A sharp gradient estimate and W^{2,q} regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space
aa r X i v : . [ m a t h . A P ] J a n A SHARP GRADIENT ESTIMATE AND W ,q REGULARITY FOR THEPRESCRIBED MEAN CURVATURE EQUATION IN THELORENTZ-MINKOWSKI SPACE
Denis BONHEURE, Alessandro IACOPETTI
Abstract.
We consider the prescribed mean curvature equation for entire spacelike hypersurfacesin the Lorentz-Minkowski space, namely − div ∇ u p − |∇ u | ! = ρ in R N , where N ≥
3. We first prove a new gradient estimate for classical solutions with smooth data ρ . Asa consequence we obtain that the unique weak solution of the equation satisfying a homogeneousboundary condition at infinity is locally of class W ,q and strictly spacelike in R N , provided that ρ ∈ L q ( R N ) ∩ L m ( R N ) with q > N and m ∈ [1 , NN +2 ]. Contents
1. Introduction 12. Preliminaries on the differential geometry of hypersurfaces in L N +1
63. Proof of the gradient estimate (1.7) 104. Haarala’s gradient estimate 175. Gradient estimates for the Born-Infeld equation 226. W ,qloc regularity of the minimizer of the Born-Infeld energy 27References 321. Introduction
The Lorentz-Minkowski space, denoted by L N +1 , is defined as the vector space R N +1 endowedwith the symmetric bilinear form(1.1) ( x, y ) L N +1 := x y + . . . + x N y N − x N +1 y N +1 . A hypersurface M ⊂ L N +1 is said to be spacelike if the induced metric on the tangent space T p M ,i.e. the restriction of ( · , · ) L N +1 to T p M , is positive definite at any point p ∈ M .We consider spacelike hypersurfaces that can be expressed globally as cartesian graphs, alsocalled vertical graphs . In particular, we focus on the case of entire vertical graphs, namely cartesiangraphs of functions defined in the whole of R N . We recall that if u ∈ C ( R N ) and M = graph ( u )is the associated cartesian graph then M is spacelike if and only if |∇ u | < R N (see Section Mathematics Subject Classification.
Key words and phrases.
Prescribed mean curvature, Lorentz-Minkowski space, Regularity, Non smooth operators,Born-Infeld electrostatic equation.
Acknowledgements.
Research partially supported by FNRS and by Gruppo Nazionale per l’Analisi Matematica,la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)..We wish to thank Akseli Haarala for the useful discussions. u is said to be strictly spacelike . The prescribed mean curvature equation forentire spacelike vertical graphs in the Lorentz-Minkowski space is(1.2) − div ∇ u p − |∇ u | ! = ρ in R N , where ρ : R N → R is a given function that prescribes the mean curvature pointwise. Maximal,that is when ρ = 0, and constant mean curvature spacelike hypersurfaces are important in generalrelativity as first emphasized in the fundamental paper of Lichnerowicz [40]. They were later usedto study dynamical aspects of general relativity or the structure of the singularities in the spaceof solutions of Einstein’s equations and in the first proof of the positivity of gravitational mass.We refer to [6, 7, 21, 26, 42, 47] and the many references therein. It is interesting to notice that thenonlinear differential operator in the left hand side of (1.2) naturally appears in other contexts, asfor instance in the nonlinear electrodynamics of Born and Infeld [15–18, 48] and in string theory,see e.g. [30]. Calabi [19] showed in 1968 that Equation (1.2) with ρ = 0 has the Bernstein propertyin dimension N
4, that is any solution u has to be affine. Cheng and Yau [20] completed this so-called Calabi-Bernstein’s theorem and proved in fact its parametric version, i.e. in every dimension,the only maximal space-like hypersurface which is a closed subset of the Lorentz-Minkowski spaceis a linear hyperplane. The validity of the Bernstein property in every dimension contrasts withthe less rigid euclidian case [3, 10, 11, 27, 46, 49]. The generalization to entire area maximizinghypersurfaces is attributed to Bartnik, see [23, Theorem F]. Later Treibergs [51] tackled the caseof entire spacelike hypersurfaces of constant mean curvature. Extensions in Lorentzian productspaces and generalized Robertson-Walker spacetimes were also considered, see for instance [1,2,45]and the citations therein.The Dirichlet problem for spacelike vertical graphs in L N +1 was solved by Bartnik and Si-mon [5], and Gerhardt [29] extended those results to the case of vertical graphs contained in aLorentzian manifolds which can be expressed as a product of a Riemannian manifold times aninterval. Solutions with singularities have been considered in [23, 34–36].Bayard [8] studied the more general problem of prescribed scalar curvature. For radial graphs,to our knowledge, the only available result concerns entire spacelike hypersurfaces with prescribedscalar curvature which are asymptotic to the light-cone [9] and our contribution [14] where weinvestigate the existence and uniqueness of spacelike radial graphs spanning a given boundarydatum lying on the hyperbolic space H N .Unbounded data ρ were first considered in [12], but only in the radial case, and in [13,32], wherethe regularity of the solution was the object of attention. As explained in [13], solutions of theregular quasilinear, possibly degenerate, equation of the form − div (cid:18) g ( |∇ u | ) |∇ u | ∇ u (cid:19) = µ in a bounded domain (see [4] for the assumptions on g ) satifsfy the estimate(1.3) g ( |∇ u ( x ) | ) c I | µ | ( x, R ) + cg − ˆ B R ( x ) |∇ u | dy ! , where