption. Similar to the proof of Lemma 5.1, we obtain first an exponential decay with respect to the distance from the boundary using Agmon’s approach [1]. Theorem 6.4 Let ⊂ Rν be a Lipschitz domain. Assume that for α large enough the problem (1) admits a minimizer u ≡ u α , which we assume non-negative and normalized by u L p () = 1, and that ≡ (, p, α) → −∞,
α → +∞. 1
Then for any τ ∈ (0, 1) and any a > 0 there holds, with γ := (−) p , ∇u(x) p − u(x) p exp τ γ dist(x, ∂) dx = O(γ p )
(39)
dist(x,∂)> γa
as α → +∞. Furthermore, if is a bounded Lipschitz domain or an admissible domain, then for any a > 0 there holds
dist(x,∂)>aα
p 2p 1 1 ∇u(x) p + α p−1 u(x) p × exp ( p − 1) p α p−1 dist(x, ∂) dx = O(α p−1 ), 1 − p−1
(40) as α → +∞.
p p p Equation (40) says that the sum ∇u + α p−1 u decays, in the integral sense, exponentially in α in any compact subset of .
Proof For x ∈ , denote ρ(x) ∂), then |∇ρ| ≤ 1. Furthermore, for large := dist(x, L > 0 denote ρ L (x) := min ρ(x), L , then we have again |∇ρ L | ≤ 1. The presence of the parameter L is only relevant for unbounded , as for a bounded domain one can take L sufficiently large to have ρ L = ρ. Let u a minimizer of (1). The regularity theory of elliptic equations, see [11] or [35], implies that u is C 1, inside . Let f be a non-negative bounded uniformly Lipschitz function defined in and vanishing in a neighborhood of ∂, then the equality (3) with φ := f p u gives u p f p dx = |∇u| p−2 ∇u · ∇( f p u) dx |∇u| p f p dx + p |∇u| p−2 f p−1 u∇u · ∇ f dx. =
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Applying the Young inequality (31) with A = |∇u| p−2 ∇u f p−1 , B = u∇ f and q = p/( p − 1) to the second term on the right-hand side we obtain, for any ε ∈ (0, 1), u p f p dx ≤ ( pε − 1) |∇u| p f p dx + pε 1− p u p |∇ f | p dx. −
Furthermore, let γ and ε0 ∈ [0, 1) be such that γ → +∞ for α → +∞ and − ≥ (1 − ε0 ) p γ p for large α.
(41)
1 p
In particular, one can simply take γ := (−) and ε0 ∈ (0, 1). We thus have (1 − ε0 ) p γ p u p f p dx + (1 − pε) |∇u| p f p dx ≤ pε 1− p u p |∇ f | p dx.
Furthermore, we may assume that δ0 := 1 − pε > 0, then pp p p p p p p u f dx + δ0 |∇u| f dx ≤ u p |∇ f | p dx. (1 − ε0 ) γ (1 − δ0 ) p−1
(42)
To estimate the term on the right-hand side, let us take a function f of a special form. Namely, we let a > 0 and χ ∈ C ∞ (R) be such that χ : R → [0, 1], χ(t) = 0 for t close to 0, χ(t) = 1 for t ≥ a, c0 := χ ∞ , and set
f (x) := χ γρ(x) ekγρ L (x) ,
where the constant k > 0 is to be chosen later. Hence
∇ f (x) = γ χ γρ(x) ekγρ L (x) ∇ρ(x) + kχ γρ(x) ekγρ L (x) ∇ρ L (x) ,
and, in particular, ∇ f (x) ≤ γ χ ρd(x) ekγρ L (x) + k f (x) . Using Proposition 6.9 we have p p ∇ f (x) p ≤ (1 + ε1 )k p γ p f (x) p + cγ χ γρ(x) e pkγρ L (x) , ε1 ∈ (0, 1), p−1 ε1 implying
u p |∇ f | p dx ≤ (1 + ε1 )k p γ p ≤ (1 + ε1 )k p γ p ≤ (1 + ε1 )k p γ p
u p f p dx + u p f p dx + u p f p dx +
cγ p p−1 ε1 cγ p
