Isovolumetric and isoperimetric problems for a class of capillarity functionals
aa r X i v : . [ m a t h . DG ] M a y Isovolumetric and isoperimetric problemsfor a class of capillarity functionals
Paolo Caldiroli ∗ Abstract
Capillarity functionals are parameter invariant functionals defined on classes of two-dimensionalparametric surfaces in R as the sum of the area integral and an anisotropic term of suitable form.In the class of parametric surfaces with the topological type of S and with fixed volume, extremalsof capillarity functionals are surfaces whose mean curvature is prescribed up to a constant. For acertain class of anisotropies vanishing at infinity, we prove existence and nonexistence of volume-constrained, S -type, minimal surfaces for the corresponding capillarity functionals. Moreover, insome cases, we show existence of extremals for the full isoperimetric inequality. Keywords:
Isovolumetric problems, isoperimetric problems, parametric surfaces, H -bubbles. In this work we deal with closed surfaces in R parametrized by mappings u : S → R . Introducingthe stereographic projection φ of S onto the compactified plane R ∪ {∞} and identifying maps u defined on S with corresponding maps u ◦ φ − on R ∪ {∞} , the area of a surface parametrized by u is given by A ( u ) := Z R | u x ∧ u y | whereas the algebraic volume enclosed by u can be computed in terms of the Bononcini-Wente integral V ( u ) := 13 Z R u · u x ∧ u y . The relationship between the area and the volume integrals is stated by the classical isoperimetricinequality, proved in [4]: S |V ( u ) | / ≤ A ( u ) ∀ u ∈ C ∞ ( S , R ) (1.1)where S = √ π . As one expects, the inequality (1.1) in fact holds true in the Sobolev space H ( S , R ) (see [25]) and the constant S = √ π is the best one and is achieved if and only if u parametrizes a round sphere with arbitrary center and radius. (This fact can be readily deduced fromthe results discussed in [5], in particular Lemma 0.1. For a self-contained and direct proof, see [11],Lemma 2.1.)The area integral A ( u ) constitutes the simplest and most relevant example of a Cartan functional.As displayed in [14], Sect. 4.13, these are integrals of the kind F ( u ) := Z R F ( u, u x ∧ u y ) ∗ Dipartimento di Matematica, Universit`a di Torino, via Carlo Alberto, 10 – 10123 Torino, Italy. Email: [email protected] F ∈ C ( R × R ) such that:( C ) F ( p, q ) is positively homogeneous of degree one with respect to q , i.e., F ( p, tq ) = tF ( p, q ) for t > p, q ) ∈ R × R ,( C ) there exist 0 < m ≤ m such that the definiteness condition m | q | ≤ F ( p, q ) ≤ m | q | holds forall ( p, q ) ∈ R × R ,( C ) F ( p, q ) is weakly elliptic, namely it is convex with respect to q , i.e. F ( p, tq + (1 − t ) q ) ≤ tF ( p, q ) + (1 − t ) F ( p, q ) for t ∈ [0 ,
1] and p, q , q ∈ R .By ( C ) and the upper bound in ( C ) any Cartan functional F turns out to be well defined in H ( S , R ) and is a parameter invariant integral, i.e., we have F ( u ◦ g ) = F ( u ) for any C diffeo-morphism g of S onto itself. This rightly reflects the geometrical character of the problem we dealwith.We point out that the framework described above admits a counterpart in the setting of GeometricMeasure Theory. In that context, surfaces are meant as boundaries of sets of finite perimeter andCartan functionals are replaced by boundary functionals defined by so-called “semielliptic” integrals(see [6], Sect. 2). Later we will come back to this aspect.Thanks to the lower positive bound in ( C ), and by (1.1), an isoperimetric-like inequality for anyCartan functional can be also written, i.e., S F |V ( u ) | / ≤ F ( u ) ∀ u ∈ H ( S , R ) (1.2)for some constant S F ∈ (0 , m S ]. The existence of extremals for (1.2) arises as a natural questionand constitutes a rather challenging target. Indeed, since in general a Cartan functional is not purelyquadratic, differently from (1.1), the inequality (1.2) is not invariant under dilation and translation(with respect to u ). These missing invariances might make difficult restoring some compactness forsequences of approximate extremals of (1.2).A way to prevent, hopefully, troubles due to dilation is to consider isovolumetric problems, i.e.,constrained minimimization problems with fixed volume, as follows. Fixing t ∈ R , study the existenceof minimizers for S F ( t ) := inf {F ( u ) | u ∈ H ( S , R ) , V ( u ) = t } . (1.3)We point out that also these minimization problems are far from being obvious because, even if Cartanfunctionals are weakly lower semicontinuous (see [14]), the constraint is not weakly closed and thevolume functional is not weakly lower semicontinuous (see [25]). In fact, as we will see in some cases,the existence or nonexistence of minimizers for (1.3) is a rather delicate issue and depends in a sensitiveway on the shape of the Lagrangian.In this paper we study problems (1.3) for a special class of Lagrangian functions. In particular weconsider F ( p, q ) = | q | + Q ( p ) · q with Q ∈ C ( R , R ) prescribed, such that k Q k ∞ <
1. Cartan functionals corresponding to such F ,which indeed satisfy ( C )–( C ), can be interpreted as modified area integrals with an anisotropy term: F ( u ) = A ( u ) + Z R Q ( u ) · u x ∧ u y and are often known as “capillarity functionals” (see [20]). They are particularly meaningful because inthis case possible minimizers for (1.3) parametrize S -type surfaces with volume t and mean curvature H ( p ) = K ( p ) − λ where K = div Q is prescribed, whereas λ is a constant corresponding to the2agrange multiplier due to the constraint. We will call such surfaces “ H -bubbles”. In the sequelthe strong relation between the isovolumetric problem for capillarity functionals and the H -bubbleproblem will become even more evident.Capillarity functionals depend on the vector field Q only by its divergence. Therefore we can statethe precise assumptions just on the scalar field K = div Q . In the present work we focus on a class ofmappings K : R → R vanishing at infinity with a suitable rate. In particular let us start by assumingthat K ∈ C ( R ) satisfies:( K ) | K ( p ) p | ≤ k < p ∈ R .( K ) K ( p ) p → | p | → ∞ .Then it is possible to construct a vector field Q K ∈ C ( R , R ) such that div Q K = K on R andenjoying the following properties:( Q ) k Q K k ∞ < Q ) | Q K ( p ) | → | p | → ∞ .These are direct consequences of ( K ) and ( K ), respectively (see Remark 2.6). Therefore the assump-tions ( K ) and ( K ) seem to be reasonably natural to deal with situations with anisotropy vanishingat infinity.In order to state a satisfactory result about the minimization problems S K ( t ) := inf (cid:8) F K ( u ) | u ∈ H ( S , R ) , V ( u ) = t (cid:9) where F K ( u ) := A ( u ) + Z R Q K ( u ) · u x ∧ u y , (1.4)in addition to the conditions ( K ) and ( K ), we introduce an extra assumption which controls theradial oscillation of K :( K ) | ( ∇ K ( p ) · p ) p | ≤ k < ∀ p ∈ R .We point out that ( K ) together with ( K ) implies ( K ) (see [13], Remark 2.2, for a proof). The firstexistence result shown in this paper can be stated as follows. Theorem 1.1
Let K ∈ C ( R ) satisfy ( K ) and ( K ) . Let t + := sup n t ≥ | K ≤ and K in some ball of radius p t/ π o t − := inf n t ≤ | K ≥ and K in some ball of radius p | t | / π o . (1.5) Then for every t ∈ ( t − , t + ) there exists U ∈ H ( S , R ) with V ( U ) = t and F K ( U ) = S K ( t ) . Moreoverwhen t = 0 such U is a ( K − λ ) -bubble, of class C ,α , for some λ = λ ( t, U ) = 0 . Notice that, in the definition of t + and t − , one could have balls with arbitrary (and in general different)centers. Moreover, excluding the trivial case K = 0, the interval ( t − , t + ) is always nonempty.In fact, the sign of K plays a crucial role in the above stated result. For example, if K <
K >
0) on the tail of some open cone, then t + = ∞ (resp., t − = −∞ ).It is not clear if the result stated in Theorem 1.1 is optimal. But in some cases we can provide somemore information. In particular, when K < R , then, according to Theorem 1.1, a minimizer forproblem (1.4) exists for every t >
0. Actually, we can show non existence of minimizers as t <
0, butjust for small | t | (see Theorem 5.1). 3he arguments of the proof make full use of refined tools already developed in the context of the H -bubble problem. In particular the study of minimizing sequences for the isovolumetric problems (1.4)exploits some deep results proved in [12] and [8], concerning the behavior of approximate solutions of( K − λ )-systems ∆ u = ( K ( u ) − λ ) u x ∧ u y on R (1.6)which, to our knowledge, are known just when the mapping K satisfies precisely ( K ) and ( K ).In fact, all the assumptions asked of K in Theorem 1.1 are the same considered in the papers [9]and [13] on the H -bubble problem for a prescribed mean curvature function H ( p ) = H ( p ) + H ∞ where H ∞ is a nonzero constant corresponding to − λ , whereas H is a C function on R , vanishingat infinity, playing a similar role of K . The only difference is a factor 2, because in [9] and [13] onewrites the prescribed mean curvature equation for parametric surfaces in the form ∆ u = 2 H ( u ) u x ∧ u y .Conditions ( K ) and ( K ) are changed accordingly.Clearly, for the H -bubble problem the volume of the solution is not prescribed. Moreover, inthe works [9] and [13], solutions are found as saddle-type critical points of the (unbounded) energyfunctional naturally associated to (1.6). Furthermore, in general, nonconstant weak solutions u ∈ H ( S , R ) of (1.6) are not necessarily minimizers for the isovolumetric problem (1.4) with t = V ( u ).Considering the set of mean curvature functions H = K − λ for which Theorem 1.1 providesexistence of a ( K − λ )-bubble, we cannot guarantee that the range of admissible values for λ does notcontain gaps. On the other hand, the occurrence of gaps would not be surprising, when the metricinduced by the anisotropy term is far from flat (see [2] and [17] for examples in this spirit, but indifferent contexts). Anyway, some information on the set-valued function t Λ( t ) := { λ ∈ R | ∃ U ∈ H ( S , R ) minimizer of S K ( t )and ( K − λ )-bubble } ( t ∈ ( t − , t + ))is available (see Theorem 4.5).A few more words can be said about the assumption ( K ). This condition, which is essential in theworks [9] and [13] about the H -bubble problem, here plays a role just in order to avoid that minimizingsequences for the isovolumetric problems (1.4) split into many ( K − λ )-bubbles (see Lemma 3.9). Itis not clear if ( K ) is a purely technical assumption. As a matter of fact, we can provide a secondexistence result for the isovolumetric problems without ( K ), just assuming ( K ) and ( K ), but witha restriction on the constant k appearing in ( K ). More precisely, we have: Theorem 1.2