c > N and I | µ | ( x, R ) is the truncated linear Riesz potential associatedwith the measure µ (see [4, Section 1], or [37]). We also refer to [22,37–39,43] for similar estimatesfor equations driven by the p -Laplacian (or more general operators). If this formal inversion of thedivergence operator could be justified also for the singular operator appearing in (1.2), namely Q − ( u ) := − div ∇ u p − |∇ u | ! , SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 3 one would derive the estimate(1.4) |∇ u ( x ) | p − |∇ u ( x ) | c ˆ ∞ ´ B t ( x ) | ρ ( y ) | dyt N dt, and therefore the norm of the gradient of the solution would stay away from 1 as soon as ρ ∈ L s ( R N )with s < N and ρ is locally L q ( R N ) for some q > N . A simple explicit example shows that thecondition ρ ∈ L qloc ( R N ) for some q > N is sharp, see [13]. It seems quite hard and unlikely toobtain the estimate (1.4), at least in a direct way. Indeed the mean curvature operator Q − doesnot satisfy the growth and ellipticity conditions appearing in the so-called Nonlinear Calder´on-Zygmund theory, see [22, 37, (1.2)] or [44, (4.9)]. Roughly speaking, the Nonlinear Calder´on-Zygmund theory is modelled around the p -Laplacian as reference operator, which is singular ordegenerate as the gradient of u is zero, a point, but Q − is singular as the gradient lies on the unitsphere, so from the structural point of view they are clearly distinct operators.The main result of the present paper is a sharp gradient estimate for smooth classical solutionsof (1.2), which extends and refines Bartnik and Simon estimates, see [5, 13]. Among the severalpossible applications, this gradient estimate is then used to deduce W ,q loc regularity and actuallysolve [13, Conjecture 1.4 ], with an explicit H¨older exponent, as further described in this section.We adopt the following notations and definitions: • f i or ∂f∂x i denote the partial derivatives of f : R N → R k , N, k ∈ N + , and if N = 1 we simplyuse D t f or f ′ to denote the standard derivative; • ∇ f , D f denote, respectively, the gradient and the Hessian matrix of f : R N → R ; • ( · , · ) R N denotes the Euclidean scalar product in R N ; • | · | denotes the Euclidean norm of a vector or a matrix; • B R ( x ) = { x ∈ R N ; | x − x | < R }} denotes the Euclidean ball of radius R centred at x ; • u ∈ C , ( R N ) is said to be: – weakly spacelike if |∇ u | R N ; – spacelike if | u ( x ) − u ( y ) | < | x − y | whenever x, y ∈ R N , x = y ; – strictly spacelike if u is spacelike, u ∈ C ( R N ) and |∇ u | < R N ; • given a strictly spacelike function u ∈ C ( R N ), we use the notation v to denote the function(1.5) v ( x ) := p − |∇ u ( x ) | x ∈ R N ; • the projection of the Lorentz ball of radius R centred at x , associated to a spacelikefunction u ∈ C , ( R N ), is denoted by K R ( x ) = { x ∈ R N ; [ | x − x | − ( u ( x ) − u ( x )) ] / < R } ; • ω N denotes the volume of the Euclidean unit ball in R N ; • ∗ denotes the conjugate H¨older exponent of the critical Sobolev exponent, namely 2 ∗ =(2 ∗ ) ′ = NN +2 , where 2 ∗ = NN − , N ≥ • | · | q denotes the standard norm in L q ( R N ), q ≥ q = ∞ , and | · | q,E denotes the standardnorm in L q ( E ), where E ⊂ R N is a Lebesgue measurable set; • D , ( R N ) is the completion of C ∞ c ( R N ) with respect to the L norm of the gradient; • X is the convex set defined by(1.6) X := D , ( R N ) ∩ (cid:8) u ∈ C , ( R N ); |∇ u | ∞ (cid:9) , and equipped with the D , ( R N ) norm; • X ∗ denotes the dual of X and h· , ·i denotes the duality pairing between X ∗ and X . Denis BONHEURE, Alessandro IACOPETTI
Theorem 1.1.
Assume N ≥ , x ∈ R N , R > , ρ ∈ C ( R N ) , q > N and β := Nq . Let u ∈ C ( R N ) be a strictly spacelike classical solution of (1.2) and suppose that K R ( x ) is bounded.Then there exist γ ∈ (cid:0) , N (cid:1) and c > , both depending on N only, such that ω N v γ ( x ) ≥ R − N ˆ K R ( x ) v γ +1 dx + cR − N ˆ K R/ ( x ) v γ − (cid:0) | D u | + |∇ v | (cid:1) dx − | ρ | q,K R ( x ) ˆ R s − β ω q N + 3 ω − q N | ρ | q,K R ( x ) q (2 − β ) s − β + | ρ | q,K R ( x ) q (1 − β ) s − β q − ds − | ρ | q,K R ( x ) ˆ R s − β ω q N + 3 ω − q N | ρ | q,K R ( x ) q (2 − β ) s − β + | ρ | q,K R ( x ) q (1 − β ) s − β q − ds. (1.7)The gradient estimate in Theorem 1.1 is sharp for two reasons. First, in contrast to whatwe proved in [13], we are able to reach the sharp regularity threshold q > N and we do notassume any global integrability assumption on ρ . In particular Theorem 1.1 can be applied toentire constant or asymptotically constant mean curvature hypersurfaces, provided that K R ( x )is a bounded subset of R N . This assumption plays a crucial role in the proof of Theorem 1.1as we consider integral quantities over the Lorentz ball involving v γ and ρ (see (3.9)). However,when u ∈ L ∞ ( R N ) then K R ( x ) is automatically bounded, for any choice of x , R , as shown inLemma 2.8-(i). The same conclusion holds true even when u is not globally bounded and possiblyasymptotic to the light-cone at infinity provided that |∇ u ( x ) | → | x | → + ∞ , and R is sufficiently small, see Lemma 2.8-(ii). Next, when considering the solution of (1.2) vanishingat infinity, i.e. the unique weak solution of the Born-Infeld equation( BI ) − div ∇ u p − |∇ u | ! = ρ in R N , lim | x |→∞ u ( x ) = 0 , we deduce the following statement. Weak solutions are further defined in Definition 1.3. Proposition 1.2.