p χ γρ(x) e pkγρ L (x) u(x) p dx
p−1 x∈:ρ(x)≤ γa ε1 p p cc0 γ pka e , p−1 ε1
p χ γρ(x) e pkγρ L (x) u(x) p dx
where we used the normalization of u on the last step. The substitution into (42) gives 1 + ε1 p p p p p γ p k u f dx + δ |∇u| p f p dx (1 − ε0 ) p − 0 (1 − δ0 ) p−1 Cγ p e pka (43) ≤ p−1 (1 − δ0 ) p−1 ε1
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H. Kovaˇ
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p
with C := cc0 p p . Note that all the estimates are uniform in the parameter L entering the definition of ρ L , hence, one can send L to +∞, which means that (43) also holds for f (x) := χ γρ(x) ekγρ(x) . Recall that ε0 ∈ [0, 1) must satisfy (41) while δ0 ∈ (0, 1), ε1 ∈ (0, 1) and k > 0 are arbitrary. In particular, if k ∈ (0, p −1 ) is fixed, then one can choose ε0 , ε1 and δ0 positive but sufficiently small to have (1 − ε0 ) p − implying
bγ p
u p f p dx + δ0
1 + ε1 p p k p =: b > 0 (1 − δ0 ) p−1
|∇u| p f p dx ≤ C γ p , C :=
C p−1
(1 − δ0 ) p−1 ε1
e pka ,
and (39) follows from
∇u(x) p + γ p u(x) p exp τ γ dist(x, ∂) dx dist(x,∂)> γa
≤
∇u(x) p + γ p u(x) p f (x) p dx,
τ := kp ∈ (0, 1).
dist(x,∂)> γa
If is a bounded Lipschitz domain, then by Proposition 6.2 the above constructions work 1
1
with γ = ( p − 1) p α p−1 and ε0 = 0, then we can set δ0 = ε1 = γ −1 and k = p −1 − Bγ −1 with B > 0 sufficiently large, which gives (1 − ε0 ) p −
1 + ε1 1 p p k p ≥ , α → +∞, (1 − δ0 ) p−1 γ
(44)
and one can proceed in the same way to obtain (40). 1
1
If is an admissible domain, then by Theorem 1.1 we can take γ = ( p − 1) p α p−1 and ε0 := Aγ −1 , with a suitable large A > 0, then by setting δ = ε1 = γ −1 and taking k = p −1 − Bγ −1 with a suitable large B > 0 we obtain the estimate (44) implying (40) again. We mention a simple but important consequence which will be used below. Recall that δ and δ are defined in (11). Corollary 6.5 Let be an admissible domain such that the problem (1) admits a minimizer u, which we assume to be non-negative and normalized by u L p () = 1, then for any δ > 0 and any N > 0 there holds uW 1, p (δ ) = o(α −N ) as α → +∞. −
1
Proof Let a > 0 be as in Theorem 6.4. Then for α large enough dist(x, ∂) > aα p−1 for any x ∈ δ and equation (40) gives
1 1 p uW 1, p ( ) exp ( p − 1) p α p−1 δ δ
p 1 1 2 ∇u(x) p + α p−1 u(x) p × exp ( p − 1) p α p−1 dist(x, ∂) dx ≤ C α p−1 . ≤ δ
This implies the claim.
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Finally we are in position to prove a weak form of a localization of the minimizer near the set at which the mean curvature of the boundary takes its maximal value. Theorem 6.6 Let be an admissible domain such that the problem (1) admits a minimizer u ≡ u α for large α, which is assumed be chosen non-negative and normalized by u L p () = 1. Define H : → R by H (x) = H s(x) , where s(x) ∈ ∂ is given by dist(x, ∂) = x − s(x), then (45) Hmax − H) u p dx = o(1), α → +∞.