Let K ∈ C ( R ) satisfy ( K ) , ( K ) , and ( K ) 2 / (2 + k ) < (2 − k ) .Moreover, let t + and t − be defined as in (1.5). Then the same conclusion of Theorem 1.1 holds true. Condition ( K ), even if somehow unnatural, is worth considering because it does not involve deriva-tives of K . Furthermore, thanks to Theorem 1.2 and to the information about Lagrange multipliers λ = λ ( t ), we obtain a new result about existence of H -bubbles with prescribed mean curvature H assuming a large constant value at infinity (see Theorem 3.15).In the second part of this work we turn attention to isoperimetric inequalities like (1.2) for cap-illarity functionals F K with K of the form considered before and, pushing on the investigation, weprove the following existence result. 4 heorem 1.3 Let K ∈ C ( R ) satisfy ( K ) – ( K ) or, as an alternative, ( K ) , ( K ) , and ( K ) . If K ≤ on R then, letting F K as in (1.4), the minimization problem S K := inf u ∈ H ( S , R ) V ( u ) > F K ( u ) V ( u ) / (1.7) admits a solution. Moreover if U ∈ H ( S , R ) is a minimizer for (1.7), then U is a ( K − λ ) -bubble,of class C ,α , with λ = S K V ( U ) − / . As mentioned at the beginning, isovolumetric-type problems, like those considered in this paper,might be tackled also using methods of Geometric Measure Theory. For example, this is carried outin [6] and [17] in case of periodic media.However, we would like to stress that we are interested in volume-constrained minimal surfaceswith the topological type of the sphere. A geometric measure-theoretic approach seems to lack inproviding this kind of information whereas, under global assumptions on the anisotropy, the approachby means of parametrizations, as followed in this work, turns out to be well suited to this purpose.Moreover, we expect that the general structure displayed here could be hopefully adapted indealing with different, maybe more general, classes of anisotropies and, in a wider perspective, couldbe possibly constitute an alternative method to tackle the H -bubble problem. Let us introduce the spaceˆ H := { u ∈ H loc ( R , R ) | Z R ( |∇ u | + µ | u | ) < ∞} where µ ( z ) = 21 + | z | for z ∈ R . (2.1)The space ˆ H is a Hilbert space with inner product h u, v i = Z R ( ∇ u · ∇ v ) + (cid:18) π Z R uµ (cid:19) · (cid:18) π Z R vµ (cid:19) and is isomorphic to the space H ( S , R ). The isomorphism is given by the correspondence ˆ H ∋ u u ◦ φ ∈ H ( S , R ), where φ is the stereographic projection of S onto the compactified plane R ∪ {∞} . As usual, we denote k u k = h u, u i / .One has that C ∞ ( S , R ) is dense in H ( S , R ) (see, e.g., [1], Ch.2). As a consequence, ˆ C ∞ := { u ◦ φ − | u ∈ C ∞ ( S , R ) } is dense in ˆ H . We point out that p + ˆ H = ˆ H for every p ∈ R . Byobvious extension, for every bounded domain Ω in R the space H (Ω , R ) can be considered as asubspace of ˆ H . Then also p + H (Ω , R ) is an affine subspace of ˆ H for every p ∈ R . Lemma 2.1
The space R + C ∞ c ( R , R ) is dense in ˆ H . In particular, for every u ∈ ˆ H ∩ L ∞ thereexists a sequence ( u n ) ⊂ R + C ∞ c ( R , R ) such that u n → u in ˆ H , in L ∞ loc and k u n k ∞ ≤ k u k ∞ . Even if this result is known, for future convenience, we sketch a proof, which contains a constructionused also in the sequel. 5 roof.
Take u ∈ ˆ C ∞ , let p = lim | z |→∞ u ( z ), and for every n ∈ N set u n ( z ) = u ( z ) as | z | ≤ n (cid:16) − log | z | log n (cid:17) u ( z ) + (cid:16) log | z | log n − (cid:17) p as n < | z | ≤ n p as | z | > n . (2.2)Setting A n = { z ∈ R | n < | z | ≤ n } , we have that Z R |∇ ( u − u n ) | = Z A n (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:20)(cid:18) − log | z | log n (cid:19) ( u ( z ) − p ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) + Z | z | >n |∇ u | ≤ Z | z | >n |∇ u | + sup | z | >n | u ( z ) − p | Z A n (cid:12)(cid:12)(cid:12)(cid:12) ∇ (log | z | )log n (cid:12)(cid:12)(cid:12)(cid:12) = o (1)as n → ∞ . Moreover (cid:12)(cid:12)(cid:12)(cid:12)Z R ( u − u n ) µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup | z | >n | u ( z ) − p | Z | z | >n µ = o (1)as n → ∞ . For every ε > n ∈ N such that k u − u n k < ε . Since u n − p ∈ H (Ω n , R ),where Ω n is the disc of radius n , we can find v ∈ C ∞ c (Ω n ) such that k u n − v k < ε . Hence theconclusion follows from the density of ˆ C ∞ in ˆ H . Since this last property can be proved by a standardregularizing technique using Friedrichs mollifiers which do not increase the L ∞ norm, also the secondpart of the lemma is true. (cid:3) Set D ( u ) := 12 Z R |∇ u | ( u ∈ ˆ H ) and V ( u ) := 13 Z R u · u x ∧ u y ( u ∈ ˆ H ∩ L ∞ ) . Lemma 2.2
The functional V admits a unique analytic extension on ˆ H . In particular for every u ∈ ˆ H V ′ ( u )[ ϕ ] = Z R ϕ · u x ∧ u y ∀ ϕ ∈ ˆ H ∩ L ∞ and there exists a unique v ∈ ˆ H ∩ L ∞ which is a (weak) solution of (cid:26) − ∆ v = u x ∧ u y on R R R vµ = 0 . (2.3) Moreover k∇ v k + k v k ∞ ≤ C k∇ u k (2.4) for a constant C independent of u . In addition, for every t = 0 the set M t := { u ∈ ˆ H | V ( u ) = t } (2.5) is a smooth manifold and, for any fixed u ∈ M t , a function ϕ ∈ ˆ H belongs to the tangent space to M t at u , denoted T u M t , if and only if V ′ ( u )[ ϕ ] = 0 . Remark 2.3
The second part of Lemma 2.2 states that there exists
C > such that kV ′ ( u ) k ˆ H − ≤ C k∇ u k for every u ∈ ˆ H , where ˆ H − denotes the dual of ˆ H . roof. All the results stated in the lemma are essentially well known; the proof displayed, e.g., in[25], Thms. 3.1 and 3.3 (see also [23], Ch. III, Thm. 2.3), considering as a domain the space H ( D , R ),where D denotes the unit disc in R , works also in ˆ H . The only additional remark regards the factthat, for fixed u ∈ ˆ H , the Riesz representative of V ′ ( u ) in ˆ H belongs to L ∞ . To prove this, weconsider a sequence of Dirichlet problems (cid:26) − ∆ v = u x ∧ u y in Ω n v = 0 on ∂ Ω n (2.6)where Ω n = { z ∈ R | | z | < n } . It is known that for every n ∈ N there exists v n ∈ H solving (2.6)and k∇ v n k + k v n k ∞ ≤ C k∇ u k with C independent of n (see [3]; see also [24] for the optimal constant C = (2 π ) − ). Then thesequence ( v n ) is bounded in ˆ H and in L ∞ , admits a subsequence which converges weakly in ˆ H tosome w ∈ ˆ H ∩ L ∞ solving Z R ∇ w · ∇ ϕ = Z R ϕ · u x ∧ u y ∀ ϕ ∈ R + C ∞ c ( R , R )(notice that R R p · u x ∧ u y = 0 for all p ∈ R ). Finally, the function v = w − π R R wµ solves (2.3)and belongs to L ∞ . (cid:3) Remark 2.4
The mapping ω ( z ) = ( µx, µy, − µ ) , with µ defined in (2.1), is a conformal parame-trization of the unit sphere. Indeed, it is the inverse of the stereographic projection from the NorthPole. Moreover A ( ω ) = D ( ω ) = 4 π and V ( ω ) = − π . If p ∈ R and r ∈ R \ { } , then u = p + rω isa parametrization of a sphere centered at p and with radius | r | , Moreover A ( u ) = D ( u ) = 4 πr and V ( u ) = − πr . Lemma 2.5 (Isoperimetric inequality)
It holds that S |V ( u ) | / ≤ A ( u ) ≤ D ( u ) ∀ u ∈ ˆ H (2.7) where S = √ π is the best constant. Moreover any extremal function for (2.7) is a conformalparametrization of a simple sphere. Inequality (2.7) for regular mappings goes back to [4]. Its extension to H ( D , R ) is proved in [25].The version for mappings in ˆ H , even not explicitly stated, can be also deduced from [25], Theorem2.5, by a density argument, by means of Lemmas 2.1 and 2.2.Fixing K ∈ C ( R ) satisfying ( K ), set m K ( p ) := Z K ( sp ) s ds and Q K ( p ) := m K ( p ) p ∀ p ∈ R and observe that div Q K = K . Then set Q ( u ) := Z R Q K ( u ) · u x ∧ u y ( u ∈ ˆ H ) . Remark 2.6
From | K ( p ) p | ≤ k for every p ∈ R , it follows that k Q K k ∞ ≤ k . In particular, by theassumption ( K ) , k Q K k ∞ < . (2.8)7 oreover the functional Q is well defined on ˆ H and |Q ( u ) | ≤ k Q K k ∞ D ( u ) ∀ u ∈ ˆ H . (2.9) One can also check that | Q K ( p ) | → as | p | → ∞ . (2.10) Indeed, for | p | > R write Q K ( p ) = ˆ p | p | Z R K ( t ˆ p ) t dt + ˆ p | p | Z | p | R K ( t ˆ p ) t dt with ˆ p = p | p | , and use ( K ) to conclude. The next result collects some useful properties of the functional Q . Lemma 2.7
Let K : R → R be a bounded continuous function. Then:(i) the functional Q is continuous in ˆ H .(ii) For every u ∈ ˆ H and ϕ ∈ ˆ H ∩ L ∞ one has Q ( u + ϕ ) − Q ( u ) = Z R Z K ( u + rϕ ) ϕ · ( u x + rϕ x ) ∧ ( u y + rϕ y ) dr dz. (iii) The functional Q admits directional derivatives at every u ∈ ˆ H along any ϕ ∈ ˆ H ∩ L ∞ , givenby Q ′ ( u )[ ϕ ] = Z R K ( u ) ϕ · u x ∧ u y . If in addition sup p ∈ R | K ( p ) p | < ∞ then for every u ∈ ˆ H the mapping s
7→ Q ( su ) is differentiableand dds [ Q ( su )] = s Z R K ( su ) u · u x ∧ u y . (2.11) Proof.