Assume N ≥ and let u ∈ X ∩ C ( R N ) be a strictly spacelike weak solutionof ( BI ) with ρ ∈ L m ( R N ) ∩ C ( R N ) , m ∈ (1 , ∗ ] and q > N (the case m = 1 requires a separatestatement, see Proposition 5.7). There exist two positive constants γ ∈ (0 , N ) depending only on N , c depending only on N and m , such that for any R > , x ∈ R N (1.8) (1 − |∇ u | ) γ ( x ) ≥ ω γ +1 N (cid:18) ω N + cR − N | ρ | NmN − m m (cid:19) γ +1 − P (cid:16) | ρ | q,K R ( x ) R q − Nq (cid:17) , where P : [0 , + ∞ ) → [0 , + ∞ ) is such that P (0) = 0 , see (5.6) for the precise definition. When ρ = 0, and since P (0) = 0, we infer thatinf R N (1 − |∇ u | ) γ ≥ , which shows that sup R N |∇ u | = 0, i.e. u = 0 due to the condition at infinity. The reason ofthe optimality of (1.7) is that, on the contrary to what is proved in [5, 13], and thanks to afiner decomposition of integral quantities, we derive a monotonicity formula, namely (3.25), whichinvolves only nonlinear powers of the auxiliary function ψ ( s ) := s − N ˆ K R ( x ) v γ +1 dx. SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 5
In particular, instead of using an integrating factor to rule out the linear term in ψ , as donein [5, 13], we prove a specific variant of Gronwall’s Lemma involving two separate powers of ψ and with two different singular weights (see Lemma 3.1), so that, at the end, there are no ex-tra factors in front of the first integral in (1.7). The quantity v appearing in Theorem 1.1 hasa significant geometrical meaning as it represents the area element of M = graph ( u ) (see Sect.2), while the terms | D u | , |∇ v | are related to the second fundamental form of M (see (2.3), (2.4)).Finally, with (1.7) at hand, we are able to improve the results of [13,32] concerning the regularityof the minimizer of the energy associated with ( BI ), i.e. the Born-Infeld electrostatic energy. Webriefly summarize the known facts about the variational formulation of ( BI ). For a given ρ ∈ X ∗ ,equation ( BI ) is, at least formally, the Euler-Lagrange equation associated to the functional I ρ : X → R defined by(1.9) I ρ ( u ) = ˆ R N (cid:16) − p − |∇ u | (cid:17) dx − h ρ, u i . We recall that X ∗ contains Radon measures as for instance linear combinations of Dirac deltas or L m ( R N ) functions, for any m ∈ [1 , ∗ ], see [12, 28, 33] and Remark 5.6. The following propertieshold true (see [12, Lemma 2.1, Lemma 2.2]): • for any u ∈ X one has lim | x |→∞ u ( x ) = 0; • X is weakly closed, it embeds continuously into W ,s ( R N ) for all s ∈ [2 ∗ , ∞ ) and in L ∞ ( R N ); • I ρ is bounded from below in X , coercive, weakly lower semi-continuous and strictly convex.In particular, from the direct methods in the Calculus of Variations the functional I ρ has a uniqueminimizer u ρ ∈ X (see [12, Proposition 2.3]) and the set of singular points E ρ = { x ∈ R N ; |∇ u ρ | = 1 } is a null set with respect to the Lebesgue measure (see [12, Proposition 2.7]). In general one has E ρ = ∅ , as it happens for instance when ρ is a Dirac mass [12, Theorem 1.6] and [17] or ρ is, ina ball B R (0), the toy radial datum 1 / | x | β +1 with β > I ρ is not C at points u ∈ X such that |∇ u | ∞ = 1, we cannot say in general that theunique minimizer is a weak solution to ( BI ). Definition 1.3.
We say that u ∈ X is a weak solution of ( BI ) if for all ψ ∈ X we have (1.10) ˆ R N ∇ u · ∇ ψ p − |∇ u | dx = h ρ, ψ i . As pointed out in [12, Section 2], if ρ is a distribution, the weak formulation of (1.10) extendsto any test function ψ ∈ C ∞ c ( R N ). Moreover, any weak solution of ( BI ) in X coincides with theminimizer of I ρ [12, Proposition 2.6], but, as pointed out before, we do not known if the converseholds true in general. This remains a challenging open question for generic data ρ ∈ X ∗ , andpositive answers have been given only for bounded or integrable distributions ρ , see [5, 12, 13, 32].In vue of the heuristic (1.4), we conjectured [13, Conjecture 1.4] that if ρ ∈ X ∗ and ρ ∈ L qloc ( R N ),with q > N , then the unique minimizer u ρ is C ,αloc ( R N ), for some α ∈ (0 , ρ , we proved C ,αloc regularity for ρ ∈ L q ( R N ) with q > N only, see [13, Theorem1.6]. With (1.8) (and the subsequent Proposition 5.4) at hand, the argument in [13, Theorem 1.6]shows there exists a positive constant c = c ( N, q, m ) such that if ρ ∈ L q ( R N ) ∩ L m ( R N ) satisfies(1.11) | ρ | qq − N q | ρ | mN − m m ≤ c, Denis BONHEURE, Alessandro IACOPETTI with q > N and m ∈ (1 , ∗ ], then the unique minimizer u ρ of (1.9) is a weak solution of ( BI ),it is strictly spacelike and u ρ ∈ C ,α loc ( R N ) for some α ∈ (0 , BI ) which preserves the modulus of the gradient (see Remark 5.5).Recently Haarala posted a preprint [32] where he claims, see [32, Theorem 1.3], C ,α regularitywithout any smallness assumptions on ρ . The strategy of [32] is based on a tricky gradient estimate,see [32, Theorem 3.5], in the spirit of [5, Theorem 3.5], and a delicate fixed point method. Withthe combination of Theorem 1.1 and [32, Theorem 3.5], it is in fact possible to prove more.Indeed, by the same proof of [13, Theorem 1.5] and exploiting Theorem 1.1 one gets immediatelythat the minimizer u ρ has a second weak derivative locally, whenever ρ ∈ L qloc ( R N ), q > N , whichis another serious issue. But this not the end, in fact, notice that in (1.7) there is a weight v γ − in front of the terms | D u | and |∇ v | , and the exponent γ − γ ∈ (0 , N ).This lead us to suspect that actually the minimizer is more than W , locally and this our secondmain result. Theorem 1.4.