Proof It is well known that s(x) is uniquely defined for almost all x ∈ . Moreover, in view of Corollary 6.5, it is sufficient to show that (46) Hmax − H) u p dx = o(1), α → +∞, δ
for some δ > 0. We assume that δ is sufficiently small such that the map (12) is bijective, then for x ∈ δ one has s = s(x) iff x = (s, t) for some t ∈ (0, δ). Furthermore, by the constructions of Sect. 3 one has p p p p (, p, α)u L p () = |∇u| dx − α u dσ ≥ |∇u| dx − α u p dσ
∂
δ
∂
p p 2 2 2 ≥ C− ∇s v(s, t) + vt (s, t) ϕ(s, t)dσ (s)dt − α v(s, 0) dσ (s),
(47)
S
where v := u ◦ and C− > 0. Using Lemma 5.1 we have p p 2 2 C− ∇s v(s, t) + vt (s, t)2 ϕ(s, t)dσ (s)dt − α v(s, 0) dσ (s) S p p vt (s, t) ϕ(s, t)dσ (s)dt − α v(s, 0) dσ (s) ≥ S p (1 − p)α p−1 − (ν − 1)H (s)α + o(α) v(s, t) p ϕ(s, t)dσ (s)dt ≥ p p p Hu p dx + o(α)u L p (δ ) . = (1 − p)α p−1 u L p (δ ) − α(ν − 1) δ
The substitution into (47) and the asymptotic expansion (5) for (, p, α) give p
p
p
p
(1 − p)α p−1 u L p () − α(ν − 1)Hmax u L p () + o(α)u L p () p p p ≥ (1 − p)α p−1 u L p (δ ) − α(ν − 1) Hu p dx + o(α)u L p (δ ) . δ
p
p
p
Using 1 = u L p () = u L p (δ ) + u L p (δ ) we arrive at δ
(Hmax − H) u p dx ≤ o(1) −
1 p − 1 p−1 p p u L p (δ ) − Hmax u L p (δ ) , α ν−1
and the result follows from Corollary 6.5.
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Acknowledgements We thank the referee for a detailed reading of our paper and for many valuable comments, in particular, on the computations of Remark 6.3. H. K. has been partially supported by Gruppo Nazionale per Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The support of MIUR-PRIN2010-11 grant for the project “Calcolo delle variazioni” (H. K.), is also gratefully acknowledged. K.P. has been partially supported by CNRS GDR 2279 DynQua. The authors thank Carlo Nitsch and Cristina Trombetti for useful comments on a preliminary version of the work.
Appendix 1: Solvable cases For the sake of completeness, let us mention some cases in which (, p, α) can be computed explicitly. p
Proposition 6.7 For any p ∈ (1, ∞) and α > 0 there holds (R+ , p, α) = (1 − p)α p−1 , 1 and the minimizer u ∗ for Eq. (1) is given by u ∗ (t) = exp − α p−1 t . Proof By computing the right-hand side of (1) for u = u ∗ we obtain the inequality p (R+ , p, α) ≤ (1 − p)α p−1 . For the reverse inequality, we remark that lim x→+∞ u(x) = 0 for any u ∈ W 1, p (R+ ). Indeed, since |u| p−1 |u| ∈ L 1 (R+ ), by the the Hölder inequality, it follows that the limit x p p−1 p u lim u(x) = lim p |u| dt + u(0) x→∞
x→∞
0
exists and therefore must be equal to zero. Hence +∞ +∞ p−1 p−1 u u |u| dt ≤ pu p−1 |u| u(0) p = − p |u| dt ≤ p p p ≤
0 p−1 pu p u p ,
and we have
+∞
0
inf
u∈W 1, p (R+ ) u ≡0
≥
inf
0
v∈W 1, p (R+ ) v p =1
u (t) p dt − α u(0) p p p = inf v p − α v(0) +∞ v∈W 1, p (R+ ) |u(t)| p dt v p =1 0
p p v p − αpv p ≥ inf (x p − pαx) = (1 − p)α p−1 ,
which gives the sought result.