The first part of the lemma is proved in [22], Proposition 3.3, whereas (2.11) is discussed in[13] (in particular, see formula (2.7) therein). (cid:3)
Remark 2.8
Let ω be the mapping introduced in Remark 2.4. For every p ∈ R and r > , one hasthat Q ( p + rω ) = − R B r ( p ) K ( p ) dp whereas if r < then Q ( p + rω ) = R B | r | ( p ) K ( p ) dp . We conclude this section with an auxiliary approximation result for conformally invariant func-tionals at a fixed u ∈ ˆ H ∩ L ∞ by means of a sequence of functions u n with prescribed compactsupport. For G ∈ C ( R , R ) let us denote G ( u ) := Z R G ( u ) · u x ∧ u y ( u ∈ ˆ H ∩ L ∞ ) . Lemma 2.9
Let D be an open disc in R .(i) For every u ∈ ˆ H ∩ L ∞ with u ( z ) = p ∈ R for | z | large, there exists a sequence ( u n ) ⊂ H ( D, R ) ∩ L ∞ such that k u n k ∞ ≤ k u k ∞ , D ( u n ) → D ( u ) , (2.12) G ( u n ) = G ( u ) ∀ G ∈ C ( R , R ) . (2.13) If in addition u ∈ R + C ∞ c ( R , R ) , then the sequence ( u n ) can be taken in C ∞ c ( D, R ) . ii) For every u ∈ ˆ H ∩ L ∞ with u ( z ) → p ∈ R as | z | → ∞ and V ( u ) = 0 , there exists a sequence ( u n ) ⊂ H ( D, R ) ∩ L ∞ satisfying (2.12), V ( u n ) = V ( u ) , and G ( u n ) → G ( u ) for finitely many G ∈ C ( R , R ) .Proof. (i) Let u ∈ ˆ H ∩ L ∞ with u ( z ) = p ∈ R for | z | ≥ R . For every integer n > max { , R } let η n : R → [0 ,
1] be a smooth decreasing function such that η n ( r ) = r ≤ n − log r log n as n + 1 < r ≤ n −
10 as r > n . Then set ˜ u n ( z ) = u ( z ) as | z | ≤ nη n ( | z | ) p as n < | z | ≤ n | z | > n . (2.14)By direct computations, one can check that (2.12) and (2.13) hold true for (˜ u n ). Notice that ˜ u n ∈ H (Ω n , R ), where Ω n denotes the disc centered at the origin and with radius n . Let D be adisc centered at some z and with radius r >
0. Setting u n ( z ) = ˜ u n (cid:16) n r ( z − z ) (cid:17) , one has that u n ∈ H ( D, R ) with k u n k ∞ = k ˜ u n k ∞ , D ( u n ) = D (˜ u n ) and G ( u n ) = G (˜ u n ). Hence (2.12) and (2.13)hold true also for ( u n ). Moreover, if u is smooth, then according to the definition (2.14), also ˜ u n andconsequently u n are so.(ii) Let u ∈ ˆ H ∩ L ∞ with u ( z ) → p ∈ R as | z | → ∞ . Consider the sequence ( u n ) defined by (2.2).Then, following the proof of Lemma 2.1, one can recognize that u n → u in ˆ H and in L ∞ loc , and k u n k ∞ ≤ k u k ∞ . (2.15)Moreover u n ( z ) = p for | z | ≥ n . Hence we are in the position to apply part (i): for every n ∈ N thereis a sequence (˜ u n,k ) k>n ⊂ ˆ H such that˜ u n,k ∈ H (Ω n , R ) , k ˜ u n,k k ∞ ≤ k u n k ∞ , D (˜ u n,k ) → D ( u n ) as k → ∞ , G (˜ u n,k ) = G ( u n ) for every G ∈ C ( R , R ) and for k > n . (2.16)Let ( ε h ) ⊂ (0 , ∞ ) be a sequence such that ε h →
0. Then, there exists a sequence n h → ∞ such thatfor every h ∈ N one has |D ( u n h ) − D ( u ) | < ε h , |V ( u n h ) − V ( u ) | < ε h , |G ( u n h ) − G ( u ) | < ε h for finitely many vector fields G in a fixed set G . (2.17)In particular the last inequality is justified as follows: for (2.15), one has that u nx ∧ u ny → u x ∧ u y in L ( R , R ) and sup n k G ◦ u n k ∞ < ∞ . Hence the dominated convergence theorem applies and onecan infer that G ( u n ) → G ( u ). From (2.16)–(2.17), for every h ∈ N one can find k h > n h such that |D ( u n h ) − D (˜ u n h ,k h ) | < ε h . Moreover we have k ˜ u n h ,k h k ∞ ≤ k u n h k ∞ and G (˜ u n h ,k h ) = G ( u n h ) for every G ∈ C ( R , R ). Hence, setting ˜ u h = ˜ u n h ,k h , one has that˜ u h ∈ H (Ω n h , R ) , k ˜ u h k ∞ ≤ k u k ∞ , D (˜ u h ) → D ( u ) , V (˜ u h ) → V ( u ) , G (˜ u h ) → G ( u ) ∀ G ∈ G . Recalling that, as in part (i), D is the disc centered at z and with radius r , and setting v h ( z ) =˜ u h (cid:16) k h r ( z − z ) (cid:17) , one has that v h ∈ H ( D, R ) , k v h k ∞ ≤ k u k ∞ , D ( v h ) = D (˜ u h ) , V ( v h ) = V (˜ u h ) , G ( v h ) = G (˜ u h ) ∀ G ∈ G . v h in order to fix the volume. To this extent, let s h = p V ( u ) / V ( v h ) and w h = s h v h . Then s h → w h ∈ H ( D, R ), V ( w h ) = V ( u ), D ( w h ) = s h D ( v h ) → D ( u ), and G ( w h ) = G ( s h ˜ u h ) = s h Z R G ( s h ˜ u h ) · ˜ u hx ∧ ˜ u hy → Z R G ( u ) · u x ∧ u y . Indeed ˜ u hx ∧ ˜ u hy → u x ∧ u y in L ( R , R ) and sup h k G ◦ ( s h ˜ u h ) k ∞ < ∞ , because k s h ˜ u h k ∞ ≤ (1 + o (1)) k u k ∞ . Moreover G ◦ ( s h ˜ u h ) → G ◦ u pointwise a.e., because s h →
1, ˜ u h ( z ) = u n h ( z ) = u ( z ) for | z | < n h and n h → ∞ . Hence the sequence ( w h ) satisfies the required properties, and the proof ofpart (ii) is complete. (cid:3) In this section we aim to investigate a family of constrained minimization problems, defined as follows.For every t ∈ R set S K ( t ) := inf u ∈ M t E ( u ) where E ( u ) := D ( u ) + Q ( u ) (3.1)and M t is defined in (2.5). Our ultimate goal is to prove Theorems 1.1 and 1.2. Hence, unlessdifferently specified, we always assume that K ∈ C ( R ) satisfy ( K ) and ( K ). The additionalassumptions ( K ) or ( K ) will be recalled when they will be needed.Firstly we point out that the mapping t S K ( t ) is well posed from R into R , in view of (2.8) and(2.9), and can be named the isovolumetric function . We also set Q t ( u ) := Z R Q K ( tu ) · u x ∧ u y ( u ∈ ˆ H )and we notice that Remark 2.6 holds true also for the functional Q t . Moreover we introduce the normalized isovolumetric function t ˜ S K ( t ) defined by˜ S K ( t ) := inf u ∈ M E t ( u ) where E t ( u ) := D ( u ) + Q t ( u ) . (3.2) Remark 3.1
For t = 0 the class M t contains the constant functions. Since ≤ (1 − k Q K k ∞ ) D ( u ) ≤E ( u ) , one infers that S K (0) = 0 and minimizers for S K (0) are exactly the constant functions. Instead,for t = 0 the vector field p Q K ( tp ) is constant and then Q t ( u ) = 0 for every u ∈ ˆ H . Hence, by(2.7), ˜ S K (0) = inf {D ( u ) | u ∈ M } = S = √ π , the isoperimetric constant.Let us examine the case K = 0 and t ∈ R fixed. Then E = D and, by (2.7), S ( t ) = inf {D ( u ) | u ∈ M t } = St / . Instead ˜ S ( t ) = S for every t ∈ R . Let us state some preliminary properties of the isovolumetric function S K ( t ). Lemma 3.2