Assume N ≥ , q > N , m ∈ [1 , ∗ ] , ρ ∈ L q ( R N ) ∩ L m ( R N ) and let u ρ ∈ X be theunique minimizer of (1.9) . Then u ρ ∈ W ,q loc ( R N ) , u ρ is strictly spacelike and it is a weak solutionof ( BI ) . We then deduce the answer to [13, Conjecture 1.4] when ρ ∈ L q ( R N ), with q > N , with theprecise value of the H¨older exponent α = 1 − Nq . The case ρ ∈ L q loc ( R N ) remains open. Corollary 1.5.
Assume N ≥ , q > N , m ∈ [1 , ∗ ] , ρ ∈ L q ( R N ) ∩ L m ( R N ) . Then the uniqueminimizer u ρ of (1.9) belongs to C ,αloc ( R N ) , with α = 1 − Nq . Our strategy is to derive a uniform control of the gradient at infinity from Proposition 1.2.In this way we somehow recover the boundary barrier construction performed by Bartnik andSimon in [5], which was crucially based on the boundedness of the mean curvature. Then, usingHaarala’s gradient estimate, namely Theorem 4.1, we capitalize on the control at infinity to deducea uniform global gradient bound. We conclude by using the universal potential estimates of Kuusiand Mingione [37] and the standard elliptic regularity theory for equations in non-divergence form.The outline of the paper is the following. In Section 2 we recall the basic facts about the geometryof spacelike hypersurfaces in the Lorentz-Minkowski space. Section 3 contains the proof of Theorem1.1. In section 4 we detail the proof of the gradient estimate due to Haarala [32, Theorem 3.5]. InSection 5 we prove our new gradient estimates for the Born-Infeld equation, namely Proposition1.2 and a similar statement when m = 1. Finally Section 6 is dedicated to the proof of Theorem1.4. 2. Preliminaries on the differential geometry of hypersurfaces in L N +1 In this section we collect, for the reader’s convenience, some definitions and results about thegeometry of spacelike hypersurfaces in the Lorentz-Minkowski space that will be used throughoutthe paper. These facts are essentially contained in [5, 12–14, 41].We denote by L N +1 the ( N + 1)-dimensional Lorentz-Minkowski space, which is defined as thevector space R N +1 equipped with the symmetric bilinear form ( · , · ) L N +1 defined by (1.1). Thebilinear form ( · , · ) L N +1 is non-degenerate and it has index one (see e.g. [50]). The modulus of x ∈ L N +1 is given by k x k L N +1 := | ( x, x ) | / L N +1 . We say that a vector x ∈ L N +1 is • spacelike if ( x, x ) L N +1 > x = 0; • timelike if ( x, x ) L N +1 < • lightlike if ( x, x ) L N +1 = 0 and x = 0. SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 7 If V is a vector subspace of L N +1 we define the induced metric ( · , · ) V in the standard way( x, y ) V := ( x, y ) L N +1 , x, y ∈ V. Definition 2.1.
We say that a hypersurface M ⊂ L N +1 is spacelike (resp. timelike, lightlike) iffor any p ∈ M the induced metric ( · , · ) T p M on the tangent space T p M is positive definite (resp. ( · , · ) T p M has index one, ( · , · ) T p M is degenerate). We say that M is a non-degenerate hypersurfaceif M is spacelike or timelike. Notice that if M is a spacelike hypersurface then it inherits a Riemannian structure in a naturalway. Moreover observe that if M is a spacelike (resp. timelike) hypersurface and p ∈ M then wecan decompose the space as L N +1 = T p M ⊕ ( T p M ) ⊥ , where ( T p M ) ⊥ is a timelike (resp. spacelike)subspace of dimension 1 (see [41]). Definition 2.2.
Let M be a non-degenerate hypersurface. A Gauss map is a differentiable map ν : M → L N +1 such that k ν ( p ) k L N +1 = 1 and ν ( p ) ∈ ( T p M ) ⊥ for all p ∈ M . For the sake of completeness we recall that closed hypersurfaces (i.e. compact hypersurfaceswithout boundary) do not play a relevant role in the geometry of L N +1 . Indeed the followingresult holds (see [41, Proposition 3.1] or [14, Proposition 2.5]) Proposition 2.3.
Let M ⊂ L N +1 be a compact spacelike, timelike or lightlike hypersurface. Then ∂M = ∅ . From now we focus only on the case of spacelike vertical graphs. Let { e , . . . , e N + } be thestandard basis of L N +1 (i.e. the standard basis of R N +1 ). According to the notations of [5], weagree that the indices i, j have the range 1 , . . . , N , while the indices I , J have the range 1 , . . . , N +1,and we observe that ( e I , e J ) L N +1 = 0 if I 6 = J , ( e i , e i ) L N +1 = 1 and ( e N + , e N + ) L N +1 = − Definition 2.4.