x∈R+
As observed in [25], the one-dimensional result can be used to study the infinite planar sectors Uθ := (x1 , x2 ) ∈ R2 : arg(x1 + i x2 ) < θ , 0 < θ < π. Proceeding literally as in Lemma 2.6 and Lemma 2.8 of [25] one arrives at the following result: π Proposition 6.8 Let α > 0 and p ∈ (1, +∞), then for θ ≥ there holds (Uθ , p, α) = 2 p π (1 − p)α p−1 . On the other hand, for θ < the infimum is attained on 2
α 1 p−1 u(x1 , x2 ) = exp − x1 , sin θ
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and satisfies α p p p−1 < (1 − p)α p−1 . (Uθ , p, α) = (1 − p) sin θ
Appendix 2: An auxiliary inequality Proposition 6.9 Let p > 1, then for any ε ∈ (0, 1] and for all a, b ≥ 0 there holds 1 1− p b p (a + b) p ≤ (1 + ε)a p + 1 − 2 1− p . ε p−1
Proof By homogenity, it is sufficient to show that (1 + t) p ≤ (1 + ε)t p + cε 1− p for all t ≥ 0. Denote f ε (t) := (1 + t) p − (1 + ε)t p , then one simply needs an upper estimate for C(ε) := ε p−1 supt∈R+ f ε (t). We have f ε (0) = 1 and f ε (+∞) = −∞. The equation f ε (t) = 0 has a unique solution t = tε =
1 (1 + ε)
1 p−1
−1
,
f (tε ) =
1+ε 1
(1 + ε) p−1 − 1
p−1 ,
implying C(ε) = max
ε p−1 (1 + ε) , 1 . 1 p−1 (1 + ε) p−1 − 1
1
Denote σ := (1 + ε) p−1 − 1, then sup
ε∈(0,1)
(1 + σ ) p − (1 + σ ) p−1 1 ε p−1 (1 + ε) , σ1 := 2 p−1 − 1. 1 p−1 = sup σ σ ∈(0,σ1 ) (1 + ε) p−1 − 1
Using the convexity of ϕ(σ ) = (1 + σ ) p − (1 + σ ) we have 1
ϕ(σ ) ≤ ϕ(0) +
2 p−1 ϕ(σ1 ) − ϕ(0) σ ≡ 1 σ, σ ∈ (0, σ1 ). σ1 2 p−1 − 1
1 1− p > 1, this gives the sought inequality. Since 1 − 2 1− p
Appendix 3: Remainder estimates for more regular domains If a stronger regularity of is imposed, the remainder in Theorem 1.1 can be made more explicit. Proposition 6.10 In Theorem 1.1 assume additionally that the boundary of it C 3 smooth and that the mean curvature attains the maximum value Hmax , then the remainder estimate in (5) can be improved to O(α 1−κ ) with ⎧ 1 ⎪ , p ∈ (1, 2], ⎨ κ = p +11 (48) ⎪ ⎩ , p ∈ (2, ∞). 3( p − 1)
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If, in addition, is C 4 smooth, then one can take ⎧ 2 ⎪ , p ∈ (1, 2], ⎨ κ = p +12 ⎪ ⎩ , p ∈ (2, ∞). 2( p − 1)
(49)
Proof Remark first that the result of Sect. 5 imply p
p
(, p, α) ≥ (1 − p)α p−1 − (ν − 1)Hmax α + O(α p−1 log α), p
and α p−1 log α = o(α 1−κ ) for κ given by (48) or (49). Therefore, it is sufficient to show the upper bound, which will be done by taking another test function in the computations of Sect. 4. Let s0 ∈ S be such that M(s0 ) = Mmax . As ∂ is C 3 smooth, then M is C 1 , and for some m ≥ 0 we have −Mmax ≤ −M(s) ≤ −Mmax + md(s, s0 ) with d(·, ·) standing for the geodesic distance on S and for s sufficiently close to s0 . Let us choose a function f ∈ Cc∞ (R) which equals 1 in a neighborhood of the origin and consider the functions v ∈ W 1, p (S) given by d(s, s ) 0 , (50) v(s) = f μ where μ is a positive parameter which tends to 0 as β → +∞ and will be chosen later. One can estimate, for large β, p A := v(s) dσ (s) = aμν−1 + O(μν ), A−1 = O(μ1−ν ), S p − M(s) v(s) dσ (s) = −Mmax A + O(μν ), (51) S ∇v(s) p dσ (s) = O(μν− p−1 ). S