For every t ∈ R the following facts hold:(i) S K ( − t ) = S − K ( t ) ;(ii) S K ( t ) = t / ˜ S K ( t / ) ;(iii) S K ( t ) = S K ( · + p ) ( t ) for every p ∈ R .(iv) S K ( t ) = inf {E ( u ) | u ∈ C ∞ c ( R , R ) , V ( u ) = t } . roof. (i) For every u ∈ ˆ H let u ( x, y ) = u ( y, x ). Then D ( u ) = D ( u ), V ( u ) = −V ( u ) and Q ( u ) = −Q ( u ) = R R Q − K ( u ) · u x ∧ u y . These identities easily imply that S K ( − t ) = S − K ( t ).(ii) For every u ∈ ˆ H and t ∈ R one has that Q ( tu ) = t Q t ( u ) whereas D ( tu ) = t D ( u ) and V ( tu ) = t V ( u ) and thus S K ( t ) = t / ˜ S K ( t / ).(iii) Fix p ∈ R . For every u ∈ ˆ H one has that D ( u + p ) = D ( u ), V ( u + p ) = V ( u ), and Q ( u + p ) = R R Q K ( · + p ) ( u ) · u x ∧ u y . Then S K ( · + p ) ( t ) = inf {E ( u ) | u ∈ p + ˆ H , V ( u ) = t } = S K ( t )because p + ˆ H = ˆ H .(iv) Fix ε > u ∈ ˆ H with V ( u ) = t and E ( u ) ≤ S K ( t ) + ε . By Lemma 2.1 and by thecontinuity of the functionals D , V , and Q , there exists a sequence ( u n ) ⊂ R + C ∞ c ( R , R ) such that E ( u n ) ≤ S K ( t ) + ε + o (1) and V ( u n ) = t + o (1) , where o (1) → n → ∞ . We can write u n = p n + ˜ u n with p n ∈ R and ˜ u n ∈ C ∞ c ( R , R ).Let w ∈ C ∞ c ( R , R ) be a mapping with V ( w ) = 1. We can find a sequence z n ∈ R such that w n = w ( · + z n ) has support with empty intersection with the support of ˜ u n . Notice that D ( w n ) = D ( w )and V ( w n ) = 1 for all n ∈ N . Finally, we define v n = u n + s n w n with s n = p t − V ( u n ) . We have that v n ∈ p n + C ∞ c ( R , R ). Moreover, using (2.9) and since s n → n → ∞ , we estimate V ( v n ) = V ( u n ) + s n V ( w n ) = t D ( v n ) = D ( u n ) + s n D ( w ) = D ( u n ) + o (1) Q ( v n ) = Q ( u n ) + Q ( s n w n ) = Q ( u n ) + o (1)where o (1) → n → ∞ . Hence for fixed n large enough,inf {E ( u ) | u ∈ p n + C ∞ c ( R , R ) , V ( u ) = t } ≤ E ( v n ) ≤ S K ( t ) + 2 ε. (3.3)Now we claim that for every p ∈ R inf M t ∩ C ∞ c ( R , R ) E ≤ ˜ S K,p ( t ) := inf {E ( u ) | u ∈ p + C ∞ c ( R , R ) , V ( u ) = t } . (3.4)Indeed, fixing ε >
0, let u ∈ C ∞ ( R , R ) be such that u ( z ) = p for | z | ≥ R , V ( u ) = t and E ( u ) ≤ ˜ S K,p ( t ) + ε. By Lemma 2.9 (i), there exists a sequence ( u n ) ⊂ C ∞ c ( R , R ) such that D ( u n ) → D ( u ), V ( u n ) = V ( u )and Q ( u n ) = Q ( u ). Hence E ( u n ) → E ( u ) and for n large enoughinf M t ∩ C ∞ c ( R , R ) E ≤ E ( u n ) ≤ ˜ S K,p ( t ) + 2 ε. Therefore by the arbitrariness of ε >
0, (3.4) follows. By (3.3) and (3.4), we conclude thatinf M t ∩ C ∞ c ( R , R ) E ≤ S K ( t ) . Since the opposite inequality is trivial, (v) is proved. (cid:3) S K ( t ). Lemma 3.3
For every t ∈ R the following facts hold:(i) For every t ∈ R one has that (1 − k Q K k ∞ ) St / ≤ S K ( t ) ≤ S ( t ) = St / .(ii) For every t , ..., t k ∈ R one has that S K ( t ) + ... + S K ( t k ) ≥ S K ( t + ... + t k ) .Proof. (i) The first inequality follows from (2.7) and (2.9). Let us show the second one. Since K satisfies ( K )–( K ) if and only if − K does so, by Lemma 3.2 (i), without loss of generality we canassume t <
0. Let p n ∈ R be such that | p n | → ∞ and let u n = rω + p n where ω is defined in Remark2.4 and r > − πr / t . Then u n ∈ M t and E ( u n ) = t / S − R B r ( p n ) K ( p ) dp (seeRemarks 2.4 and 2.8) and the conclusion follows from the fact that, by ( K ), K ( p ) → | p | → ∞ .(ii) Let t , ..., t k ∈ R be given and fix an arbitrary ε >
0. By Lemma 3.2 (iv) there exist u , ..., u k ∈ C ∞ c ( R , R ) such that V ( u i ) = t i , D ( u i ) + Q ( u i ) ≤ S K ( t i ) + εk ∀ i = 1 , ..., k. Up to translation we can assume that the supports of the mappings u i are pairwise disjoint. Then V (cid:0)P i u i (cid:1) = P i t i and S K ( P i t i ) ≤ E (cid:0)P i u i (cid:1) = P i E ( u i ) ≤ P i S K ( t i ) + ε . By the arbitrariness of ε >
0, (ii) holds. (cid:3)
The next result contains some properties about minimizing sequences for the isovolumetric problemdefined by (3.1). In particular we state a bound from above and from below on the Dirichlet norm,and we show that every minimizing sequence shadows another minimizing sequence consisting ofapproximating solutions for some prescribed mean curvature equation.
Lemma 3.4
Let t ∈ R be fixed. Then:(i) D ( u ) ≥ S K ( t )1+ k Q K k ∞ for every u ∈ M t .(ii) If ( u n ) ⊂ M t is a minimizing sequence for S K ( t ) then lim sup D ( u n ) ≤ St / −k Q K k ∞ .(iii) For every minimizing sequence (˜ u n ) ⊂ M t for S K ( t ) there exists another minimizing sequence ( u n ) ⊂ M t such that k u n − ˜ u n k → and with the additional property that ∆ u n − K ( u n ) u nx ∧ u ny + λu nx ∧ u ny → in ˆ H − (= dual of ˆ H ) (3.5) for some λ ∈ R .Proof. (i) and (ii) can be easily obtained by (2.9) and by Lemma 3.3 (i).(iii) Assume t = 0. Then S K (0) = 0 (see Remark 3.1) and if (˜ u n ) ⊂ M is a minimizing sequence for S K (0) then D (˜ u n ) → u n = π R R ˜ u n µ , one easily checks that ( u n ) satisfiesthe thesis by the Poincar´e inequality which leads to k u n − ˜ u n k → n → ∞ . (Indeed each u n isa constant and is a minimizer for S K (0)). Now let us examine the case t = 0. Since, in general, thefunctional E is not differentiable everywhere in M t , the proof of (iii) needs some care. Since t = 0,the set M t constitutes a smooth closed manifold (see Lemma 2.2). Let (˜ u n ) ⊂ M t be such that E (˜ u n ) → S K ( t ) and fix a sequence ( ε n ) ⊂ (0 , ∞ ) with ε n →
0. By Ekeland’s variational principle (see,e.g., [15]), there exists a sequence ( u n ) ⊂ M t such that k u n − ˜ u n k ≤ ε n , E ( u n ) ≤ E (˜ u n ) , E ( u n ) ≤ E ( u ) + ε n k u − u n k ∀ u ∈ M t . ϕ ∈ T u n M t ∩ L ∞ and for s > τ n ( s ) = p t/ V ( u n + sϕ ). Then τ n ( s )( u n + sϕ ) ∈ M t and E ( τ n ( s )( u n + sϕ )) − E ( u n ) s ≥ − ε n (cid:13)(cid:13)(cid:13)(cid:13) τ n ( s )( u n + sϕ ) − u n s (cid:13)(cid:13)(cid:13)(cid:13) . (3.6)We compute the limit as s → + in the following separate auxiliary Lemma. Lemma 3.5
Let t ∈ R \ { } be fixed. For every u ∈ M t and ϕ ∈ ˆ H ∩ L ∞ with V ′ ( u )[ ϕ ] = 0 , it holdsthat: lim s → + (cid:13)(cid:13)(cid:13)(cid:13) τ ( s )( u + sϕ ) − us − ϕ (cid:13)(cid:13)(cid:13)(cid:13) = 0 (3.7)lim s → + Q ( τ ( s )( u + sϕ )) − Q ( u ) s = Z R K ( u ) ϕ · u x ∧ u y (3.8)lim s → + E ( τ ( s )( u + sϕ )) − E ( u ) s = Z R ( ∇ u · ∇ ϕ + K ( u ) ϕ · u x ∧ u y ) =: E ′ ( u )[ ϕ ] . (3.9) where τ ( s ) = p t/ V ( u + sϕ ) . Hence, passing to the limit as s → + in (3.6), by Lemma 3.5 we obtain that E ′ ( u n )[ ϕ ] ≥ − ε n k ϕ k .Taking now − ϕ instead of ϕ we get E ′ ( u n )[ ϕ ] ≤ ε n k ϕ k . Then, since ˆ H ∩ L ∞ is dense in ˆ H weconclude that sup ϕ ∈ T un M t ϕ =0 |E ′ ( u n )[ ϕ ] |k ϕ k ≤ ε n . (3.10)Now let v n ∈ ˆ H be the Riesz representative of V ′ ( u n ). Set λ n = E ′ ( u n )[ v n ] k v n k (notice that λ n is well defined because v n ∈ L ∞ , see Lemma 2.2). For every ϕ ∈ ˆ H ∩ L ∞ theprojection of ϕ on T u n M t is given by ˜ ϕ = ϕ − h v n , ϕ ik v n k v n and, by (3.10), |E ′ ( u n )[ ϕ ] − λ n V ′ ( u n )[ ϕ ] | = |E ′ ( u n )[ ˜ ϕ ] | ≤ ε n k ˜ ϕ k ≤ ε n k ϕ k , and then, by density, E ′ ( u n ) − λ n V ′ ( u n ) → H − . Now we show that the sequence ( λ n ) is bounded.Indeed, by (2.4) and by Lemma 3.4, part (ii), we know that k∇ v n k + k v n k ∞ ≤ C k∇ u n k ≤ C. (3.11)Then |E ′ ( u n )[ v n ] | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z R ( ∇ u n · ∇ v n + K ( u n ) v n · u nx ∧ u ny ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ u n k k∇ v n k + k K k ∞ k v n k ∞ k∇ u n k ≤ C. (3.12)Moreover, keeping into account that R R v n µ = 0 and using again Lemma 3.4, part (ii), we have that | t | = |V ′ ( u n )[ u n ] | = |h v n , u n i| = (cid:12)(cid:12)(cid:12)(cid:12)Z R ∇ v n · ∇ u n (cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ v n k k∇ u n k ≤ C k∇ v n k = C k v n k . (3.13)13hen (3.12) and (3.13) imply that ( λ n ) is bounded, because t = 0. Hence, for a subsequence λ n → λ ∈ R and since ( v n ) is bounded in ˆ H (use (3.11)), we conclude that E ( u n ) − λ V ′ ( u n ) → H − . (cid:3) Proof of Lemma 3.5.