We say that a timelike vector x ∈ L N +1 is future-directed (resp. past-directed) if ( x, e N + ) L N +1 < (resp. ( x, e N + ) L N +1 > ). Let Ω ⊂ R N be a domain (bounded or unbounded) and let u ∈ C (Ω), the associated verticalgraph is M = { ( x, u ( x )) ∈ L N +1 ; x ∈ Ω } . An obvious parametrization is given by Φ : Ω → L N +1 , Φ( x ) = ( x, u ( x )) and thus(2.1) X i := ∂ Φ ∂x i = e i + u i e N + , i = 1 , . . . , N, is a basis of tangent vectors for T p M , where p = ( x, u ( x )). The induced metric on M is given by g = ( g ij ) i,j =1 ,...,N , where g ij = ( X i , X j ) L N +1 = δ ij − u i u j . Since the determinant of the principal minor of order k of the matrix ( g ij ) i,j =1 ,...,N , denoted by g k , is Det ( g k ) = 1 − P ki =1 u i , it is clear that M is spacelike (resp. timelike, lightlike) if and onlyif |∇ u | < |∇ u | >
1, lightlike if and only if |∇ u | = 1), for all x ∈ Ω.Accordingly, we say that u ∈ C (Ω) is strictly spacelike if |∇ u | < u ∈ C (Ω) be a strictly spacelike function and let M be the associated cartesian graph. Thefuture-directed Gauss map is expressed by ν = ( ∇ u, p − |∇ u | . Recalling the notation v = p − |∇ u | introduced in (1.5), we can then write(2.2) ν = N X i =1 ν i e i + ν N +1 e N + = N X i =1 u i v e i + 1 v e N + . Denis BONHEURE, Alessandro IACOPETTI
The coefficients and the norm of the second fundamental form of M are given, respectively, by II ij = ( X i , ∇ X j ν ) L N +1 = 1 v u ij , and(2.3) k II k := N X i,j,k,l =1 g ij g kl II ik II jl = 1 v N X i,j,k,l =1 g ij g kl u ik u jl , where { X i } i =1 ,...,N is the basis of T p M found in (2.1), g − = ( g ij ) ij =1 ,...,N , g ij = δ ij + ν i ν j is theinverse matrix of g . Notice that from (2.3), see also (3.34), it follows that(2.4) v k II k = N X i,k =1 u ik + 2 N X i =1 v − N X k =1 u k u ik ! + v − v − N X i,k =1 u i u k u ik = | D u | + 2 |∇ v | + v − | ( ∇ u, ∇ v ) R N | . The mean curvature of M (see [41] for more details) is defined as(2.5) H := − N X i,j =1 g ij II ij = − v N X i,j =1 g ij u ij . Notice that for notational convenience, and differently from [41] (and other references), we do notdivide by N in the definition of H .We recall now some explicit formulas for the differential operators on a spacelike vertical graph M (we refer to [5, Section 2] for more details). Let δ = grad M , div M , ∆ M be, respectively, thegradient, the divergence and the Laplace-Beltrami operators on M . Assume W ⊂ L N +1 is an openneighborhood of M , Y is a C vector field on W , and f ∈ C ( W ) is such that ∂f∂x N +1 = 0. Wehave(2.6) δf = N +1 X I =1 ( δ I f ) e I , where δ i f = N X j =1 g ij ∂f∂x j , δ N +1 f = 1 v N X i =1 ν i ∂f∂x i , k δf k L N +1 = N +1 X i,j =1 g ij ∂f∂x i ∂f∂x j = |∇ f | + N X i =1 (cid:18) ν i ∂f∂x i (cid:19) , div M Y = N X i,j =1 g ij ( X i , ∇ X j Y ) L N +1 . Moreover, if in addition f ∈ C ( W ) then, recalling (2.5), we have(2.7) ∆ M f = div M (grad M f ) = N X i,j =1 g ij ∂ f∂x i ∂x j − N X i =1 Hν i ∂f∂x i . Another tool that will be used in the next section is the following corollary of Stokes’s theorem(for the proof see [5, Sect. 2], with the only caution that H has the opposite sign). Proposition 2.5 (Integration by parts) . For any f ∈ C c ( M ) , g ∈ C ( M ) one has (2.8) ˆ M f δ N +1 g dA = ˆ M v f gH dA − ˆ M gδ N +1 f dA, ˆ M f δ i g dA = ˆ M ν i f gH dA − ˆ M gδ i f dA, i = 1 , . . . , N, where dA = vdx is the induced volume form on M , dx being the Lebesgue measure on R N . SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 9
We conclude this section by recalling some properties and two important identities involvingthe gradient and the Laplacian of the Lorentz distance that will play a crucial role in the proof ofTheorem 1.1. We begin with a definition.
Definition 2.6.
Let u ∈ C , ( R N ) be a spacelike function and let x ∈ R N . We define the Lorentzdistance from ( x , u ( x )) as l ( x, x ) := [ | x − x | − ( u ( x ) − u ( x )) ] / . Given
R > and M = graph ( u ) , we define the Lorentz ball of radius R centred at ( x , u ( x )) as L R ( x ) := { ( x, u ( x )) ∈ M ; l ( x, x ) < R } and its projection on R N as K R ( x ) := { x ∈ R N ; l ( x, x ) < R } . For simplicity, when x ∈ R N is fixed and there is no possible confusion we will adopt thesimpler notations l = l ( x ), L R and K R . Remark 2.7.