Consider first the case p ∈ (1, 2]. The substitution into (18) gives 1 1 + ( p, α) ≤ O(μν− p−1 β −1 ) + β p−1 A − 2 β p−2 Mmax A + O(μν β p−2 ) p p + O(μν−1 β p−3 ) − β p−1 A −1 1 Mmax A − 2 2 A + O(μν β −2 ) + O(μν−1 β −3 ) pβ p β 1 − p 1 β p−1 A − 2 Mmax β p−2 A + O(μν− p−1 β −1 + μν β p−2 + μν−1 β p−3 ) = p p
−1 A Mmax 1− × + O(μβ −1 + β −2 ) pβ pβ 1 = (1 − p)β p − Mmax β p−1 + O μ− p + μβ p−1 + β p−2 p
Mmax × 1+ + O(μβ −1 + β −2 ) pβ = (1 − p)β p − Mmax β p−1 + O μ− p + μβ p−1 + β p−2 . ×
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The remainder is optimized by μ = β
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, and we arrive p−1− p−1
p+1 ) + ( p, α) ≤ (1 − p)β p − Mmax β p−1 + O(β p 1− 1 = (1 − p)α p−1 − Mmax α + O α p+1 .
If, in addition, is C 4 smooth, then M is C 2 smooth, and for some m ≥ 0 we have −Mmax ≤ −M(s) ≤ −Mmax + md(s, s0 )2 as s is sufficiently close to s0 , which allows one to replace the second estimate in (51) by p (52) − M(s) v(s) dσ (s) = −Mmax A + O(μν+1 ), S
and a similar computation gives
+ ( p, α) ≤ (1 − p)β p − Mmax β p−1 + O μ− p + μ2 β p−1 + β p−2 .
Hence, taking μ = β
p−1 − p+2
we arrive at p−1− 2( p−1)
p+2 ) + ( p, α) ≤ (1 − p)β p − Mmax β p−1 + O(β p 1− 2 = (1 − p)α p−1 − Mmax α + O α p+2 .
For the case p ∈ (2, +∞), the substitution of (51) into (23) gives 1 − p 2− p Mmax p−2 + ( p, α) ≤ β p−1 A − β A + O ε0 2 μν− p−1 β −1 + μν β p−2 2 p p
−1 A Mmax ν−1 p−3 ν−1 p−1 1− × +μ β + ε0 μ β + O(μβ −1 + β −2 ) pβ pβ 2− p 1 = (1 − p)β p − Mmax β p−1 + O ε0 2 μ− p + μβ p−1 + β p−2 + ε0 β p p
Mmax + O(μβ −1 + β −2 ) × 1+ pβ 2− p = (1 − p)β p − Mmax β p−1 + O ε0 2 μ− p + μβ p−1 + β p−2 + ε0 β p . 2− p
In order to optimize we solve ε0 2 μ− p = μβ p−1 = ε0 β p , which gives 2− p 4 μ = β −1/3 , ε0 = β −4/3 , O ε0 2 μ− p + μβ p−1 + β p−2 + ε0 β p = O(β p− 3 ), and, therefore, 4
p
+ ( p, α) ≤ (1 − p)β p − Mmax β p−1 + O(β p− 3 ) = (1 − p)α p−1 1− 1 −Mmax α + O α 3( p−1) . For C 4 domains, a similar computation using (52) gives 2− p + ( p, α) ≤ (1 − p)β p − Mmax β p−1 + O ε0 2 μ− p + μ2 β p−1 + β p−2 + ε0 β p . The remainder is optimized by 2− p 3 μ = β −1/4 , ε0 = β −3/2 , O ε0 2 μ− p + μβ p−1 + β p−2 + ε0 β p = O(β p− 2 ),
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which gives the sought result. We remark that for p = 2 in (49) we obtain κ = 1/2, which is optimal, see [23,32]. Finally, an easy revision of the proof of Theorem 6.6 gives the following result: Corollary 6.11 Under the assumptions of Proposition 6.10, the right-hand-side of (45) can be improved to O(α −κ ).
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