First of all we observe that, by Lemma 2.2, the mapping s τ ( s ) is smooth,with τ ′ ( s ) = − √ t V ′ ( u + sϕ )[ ϕ ] V ( u + sϕ ) / . In particular, τ (0) = 1 , lim s → τ ( s ) − s = τ ′ (0) = 0 (3.14)because V ′ ( u )[ ϕ ] = 0. Hence (3.7) easily follows from (3.14). In order to prove (3.8) we write Q ( τ ( s )( u + sϕ )) − Q ( u ) s = Q ( τ ( s )( u + sϕ )) − Q ( τ ( s ) u ) s + Q ( τ ( s ) u ) − Q ( u ) s . Using (2.11), we have that lim τ → Q ( τ u ) − Q ( u ) τ − Z R K ( u ) u · u x ∧ u y ∈ R and consequently lim s → Q ( τ ( s ) u ) − Q ( u ) s = lim s → τ ( s ) − s lim τ → Q ( τ u ) − Q ( u ) τ − . (3.15)Then, writing u s = τ ( s ) u and ϕ s = τ ( s ) ϕ , by Lemma 2.7, part (ii), Q ( τ ( s )( u + sϕ )) − Q ( τ ( s ) u ) s = Z R Z K ( u s + rsϕ s ) ϕ s · ( u sx + rsϕ sx ) ∧ ( u sy + rsϕ sy ) dr dz. We point out that if s → u s → u and ϕ s → ϕ in ˆ H and pointwise a.e. Hence( u sx + rsϕ sx ) ∧ ( u sy + rsϕ sy ) → u x ∧ u y in L ,K ( u s + rsϕ s ) → K ( u ) pointwise a.e.and k K ( u s + rsϕ s ) ϕ s k ∞ ≤ C for s close to 0, because K and ϕ are bounded functions. Thenlim s → Q ( τ ( s )( u + sϕ )) − Q ( τ ( s ) u ) s = Z R K ( u ) ϕ · u x ∧ u y (3.16)and (3.8) follows from (3.15) and (3.16). Finally, since E = D + Q , (3.9) is an obvious consequence of(3.8) and of the fact that the functional D is analytic. (cid:3) As a next step, we provide a precise description of the specific minimizing sequences for theisovolumetric problems (3.1). To this purpose, it is convenient to introduce the definition of a bubble:
Definition 3.6
Let H ∈ C ( R ) be a given function. We call U ∈ ˆ H an H -bubble if it is a noncon-stant solution to ∆ U = H ( U ) U x ∧ U y on R (3.17) in the distributional sense. If H is constant, an H -bubble will be named H -sphere. The system (3.17)is called H -system. H -bubbles, for a class of mappings H of our interest. Lemma 3.7
Let H ( p ) = K ( p ) − λ with λ ∈ R and K ∈ C ( R ) satisfying ( K ) . If U ∈ ˆ H is an H -bubble, then U ∈ L ∞ , and λ V ( U ) > . If, in addition, K ∈ C ( R ) then U is of class C ,α as amap on S .Proof. Multiplying ∆ U = H ( U ) U x ∧ U y by U and integrating, by ( K ), one obtains that0 < (2 − k ) D ( U ) ≤ λ V ( U )because U is nonconstant. The fact that U ∈ L ∞ has been proved in [12], Theorem 2.1 and Remark2.5. When K , and then H , is of class C , the regularity theory for H -systems (3.17) applies (see, forinstance, [19] or [3]) and one infers that U ∈ C ,α ( R , R ). By the invariance of H -systems and ofthe Dirichlet integral with respect to the transformation ( x, y ) (ˆ x, ˆ y ) := ( xx + y , − yx + y ), also themapping ˆ U ( x, y ) := U (ˆ x, ˆ y ) is an H -bubble. From this one infers that U is of class C ,α as a map on S . (cid:3) According to the following crucial result, minimizing sequences for problems (3.1) admit a limitconfiguration made by bubbles. More precisely:
Lemma 3.8 (Decomposition Theorem)
Let K : R → R be a continuous function satisfying ( K ) and ( K ) . If ( u n ) ⊂ ˆ H is a sequence satisfying (3.5) for some λ ∈ R and such that c ≤ k∇ u n k ≤ c for some < c ≤ c < ∞ and for every n , then there exist a subsequence of ( u n ) , still denoted ( u n ) ,finitely many ( K − λ ) -bubbles U i ( i ∈ I ), finitely many ( − λ ) -spheres U j ( j ∈ J ) such that, as n → ∞ : D ( u n ) → P i ∈ I D ( U i ) + P j ∈ J D ( U j ) V ( u n ) → P i ∈ I V ( U i ) + P j ∈ J V ( U j ) Q ( u n ) → P i ∈ I Q ( U i ) (3.18) where I or J can be empty but not both. In particular, if J = ∅ then the subsequence ( u n ) is boundedin ˆ H .Proof. This result is obtained by combining Theorem 0.1 of [8] with the proof of Theorem A.1 of [13],which in fact holds true assuming just ( K ) and ( K ). See also [12] for previous partial, fundamentalresults. (cid:3) The next result will be used to show that minimizing sequences for the isovolumetric problems(3.1) do not split in two or more ( K − λ )-bubbles. We point out that assumption ( K ) plays a rolejust at this point of the argument. We also recall that, since ( K ), together with ( K ), implies ( K ),Lemma 3.7 applies and, in particular, if U ∈ ˆ H is a ( K − λ )-bubble, then λ = 0. Lemma 3.9
Assume ( K ) – ( K ) . If U , U ∈ ˆ H are two ( K − λ ) -bubbles, for a common λ ∈ R , with V ( U i ) = t i > ( i = 1 , ), then S K ( t + t ) < E ( U ) + E ( U ) .Proof. Let D and D be two disjoint discs, and for i = 1 , u i ∈ H ( D i , R ) with V ( u i ) = t i .Then set τ = t t and for s ∈ [0 , τ − ] v s = √ s u + p (1 − s ) τ + 1 u . Note that V ( v s ) = s V ( u ) + ((1 − s ) τ + 1) V ( u ) = t + t . Moreover the mapping f ( s ) := E ( v s )15s continuous in [0 , τ − ] and since f ( s ) = E ( √ s u ) + E ( p (1 − s ) τ + 1 u ), by means of Lemma2.7 (iii), one can compute the derivatives f ′ ( s ) = s − Z R ( |∇ u | + G ( √ s u ) · u x ∧ u y ) − τ [(1 − s ) τ + 1] − Z R ( |∇ u | + G ( p (1 − s ) τ + 1 u ) · u x ∧ u y ) , (3.19) f ′′ ( s ) = − s − Z R ( |∇ u | − G ( √ s u ) · u x ∧ u y ) − τ ((1 − s ) τ + 1) − Z R ( |∇ u | − G ( p (1 − s ) τ + 1 u ) · u x ∧ u y )where G ( u ) = K ( u ) u and G ( u ) = ( ∇ K ( u ) · u ) u . From ( K ) it follows that Z R |∇ u | + Z R G ( √ s u ) · u x ∧ u y ≥ (2 − k ) D ( u ) Z R |∇ u | + Z R G ( p (1 − s ) τ + 1 u ) · u x ∧ u y ≥ (2 − k ) D ( u ) . In particular f ′ ( s ) → ∞ as s → f ′ ( s ) → −∞ as s → τ − . From ( K ) it follows that Z R |∇ u | − Z R G ( √ s u ) · u x ∧ u y ≥ (2 − k ) D ( u ) Z R |∇ u | − Z R G ( p (1 − s ) τ + 1 u ) · u x ∧ u y ≥ (2 − k ) D ( u ) . In particular f is strictly concave in [0 , τ − ] and, by Lemma 3.3 (i) and Lemma 3.4, there existsa constant c > u and u , such that f ′′ ( s ) ≤ − c for every s ∈ (0 , τ − ). As aconsequence, with elementary arguments, one obtains thatmin { f (0) , f (1 + τ − ) } ≤ max s ∈ [0 , τ − ] f ( s ) − δ where δ = c (1 + τ − ) >
0. Hence, S K ( t + t ) ≤ max s ∈ [0 , τ − ] E ( √ s u + p (1 − s ) τ + 1 u ) − δ. (3.20)Now let us prove the strict inequality S K ( t + t ) < S K ( t ) + S K ( t ). For i = 1 , U i ∈ M t i be( K − λ )-bubbles. According to Lemma 3.7, U and U are bounded and, since we are assuming K of class C , again by Lemma 3.7, there exists lim | z |→∞ U i ( z ) ∈ R ( i = 1 , D and D , by Lemma 2.9 (ii), there exist sequences ( u i,n ) n ⊂ H ( D i , R ) ∩ L ∞ ( i = 1 ,
2) with V ( u i,n ) = t i , D ( u i,n ) → D ( U i ), E ( u i,n ) → E ( U i ), and Z R G ( u i,n ) · u i,nx ∧ u i,ny → Z R G ( U i ) · U ix ∧ U iy . For every n ∈ N set f n ( s ) = E ( √ s u ,n + p (1 − s ) τ + 1 u ,n ) ( s ∈ [0 , τ − ]) . By (3.20), for every n we have that S K ( t + t ) ≤ f n ( s n ) − δ s n is the (unique) value in [0 , τ − ] such that f n ( s n ) = max s ∈ [0 , τ − ] f n ( s ). We also compute(see (3.19)) f ′ n (1) = 13 Z R (cid:0) |∇ u ,n | + G ( u ,n ) · u ,nx ∧ u ,ny (cid:1) − τ Z R (cid:0) |∇ u ,n | + G ( u ,n ) · u ,nx ∧ u ,ny (cid:1) = 13 Z R (cid:0) |∇ U | + G ( U ) · U x ∧ U y (cid:1) − τ Z R (cid:0) |∇ U | + G ( U ) · U x ∧ U y (cid:1) + o (1)= λ V ( U ) − τ λ V ( U ) + o (1) = o (1) as n → ∞ where in the last line we exploit the fact that U and U are ( K − λ )-bubbles and then Z R ( |∇ U i | + K ( U i ) U i · U ix ∧ U iy ) = λ Z R U i · U ix ∧ U iy = 3 λ V ( U i ) ( i = 1 , . From this and using the fact that f ′′ n ( s ) ≤ − c for every s ∈ (0 , τ − ) with c > n , weinfer that s n →
1. Hence, setting ρ = min { τ − , } , for n large enough s n ∈ [1 − ρ, ρ ] ⊂ (0 , τ − )and | f n ( s n ) − f n (1) | ≤ | s n − | max | s − |≤ ρ | f ′ n ( s ) | = o (1) as n → ∞ .Indeed max | s − |≤ ρ | f ′ n ( s ) | ≤ c with c independent of n . To check this, just use (3.19) with u i,n instead of u i , and the fact that the sequences ( u i,n ) n are bounded in ˆ H and in L ∞ . Hence S K ( t + t ) ≤ f n (1) + o (1) − δ = E ( u ,n ) + E ( u ,n ) + o (1) − δ = E ( U ) + E ( U ) − δ as n → ∞ and this completes the proof. (cid:3) Finally we can show the existence of minimizers for problems (3.1).