From Definition 2.6 it is clear that the notions of Lorentz distance and Lorentz ballcan be extended to the class of weakly spacelike functions. However, observe that if u ∈ C , ( R N ) contains a light ray, namely if there exist x , x ∈ R N such that for all t ≥ , | u ( x + t ( x − x )) − u ( x ) | = t | x − x | , then l ( x, x ) = 0 for any x = x + t ( x − x ) , t ≥ , so that l is not a distance in the usual meaningand K R ( x ) turns out to be an unbounded subset of R N . Nevertheless, if u is weakly spacelikeand u ∈ L ∞ ( R N ) then K R ( x ) is bounded. The same conclusion holds true also for (possibly)unbounded strictly spacelike functions u whose gradient (in modulus) goes to one at infinity butnot too fast. This is the content of the next lemma. Lemma 2.8.
The following facts hold:i) if u ∈ C , ( R N ) ∩ L ∞ ( R N ) is a weakly spacelike function, then for any R > , x ∈ R N one has that K R ( x ) is bounded and K R ( x ) ⊂ B R ′ ( x ) , with R ′ = p R + 4 | u | ∞ ;ii) if u ∈ C ( R N ) is strictly spacelike and (2.9) lim inf | x |→ + ∞ (1 − |∇ u ( x ) | ) | x | > , then for any x ∈ R N there exists R > such that K R ( x ) is bounded.Proof. For i), given x ∈ R N , R > K R ( x ) it is clear that for any x ∈ K R ( x ), one has | x − x | < R + ( u ( x ) − u ( x )) ≤ R + 4 | u | ∞ , and this implies that K R ( x ) ⊂ B R ′ ( x ), where R ′ = p R + 4 | u | ∞ .For ii), fix x ∈ R N and assume by contradiction that the thesis is false. Then we find a sequenceof positive radii ( R n ) n and a sequence of points ( x n ) n ⊂ R N such that x n ∈ K R n ( x ), for all n ∈ N ,and R n → | x n | → + ∞ , as n → + ∞ . By the mean value theorem, for any n ∈ N there exists ξ n ∈ x x n , such that u ( x n ) − u ( x ) = ∇ u ( ξ n ) · ( x n − x ). Hence, by definition of K R n ( x ) we inferthat(2.10) | x n − x | (1 − |∇ u ( ξ n ) | ) < R n ∀ n ∈ N . Now, up to a subsequence (still indexed by n ∈ N ), as n → + ∞ there are only two possibilities: ξ n → ξ , for some ξ ∈ R N , or | ξ n | → + ∞ . We claim that both cases lead to a contradiction. Indeed, assume that ξ n → ξ , for some ξ ∈ R N , and observe that, as u ∈ C ( R N ) is strictlyspacelike and ξ ∈ R N then |∇ u ( ξ ) | = 1 − δ <
1, for some δ ∈ (0 , | x n − x | → + ∞ , R n →
0, then, passing to the limit as n → + ∞ we reach a contradiction asthe left-hand side diverges and the right-hand side goes to zero. For the latter case, assume that | ξ n | → + ∞ . In particular, from (2.10) we can write, for all sufficiently large n , that(2.11) | x n − x | | ξ n | (1 − |∇ u ( ξ n ) | ) | ξ n | < R n . We notice that since | x n | → + ∞ and ξ n ∈ x x n , where x x n denotes the line segment joining x and x n , it follows that | x n − x | | ξ n | ≥ for all sufficiently large n ∈ N . In addition, since | ξ n | → + ∞ and thanks to (2.9) we infer that(2.12) lim inf n → + ∞ (1 − |∇ u ( ξ n ) | ) | ξ n | =: c > . Hence, taking the lim inf of both sides of (2.11) we reach again a contradiction and the proof iscomplete. (cid:3)
Let us fix x ∈ R N , assume that u ∈ C ( R N ) is strictly spacelike, set X := ( x , u ( x )) ∈ L N +1 and let l be the Lorentz distance (from X ) associated to u . We notice that since l = l ( x ) doesnot depend on the variable x N +1 and u is strictly spacelike then l is naturally defined and ofclass C in a neighborhood W ⊂ L N +1 of M . In particular, using (2.6)-(2.7) we can prove that(see [5, (2.10)])(2.13) k δl k L N +1 = 1 + l − ( ν, X − X ) L N +1 , (2.14) ∆ M (cid:18) l (cid:19) = N − ( Hν, X − X ) L N +1 . Finally, we recall the following special case of Federer’s coarea formula (see [5, (2.14)], [25, Theorem3.2.12] or [24, Section 3.4.4])
Theorem 2.9.
Let u ∈ C ( R N ) be a strictly spacelike function, M = graph ( u ) , x ∈ R N , s > and assume that K s ( x ) is bounded. For any h ∈ C ( M ) we have (2.15) D s " ˆ L s ( x ) h dA = ˆ ∂L s ( x ) h k δl k − L N +1 dµ, where dµ is the surface measure on ∂L s ( x ) . Proof of the gradient estimate (1.7)In this section we prove Theorem 1.1. We begin with a preliminary technical result, which is ageneralization of the classical Gronwall’s lemma.
Lemma 3.1.
Let
T > , β ∈ (0 , , q > and assume ψ : [0 , T ] → [0 , + ∞ [ is a continuousfunction such that (3.1) ψ ( t ) ≤ C + C ˆ t s − β ( ψ ( s )) q − q ds + C ˆ t s − β ( ψ ( s )) q − q ds ∀ t ∈ [0 , T ] for some constants C > , C , C ≥ . Then (3.2) ψ ( t ) ≤ C q + C − q C q (2 − β ) t − β + C q (1 − β ) t − β q ∀ t ∈ [0 , T ] . SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 11
Proof.