Lemma 3.10
Assume ( K ) – ( K ) or, as an alternative, ( K ) – ( K ) and ( K ) . If t > is such that ( ∗ ) S K ( τ ) < S ( τ ) ∀ τ ∈ (0 , t ] , then there exists U ∈ M t such that E ( U ) = S K ( t ) . Moreover such U is a ( K − λ ) -bubble, for some λ > . Remark 3.11
Recall that S ( τ ) is the infimum value for the Dirichlet integral in the class M τ ofmappings in ˆ H parametrizing surfaces with volume τ . We know that S ( τ ) is attained by a conformalparametrization of a round sphere of volume τ with arbitrary center (Lemma 2.5). On the other hand, S K ( τ ) is is the infimum value for the functional E = D + Q in the same class M τ , and Q has themeaning of K -weighted algebraic volume (see Remark 2.8; see also [11], Sect. 2.3). Hence, roughlyspeaking, the inequality S K ( τ ) < S ( τ ) means that one can find a ball B of volume τ , with centerpossibly depending on τ and K , such that R B K ( p ) dp < (see Lemma 3.12, later on).Proof. By Lemma 3.4 there exists a minimizing sequence ( u n ) ⊂ M t satisfying (3.5) for some λ ∈ R .By Lemma 3.8 there exist finitely many ( K − λ )-bubbles U i ∈ ˆ H ( i ∈ I ) and finitely many ( − λ )-spheres U j ∈ ˆ H ( j ∈ J ) such that, for a subsequence, (3.18) holds. Recall that I or J (but not both)17an be empty. Thus, setting t i = V ( U i ) for i ∈ I ∪ J , t = P i ∈ I t i + P j ∈ J t j S K ( t ) = E ( u n ) + o (1) = P i ∈ I E ( U i ) + P j ∈ J D ( U j ) + o (1) as n → ∞ . (3.21)By Lemma 3.7 one has that t i λ > i ∈ I and t j λ > j ∈ J . Hence t i , t j ∈ (0 , t ] for all i ∈ I and j ∈ J . If J = ∅ then, by ( ∗ ), (3.21), and by Lemma 3.3, we obtain S K ( t ) ≥ P i ∈ I S K ( t i ) + P j ∈ J S ( t j ) > P i ∈ I S K ( t i ) + P j ∈ J S K ( t j ) ≥ S K (cid:16)P i ∈ I t i + P j ∈ J t j (cid:17) = S K ( t ) , a contradiction. Therefore J = ∅ and, by (3.21) and Lemma 3.3 (ii),0 ≤ P i ∈ I ( E ( U i ) − S K ( t i )) = S K ( t ) − P i ∈ I S K ( t i ) ≤ E ( U i ) = S K ( t i ) for all i ∈ I . Now we claim that I is a singleton. We prove this in twocases, as follows. Case 1: K satisfies ( K ).If I is not a singleton, by Lemma 3.9 and by (3.22), we reach a contradiction. Case 2: K satisfies ( K ).Since J = ∅ , by Lemma 3.8, the sequence ( u n ) is bounded in ˆ H . Then, testing (3.5) with u n wehave Z |∇ u n | + K ( u n ) u n · u nx ∧ u ny = 3 λt + o (1) as n → ∞ and consequently 3 λt ≤ (2 + k ) D ( u n ) + o (1) . Using Lemma 3.4 (ii) and Remark 2.6, we infer that S λ ≥ − k k ) √ t. (3.23)Now assume that there exist at least two ( K − λ )-bubbles U and U in the decomposition of ( u n ).From Z R |∇ U i | + Z R K ( U i ) U i · U ix ∧ U iy = 3 λt i it follows that 3 λt i − k ≥ D ( U i ) ≥ St / i , (3.24)having used ( K ) and the isoperimetric inequality (2.7). Thus (3.24) yields S λ ≤ √ t i − k . (3.25)Since 0 < t + t ≤ t , using (3.23) and (3.25) and applying the elemantary estimate √ r + √ − r ≤ / (take r = t / ( t + t )), we obtain 2 / (2 + k ) ≥ (2 − k ) , contrary to ( K ). Conclusion.
In both cases, there exists just one U ∈ ˆ H such that E ( u n ) → E ( U ) and V ( U ) = V ( u n ) + o (1) = t . This means that U is a minimizer for E in M t . Moreover U is a ( K − λ )-bubbleand, by Lemma 3.7, λ > t > (cid:3)
18s suggested by Remark 3.11, condition ( ∗ ) is connected to the sign of K . More precisely: Lemma 3.12
For p ∈ R and r > let B r ( p ) = { p ∈ R | | p − p | < r } . If K ≤ and K in B r ( p ) , then S K ( t ) < S ( t ) for every t ∈ (0 , πr / . In particular, if K ( p ) ≤ for every p ∈ R and K , then S K ( t ) < S ( t ) for every t > . Remark 3.13
The global negativeness of K is not a necessary condition to ensure the strict inequality S K ( t ) < S ( t ) for every t > . For example, it is enough that K is negative on the tail of some cone,that is, K ( p ) < for every p = rσ with r large enough and σ ∈ Σ where Σ is an open domain in S .Proof. Fix t ∈ (0 , πr /
3] and let δ > πδ / t . Let u = p − δω where ω is thestandard conformal parametrization of the unit sphere, defined in Remark 2.4. Then V ( u ) = t and E ( u ) = 4 πδ + R B δ ( p ) K ( p ) dp < St / = S ( t ) (see Remarks 2.4, 2.8 and 3.1). (cid:3) Proof of Theorems 1.1 and 1.2.
Assume that t + > r + = p t + / π . Fix t ∈ (0 , t + ). Thenthere exists p ∈ R , possibly depending on t , such that K ≤ K B r ( p ), where r = p t/ π .Then, by Lemma 3.12, S K ( τ ) < S ( τ ) for every τ ∈ (0 , t ]. Therefore we can apply Lemma 3.10 inorder to infer that there exists a minimizer U ∈ ˆ H for the minimization problem defined by (3.1).Moreover such U is a ( K − λ )-bubble, for some λ >
0. By Lemma 3.7, U is of class C ,α as a mapon S . Then, with a standard procedure (e.g., considering the weak formulation of (3.17) and takingvariations of the form U ◦ Φ t , where Φ t is a smooth flow on S ), one also infers that U satisfies theconformality conditions. Hence A ( U ) = D ( U ) and U turns out to be a minimizer also for F , namelyis a solution of the original isovolumetric problem (1.4). Thus the proof for t ∈ (0 , t + ) is complete.The case t = 0 is trivial and already discussed in Remark 3.1. Lastly, if t − <
0, one can conclude for t ∈ ( t − ,
0) by changing sign to K and using Lemma 3.2 (i) and what we just proved for t > (cid:3) Remark 3.14
The ( K − λ ) -bubble U found as minimizer of the isovolumetric problem (1.4) describesa parametric surface S = U ( R ∪ {∞} ) such that K ( p ) − λ equals the mean curvature of S at anyregular point p ∈ S (see, for instance, [9]). In addition, S has at most finitely many branch points (see[18]). We also notice that U is simple in the sense that it cannot be expressed in the form U ( z ) = u ( z n ) for some u ∈ ˆ H and n > integer (here we use complex notation). Indeed, otherwise we should have V ( u ) = t/n and S K ( t/n ) ≤ E ( u ) = n − E ( U ) . But in the first part of the proof of Lemma 3.10 wehave shown that no decomposition of the form S K ( t ) ≥ S K ( t ) + ... + S K ( t n ) with t = t + ... + t n and < t i < t ( i = 1 , ..., n ) can occur. Useful bounds on the Lagrange multiplier λ can be easily deduced. Indeed, multiplying the system∆ U = ( K ( U ) − λ ) U x ∧ U y by U , integrating, and exploiting ( K ), one infers that(2 − k ) D ( U ) ≤ λt ≤ (2 + k ) D ( U ) . Then, by Lemma 3.4, 2(2 − k )3(2 + k ) S K ( t ) ≤ λt ≤ k )3(2 − k ) S K ( t )and finally, by Lemma 3.3 (i), for t > − k ) S k ) √ t ≤ λ ≤ k ) S − k ) √ t . (3.26)Let us point out that, as a by-product of Theorem 1.2 and with the estimates (3.26) we obtain anew existence result for the H -bubble problem. This result has a perturbative character, in the samedirection of other works like [7], [10], [16], [21]. 19 heorem 3.15 Let K ∈ C ( R ) satisfy ( K ) , ( K ) , and ( K ) . Then there exists a sequence ( λ n ) ⊂ R with | λ n | → ∞ such that for every n there exists a ( K − λ n ) -bubble.Proof. If K ≡ K t − , t + ) is nonempty, with t − ≤ ≤ t + .Suppose t + >
0. Then for every t ∈ (0 , t + ) there exists a ( K − λ t )-bubble, for some λ t satisfying(3.26). In particular λ t → ∞ as t → + . Thus the result is proved. If t + = 0 then t − < (cid:3) Theorem 3.15 holds just for a sequence | λ n | → ∞ and not for every large | λ | because the set ofLagrange multipliers for constrained minimizers of the isovolumetric problems (3.1) in principle couldcontain gaps. In this section we are mainly interested in the study of the minimization problem defined by (1.7). Weassume that K ∈ C ( R ) satisfies ( K )–( K ) or, as an alternative, ( K ), ( K ), and ( K ). Moreoverwe suppose that K ≤ R . If K ≡ K
0. Then by Lemma 3.12, we have that S K ( t ) < S ( t ) for all t >
0. Hence, byTheorem 1.1, for every t > S K = inf t> ˜ S K ( t )(1 − k ) S ≤ S K ≤ S, (4.1)where S K and ˜ S K ( t ) are defined in (1.7) and (3.2), respectively. In the following we study the regularityand the asymptotic behavior of the normalized isovolumetric function t ˜ S K ( t ) for t ∈ (0 , ∞ ). Lemma 4.1
The function t ˜ S K ( t ) is locally Lipschitz-continuous in (0 , ∞ ) . Hence it is differen-tiable almost everywhere.Proof. Let t , t > u ∈ M . Then E t ( u ) − E t ( u ) = Z R ( Q K ( t u ) − Q K ( t u )) · u x ∧ u y = Z R (cid:18)Z t t ∂∂t [ Q K ( tu )] dt (cid:19) · u x ∧ u y . (4.2)Recalling that Q K ( p ) = m K ( p ) p , one has that ∂∂t [ Q K ( tp )] = m K ( tp ) p + ( ∇ m K ( tp ) · p ) tp and, since K ( p ) = div Q K ( p ) = ∇ m K ( p ) · p + 3 m K ( p ), one gets ∂∂t [ Q K ( tp )] = 1 t ( K ( tp ) tp − Q K ( tp )) . Then, taking into account that | K ( p ) p | ≤ k and | Q K ( p ) | ≤ k (see Remark 2.6), one infers that (cid:12)(cid:12)(cid:12)(cid:12)Z t t ∂∂t [ Q K ( tu )] dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ k (cid:12)(cid:12)(cid:12)(cid:12) log t t (cid:12)(cid:12)(cid:12)(cid:12) . (4.3)20ence, from (4.2) and (4.3) it follows that E t ( u ) − E t ( u ) ≤ k D ( u ) (cid:12)(cid:12)(cid:12)(cid:12) log t t (cid:12)(cid:12)(cid:12)(cid:12) . (4.4)Now take a sequence ( u n ) ⊂ M such that E t ( u n ) → ˜ S K ( t ). Since ˜ S K ( t ) = S K t (1) with K t ( p ) =div Q K ( t p ), using Lemma 3.4 (ii), we have that D ( u n ) ≤ S − k Q K k ∞ + o (1) (4.5)where o (1) → n → ∞ . Then, by (4.4)–(4.5), ˜ S K ( t ) ≤ ˜ S K ( t ) + C (cid:12)(cid:12)(cid:12) log t t (cid:12)(cid:12)(cid:12) for some constant C > t and t . Exchanging t with t we finally obtain | ˜ S K ( t ) − ˜ S K ( t ) | ≤ C (cid:12)(cid:12)(cid:12)(cid:12) log t t (cid:12)(cid:12)(cid:12)(cid:12) which implies local Lipschitz-continuity in (0 , ∞ ). (cid:3) Lemma 4.2
One has that ˜ S K ( t ) → S as t → .Proof. Let us recall the following inequality, due to Steffen [22] (see also [11] § |Q t ( u ) | ≤ k K t k ∞ S / D ( u ) / ∀ u ∈ ˆ H (4.6)where K t ( p ) = div Q K ( tp ) = tK ( tp ). If ( u n ) is a minimizing sequence for ˜ S K ( t ) then D ( u n ) ≤ ˜ S K ( t ) + o (1)1 − k Q K k ∞ ≤ S + o (1)1 − k Q K k ∞ (4.7)where o (1) → n → ∞ . Moreover, by (4.6) and (4.7)˜ S K ( t ) ≥ D ( u n ) − t k K k ∞ S / D ( u n ) / + o (1) ≥ S − t k K k ∞ (1 − k Q K k ∞ ) / + o (1) . Hence the conclusion follows immediately, using also (4.1). (cid:3)
Lemma 4.3
Let t n → ∞ be such that for every n there exists a minimizer for ˜ S K ( t n ) . Then ˜ S K ( t n ) → S .Proof. For every n let U n ∈ ˆ H be a minimizer for ˜ S K ( t n ), i.e., V ( U n ) = 1 and E t n ( U n ) = ˜ S K ( t n ).In particular D ( U n ) ≤ ˜ S K ( t n )1 − k Q K k ∞ ≤ S − k Q K k ∞ . (4.8)Since 1 = V ( U n ) ≤ k U n k ∞ D ( U n ) , one has that k U n k ∞ ≥ S − k Q K k ∞ =: δ > . Recall that U n is a ( K n − λ n )-bubble, with K n ( p ) = div Q K ( t n p ) and λ n ∈ R . In particular U n isbounded and regular as a map on S and there exists U n ( ∞ ) = lim | z |→∞ U n ( z ). By the conformal21nvariance, without changing notation, we may assume that | U n ( ∞ ) | = k U n k ∞ and that {| U n | <δ / } ⊂ D where in general {| U n | < δ } := { z ∈ R | | U n ( z ) | < δ } and D denotes the open unit disc. For every δ ∈ (0 , δ ) let A n ( δ ) := Z {| U n | <δ } |∇ U n | ( n ∈ N ) and A ( δ ) := lim inf n →∞ A n ( δ ) . (4.9)The following technical result holds. Lemma 4.4
One has that A ( δ ) → as δ → + . Let us complete the proof of Lemma 4.3. Since ˜ S K ( t ) ≤ S for every t >
0, it is enough to show thatlim inf ˜ S K ( t n ) ≥ S . Fix ε >
0. By Lemma 4.4, there exists δ ε > A ( δ ε ) ≤ ε , namely,lim inf n →∞ Z {| U n | <δ ε } |∇ U n | ≤ ε. (4.10)Then we estimate |Q t n ( U n ) | ≤ Z {| U n |≥ δ ε } | Q K ( t n U n ) | | U nx ∧ U ny | + k Q K k ∞ Z {| U n | <δ ε } |∇ U n | . Using (2.10), (4.8), and (4.10), we obtain thatlim inf n →∞ |Q t n ( U n ) | ≤ k Q K k ∞ ε . Thus, by the arbitrariness of ε >
0, one deduces that Q t n ( U n ) → V ( U n ) = 1,˜ S K ( t n ) = D ( U n ) + Q t n ( U n ) ≥ S − |Q t n ( U n ) | = S + o (1) as n → ∞ and we are done. (cid:3) Proof of Lemma 4.4.
We know that U n is a minimizer for ˜ S K ( t n ) and solves − ∆ U n + t n K ( t n U n ) U nx ∧ U ny = λ n U nx ∧ U ny on R (4.11)for some λ n . Then t n U n is a minimizer for S K ( t n ) and is a (cid:0) K − λ n t n (cid:1) -bubble. By (3.26) we infer that c := (2 − k ) S k ) ≤ λ n ≤ k ) S − k ) =: c . (4.12)For every δ ∈ (0 , δ /
2) let φ δ ( s ) := ≤ s ≤ δ δs − δ < s ≤ δ s > δ. Set u n := φ δ ( | U n | ) U n . Since φ δ is Lipschitz-continuous and {| U n | < δ / } ⊂ D , one has that u n ∈ H ( D , R ). We test (4.11) with u n and we find Z D ∇ U n · ∇ u n + Z D φ δ ( | U n | ) K ( t n U n ) t n U n · U nx ∧ U ny = λ n Z D φ δ ( | U n | ) U n · U nx ∧ U ny . (4.13)22et us estimate each term in (4.13) as follows. Firstly Z D ∇ U n · ∇ u n = Z {| U n | <δ } |∇ U n | + Z { δ< | U n | < δ } ∇ U n · ∇ u n . On the set { δ < | U n | < δ } , with direct computations, one finds that ∇ U n · ∇ u n = − δ ( U n · U nx ) + ( U n · U ny ) | U n | + 2 δ |∇ U n | | U n | − |∇ U n | ≥ −|∇ U n | and then Z D ∇ U n · ∇ u n ≥ Z {| U n | <δ } |∇ U n | − Z { δ< | U n | < δ } ∇ U n · ∇ u n = 2 A n ( δ ) − A n (2 δ ) , (4.14)according to the notation introduced in (4.9). Secondly, by ( K ), (cid:12)(cid:12)(cid:12)(cid:12)Z D φ δ ( | U n | ) K ( t n U n ) t n U n · U nx ∧ U ny (cid:12)(cid:12)(cid:12)(cid:12) ≤ k Z {| U n | < δ } | U nx ∧ U ny | ≤ k A n (2 δ ) . (4.15)In addition (cid:12)(cid:12)(cid:12)(cid:12)Z D φ δ ( | U n | ) U n · U nx ∧ U ny (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z {| U n | < δ } | U n | | U nx ∧ U ny | ≤ δ A n (2 δ ) . (4.16)Hence, from (4.13)–(4.16) it follows that2 A n ( δ ) ≤ (cid:18) k λ n δ (cid:19) A n (2 δ ) ∀ n ∈ N and then, using also (4.12), 2 A ( δ ) ≤ (cid:18) k λδ (cid:19) A (2 δ ) (4.17)for some λ > δ ∈ (0 , δ / δ A n ( δ ) are non-negative and non-decreasing, and the same holds for the mapping δ A ( δ ). In particular there existslim δ → + A ( δ ) =: A ≥ A ≤ (cid:0) k (cid:1) A . Since k < A = 0. (cid:3) Theorem 4.5
For every t ∈ (0 , t + ) let Λ( t ) be the set of Lagrange multipliers for minimizers of S K ( t ) , i.e., Λ( t ) = { λ ∈ R | ∃ ( K − λ ) -bubble U ∈ M t such that E ( U ) = S K ( t ) } . Then:(i) for every t ∈ (0 , t + ) the set Λ( t ) is compact, Λ( t ) ⊂ [ c t − / , c t − / ] with c and c defined in(4.12). Moreover lim sup ε → + S K ( t + ε ) − S K ( t ) ε ≤ min Λ( t ) , lim inf ε → − S K ( t + ε ) − S K ( t ) ε ≥ max Λ( t ) . (4.18) (ii) For a.e. t ∈ (0 , t + ) there exists the derivative S ′ K ( t ) and Λ( t ) = { S ′ K ( t ) } . roof. (i) By (3.26), Λ( t ) is bounded. Let ( λ n ) ⊂ Λ( t ) be such that λ n → λ , λ n = λ . Then there is asequence ( U n ) ⊂ ˆ H of minimizers for S K ( t ) and each U n is a ( K − λ n )-bubble. Since t ∈ (0 , t + ), wehave that S K ( τ ) < S ( τ ) for every τ ∈ (0 , t ] (Lemma 3.12). The sequence ( U n ) satisfies (3.5). Indeed |E ′ ( U n )[ ϕ ] − λ V ′ ( U n )[ ϕ ] | = | λ n − λ | |V ′ ( U n )[ ϕ ] |≤ C | λ n − λ | k∇ U n k k∇ ϕ k ∀ ϕ ∈ C ∞ c ( R , R )with C > n (see Remark 2.3). Moreover c ≤ k∇ U n k ≤ c for some constants0 < c ≤ c < ∞ (Lemma 3.4), and C ∞ c ( R , R ) is dense in ˆ H . Hence kE ′ ( U n ) − λ V ′ ( U n ) k ˆ H − →
0. Therefore we can apply Lemma 3.8 and repeating the proof of Lemma 3.10, we infer that thedecomposition of ( U n ) according to Lemma 3.8 in fact is made by just one ( K − λ )-bubble U ∈ ˆ H .Moreover E ( U n ) → E ( U ) and V ( U n ) → V ( U ). In particular U is a minimizer for S K ( t ). Thus weproved that λ ∈ Λ( t ), namely Λ( t ) is closed. Now let us prove (4.18). For every λ ∈ Λ( t ) there existsa ( K − λ )-bubble U ∈ ˆ H which is a minimizer for S K ( t ). Let u ε = p εt U . Then V ( u ε ) = t + ε , S K ( t + ε ) ≤ E ( u ε ), andlim sup ε → + S K ( t + ε ) − S K ( t ) ε ≤ lim ε → E ( u ε ) − E ( U ) ε = lim ε → p εt − ε lim s → E ( sU ) − E ( U ) s − E ′ ( U )[ U ]3 t Since U is a ( K − λ )-bubble, we have that E ′ ( U )[ U ] = 3 λt and thus we getlim sup ε → + S K ( t + ε ) − S K ( t ) ε ≤ λ. In a similar way we can show the opposite inequality for the lim inf as ε → − . Thus (4.18) holds.(ii) By Lemma 4.1, the isovolumetric mapping t S K ( t ) = t / ˜ S K ( t / ) is differentiable a.e. Hence,if there exists the derivative S ′ K ( t ), by (4.18) the set Λ( t ) is a singleton and its unique element is S ′ K ( t ) . (cid:3) Proof of Theorem 1.3.
The normalized isovolumetric function t ˜ S K ( t ) is continuous in (0 , ∞ ),as stated in Lemma 4.1. Then, by (4.1) and by Lemmas 4.2 and 4.3, there exists t > S K ( t / ) = inf t> ˜ S K ( t ) = S K . Let U ∈ ˆ H be a minimizer for the isovolumetric problem (1.4) with t = t . Then one easily checks that E ( U ) V ( U ) / = S K namely U is a minimizer for (1.7). In particular U is a ( K − λ )-bubble for some λ >
0. Since t ˜ S K ( t ) = S K ( t ) (Lemma 3.2 (ii)), we have thatlim sup ε → + ˜ S K ( t / + ε ) − ˜ S K ( t / ) ε + 2 ˜ S K ( t / ) t / ≤ δ → + S K ( t + δ ) − S K ( t ) δ ≤ t ) (4.19)in view of (4.18). Similarly one can obtainlim inf ε → − ˜ S K ( t / + ε ) − ˜ S K ( t / ) ε + 2 ˜ S K ( t / ) t / ≥ t ) . (4.20)Since ˜ S K ( t / ) = min t> ˜ S K ( t ) = S K , from (4.19) and (4.20) it follows that min Λ( t ) ≥ S K t − / and max Λ( t ) ≤ S K t − / , respectively. Thus Λ( t ) = (cid:8) S K t − / (cid:9) , that is λ = S K V ( U ) − / . (cid:3) A nonexistence result for isovolumetric problems
Some tools introduced in the previous section can be also used to show a nonexistence result for theisovolumetric problems (3.1). Such a result has a counterpart in the context of the H -bubble problem(see [11], § K < R . Theorem 5.1
Let K ∈ C ( R ) satisfy ( K ) – ( K ) . If K < on R , then there exists ε > such that ˜ S K ( t ) = S and the isovolumetric problem (3.1) has no minimizer for all t ∈ ( − ε, .Proof. Firstly let us prove that no minimizer for S K ( t ) exists as t < | t | . Arguingby contradiction, assume that in correspondence of a sequence t n → − , for every n there existsa minimizer U n ∈ ˆ H for S K ( t n ). Setting τ n = √ t n and u n = τ n U n , each u n turns out to be aminimizer for ˜ S K ( τ n ). Moreover, since τ n →
0, ( u n ) is a minimizing sequence for the isoperimetricproblem defined by S = inf {D ( u ) | u ∈ ˆ H , V ( u ) = 1 } because V ( u n ) = 1 and by an application of (4.6) and (4.7). By known results (see Lemma 2.1 in[11]), there exist a sequence of conformal mappings g n : S → S and a sequence ( p n ) ⊂ R suchthat u n ◦ g n − p n → − ω strongly in ˆ H , where ω is the standard parametrization of the unit sphere,defined in Remark 2.4. In fact, by conformal invariance, the function ˜ U n = u n ◦ g n is also a minimizerfor ˜ S K ( τ n ), hence is a ( ˜ K n − λ n )-bubble, where ˜ K n ( p ) = τ n K ( τ n p ) and λ n is bounded. Using an ε -regularity argument (see, e.g., the last part of the proof of Theorem 6.3 in [11] and the referencestherein), we can show that ˜ U n − p n → − ω in C ( S , R ). This implies that for n large enough, ˜ U n isan embedded parametric surface, bounding a domain A n ⊂ R and Q t n ( ˜ U n ) = Z A n ˜ K n ( p ) dp. Since
K < R and τ n <
0, we obtain that˜ S K ( τ n ) = D ( ˜ U n ) + Q t n ( ˜ U n ) > D ( ˜ U n ) ≥ S, namely S K ( t n ) > St / n , contrary to Lemma 3.3 (i). Now we show that ˜ S K ( t ) = S as t < | t | . Again we argue by contradiction, assuming that ˜ S K ( τ n ) < S along a sequence τ n → − . Then S K ( t n ) < St / n for every n , where t n = τ n . Reasoning as in the proof of Lemma 3.10, for fixed n , wefind a decomposition of t n = P i ∈ I n t n,i + P j ∈ J n t n,j with t n,i , t n,j ∈ [ t n , I n and J n finite sets ofindices, with S K ( t n,i ) admitting a minimizer, and S K ( t n ) = X i ∈ I n S K ( t n,i ) + X j ∈ J n S ( t n,j ) . Notice that the assumptions ( K ) and ( K ) are enough for this part of the argument. We claim that I n = ∅ . If not, then we reach a contradiction because St / n > S K ( t n ) = X j ∈ J n S ( t n,j ) = X j ∈ J n St / n,j ≥ S (cid:18) X j ∈ J n t n,j (cid:19) / = St / n . Thus we proved that there exists t ′ n ∈ ( t n ,
0) such that the isovolumetric problem defined by S K ( t ′ n )admits a minimizer. Then we apply the first part of the proof to reach a contradiction. (cid:3) cknowledgements. Work partially supported by the PRIN-2012-74FYK7 Grant “Variational andperturbative aspects of nonlinear differential problems”, by the project ERC Advanced Grant 2013n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”, and by theGruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of theIstituto Nazionale di Alta Matematica (INdAM).
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