Let
T > β ∈ (0 ,
2) and consider the auxiliary function y : [0 , T ] → [0 , + ∞ [ defined by y ( t ) := C ˆ t s − β ( ψ ( s )) q − q ds + C ˆ t s − β ( ψ ( s )) q − q ds. By definition and thanks to (3.1) we infer that y ′ ( t ) = C t − β ( ψ ( t )) q − q + C t − β ( ψ ( t )) q − q ≤ C t − β ( C + y ( t )) q − q + C t − β ( C + y ( t )) q − q ∀ t ∈ (0 , T ] . (3.3)Since y is a non-negative function and C >
0, dividing each side of (3.3) by ( C + y ( t )) q − q weget, for any t ∈ (0 , T ], y ′ ( t )( C + y ( t )) q − q ≤ C t − β ( C + y ( t )) q + C t − β ≤ C − q C t − β + C t − β , and integrating on (0 , t ) we obtain ˆ t y ′ ( s )( C + y ( t )) q − q ds ≤ C − q C − β t − β + C − β t − β ∀ t ∈ (0 , T ] . Changing variable in the first integral and taking into account that β ∈ (0 , q ( C + y ( t )) q − qC q = ˆ C + y ( t ) C k − q − q dk ≤ C − q C − β t − β + C − β t − β ∀ t ∈ [0 , T ] , since by definition y (0) = 0. Finally, recalling that ψ ( t ) ≤ C + y ( t ), from (3.4) we readily obtain ψ ( t ) ≤ C q + C − q C q (2 − β ) t − β + C q (1 − β ) t − β q ∀ t ∈ [0 , T ] . (cid:3) Proof of Theorem 1.1.
Let N ≥ q > N , x ∈ R N , R > ρ , u , K R ( x ) are as in thestatement of Theorem 1.1. Observe first that since ˆ u ( x ) := u ( x + x ) − u ( x ) is still a solutionto (1.2) with ρ replaced by ˇ ρ ( x ) := ρ ( x + x ), and since ∇ ˆ u (0) = ∇ u ( x ), | ρ | q,K R ( x ) = | ˇ ρ | q, ˆ K R (0) ,where ˆ K R (0) is the projection of the Lorentz ball associated to ˆ u , we can assume without loss ofgenerality that X := ( x , u ( x )) = (0 , X := ( x, u ( x )) ∈ L N +1 be the position vector, M = graph ( u ) and s ∈ (0 , R ]. Since u is astrictly spacelike classical solution to (1.2), M has mean curvature ρ . Therefore, as a consequenceof Green’s formula, Proposition 2.8, (2.13),(2.14) and Federer’s coarea formula (2.15), for any f ∈ C ( M ) we have D s (cid:20) s − N ˆ L s f dA (cid:21) = ˆ L s s − N − (cid:18)
12 ( s − l )∆ M f − f ρ ( X, ν ) L N +1 (cid:19) dA − D s (cid:20) ˆ L s f l − N − ( X, ν ) L N +1 dA (cid:21) , (3.5)where L s , K s denote, respectively, the Lorentz ball and its projection on R N associated to u ,centred at the origin and of radius s (notice that L s is a bounded subset of M , for any s ∈ (0 , R ]because by assumption K R is bounded). We refer to [5, (2.15)] or [13, (3.4)] for the proof of (3.5).Let γ be a positive number to be determined later and set v := p − |∇ u | . Notice that since u ∈ C ( R N ) is strictly spacelike then v γ ∈ C ( R N ). Moreover, since v γ does not depend on the variable x N +1 , it is naturally defined and of class C in a whole neighborhood of M . Exploiting(2.6) and (2.7), we check that∆ M v γ = γv γ − ∆ M v + γ ( γ − v γ − k δv k L N +1 = − γv γ − N X i,j =1 u ij − γ N X i =1 u ii ! + (1 − γ ) vρ N X i =1 u ii + v ρ + (1 − γ ) N X j =1 N X i =1 ν i u ij ! + γδ N +1 (cid:0) v γ +1 ρ (cid:1) . (3.6)By simple algebraic manipulations and using the well known trace inequality N X i =1 u ii ! ≤ N N X i,j =1 u ij , and Young inequality(3.7) ab ≤ ǫa + 14 ǫ b ∀ a, b ≥ , ∀ ǫ > , we find two constants γ ∈ (0 , N ) and C >
0, both depending on N only, such that(3.8) ∆ M v γ ≤ − Cv γ − N X i,j =1 u ij + N X j =1 N X i =1 ν i u ij ! + 14 v γ ρ + γδ N +1 (cid:0) v γ +1 ρ (cid:1) . Applying (3.5) with f = v γ and exploiting (3.8), we get D s (cid:26) s − N ˆ L s v γ dA (cid:27) ≤ − C ˆ L s s − N − ( s − l ) N X i,j =1 u ij + N X j =1 N X i =1 ν i u ij ! v γ − dA + ˆ L s s − N − ( s − l ) (cid:18) v γ ρ + γδ N +1 ( v γ +1 ρ ) (cid:19) dA | {z } ( I ) + ˆ L s s − N − ρ ( X, ν ) L N +1 v γ dA | {z } ( I ) − D s (cid:26) ˆ L s ( X, ν ) L N +1 l − N − v γ dA (cid:27) . (3.9)Let us analyze the terms ( I ) and ( I ). For ( I ), we write(3.10) ( I ) = 18 s − N − ˆ L s ( s − l ) v γ ρ dA + γ s − N − ˆ L s ( s − l ) δ N +1 ( v γ +1 ρ ) dA. Now, by definition of L s one has s − l ≤ s in L s , then, thanks to H¨older’s inequality, takinginto account that v ≤ dA = vdx and ρ ∈ L qloc ( R N ), we have18 s − N − ˆ L s ( s − l ) v γ ρ dA ≤ s − N +1 (cid:18) ˆ L s v γ qq − dA (cid:19) q − q (cid:18) ˆ L s | ρ | q dA (cid:19) q ≤ s − N +1+ N q − q (cid:18) s − N ˆ L s v γ dA (cid:19) q − q (cid:18) ˆ L s | ρ | q dA (cid:19) q ≤ | ρ | q,K s s − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q , (3.11) SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 13 where β := Nq . For the second integral term in ( I ), using the first relation in (2.8) with f = s − l (notice that f = 0 on ∂L s ), g = v γ +1 ρ , H = ρ , we deduce that γ s − N − ˆ L s ( s − l ) δ N +1 ( v γ +1 ρ ) dA = γ s − N − ˆ L s ( s − l ) v γ ρ dA | {z } ( I ) + γs − N − ˆ L s v γ +1 ρlδ N +1 l dA | {z } ( I ) . (3.12)We notice that ( I ) is of the same form as the first term in the right-hand side of (3.10) and thus,arguing as in (3.11), we deduce that(3.13) ( I ) ≤ γ | ρ | q,K s s − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q . For ( I ), recalling that ∂∂x N +1 l = 0, we deduce from the first relation of (2.6) that vδ N +1 l = N X i =1 ν i ∂∂x i l, which in particular implies(3.14) | vδ N +1 l | ≤ k δl k L N +1 . Then, using (3.14), recalling that l ≤ s in L s , exploiting (2.13) and the elementary inequality(3.15) ( a + b ) α ≤ a α + b α ∀ a, b > , ∀ α ∈ (0 , , with α = 1 /
2, we obtain( I ) = γs − N − ˆ L s v γ ρlvδ N +1 l dA ≤ γs − N ˆ L s v γ | ρ |k δl k L N +1 dA ≤ γs − N ˆ L s v γ | ρ | dA + γs − N ˆ L s v γ | ρ | l − | ( X, ν ) L N +1 | dA (3.16)For the first integral in the right-hand side of (3.16), applying H¨older’s inequality and arguing asin (3.11) we readily obtain(3.17) γs − N ˆ L s v γ | ρ | dA ≤ γs − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q | ρ | q,K s . For the second integral in the right-hand side of (3.16), using the elementary inequality(3.18) | ρ || ( X, ν ) L N +1 | ≤ l ρ + 12 l − ( X, ν ) L N +1 , taking into account that l ≤ s in L s and (3.13), we infer that γs − N ˆ L s v γ | ρ | l − | ( X, ν ) L N +1 | dA ≤ γ s − N ˆ L s v γ l | ρ | dA + γ s − N ˆ L s v γ l − ( X, ν ) L N +1 dA ≤ γ | ρ | q,K s s − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q + γ s − N ˆ L s v γ l − ( X, ν ) L N +1 dA. (3.19)This completes the study of the term ( I ). For ( I ), using again (3.18) and arguing as in (3.19),we deduce that s − N − ˆ L s ρ ( X, ν ) L N +1 v γ dA ≤ | ρ | q,K s s − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q + 12 s − N − ˆ L s l − ( X, ν ) L N +1 v γ dA, (3.20)Summing up, from (3.9), taking into account (3.10)–(3.13), (3.16)–(3.19), (3.20) and recallingthat 0 < γ < N and N ≥
3, we have obtained the estimate D s (cid:26) s − N ˆ L s v γ dA (cid:27) ≤ − C ˆ L s s − N − ( s − l ) N X i,j =1 u ij + N X j =1 N X i =1 ν i u ij ! v γ − dA + 32 | ρ | q,K s s − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q + 12 | ρ | q,K s s − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q + 14 s − N ˆ L s l − ( X, ν ) L N +1 v γ dA | {z } ( I ) + 12 s − N − ˆ L s l − ( X, ν ) L N +1 v γ dA | {z } ( I ) − D s (cid:26) ˆ L s ( X, ν ) L N +1 l − N − v γ dA (cid:27) . (3.21)We now study the terms ( I ) and ( I ). For ( I ), by elementary algebraic manipulations and usingFederer’s coarea formula (2.15), taking into account that l = s on ∂L s , we write14 s − N ˆ L s l − ( X, ν ) L N +1 v γ dA = − N − D s (cid:20) s − N +1 ˆ L s l − ( X, ν ) L N +1 v γ dA (cid:21) + 14( N − s − N +1 D s (cid:20) ˆ L s l − ( X, ν ) L N +1 v γ dA (cid:21) = − N − D s (cid:20) s − N +1 ˆ L s l − ( X, ν ) L N +1 v γ dA (cid:21) + 14( N − ˆ ∂L s l − N − ( X, ν ) L N +1 v γ k δl k − L N +1 dµ = − N − D s (cid:20) s − N +1 ˆ L s l − ( X, ν ) L N +1 v γ dA (cid:21) + 14( N − D s (cid:20) ˆ L s l − N − ( X, ν ) L N +1 v γ dA (cid:21) . (3.22) SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 15
Treating ( I ) similarly yields12 s − N − ˆ L s l − ( X, ν ) L N +1 v γ dA = − N D s (cid:20) s − N ˆ L s l − ( X, ν ) L N +1 v γ dA (cid:21) + 12 N D s (cid:20) ˆ L s l − N − ( X, ν ) L N +1 v γ dA (cid:21) . (3.23)Therefore, from (3.21)–(3.23), we readily obtain the estimate(3.24) D s (cid:26) s − N ˆ L s v γ dA (cid:27) ≤ − C ˆ L s s − N − ( s − l ) N X i,j =1 u ij + N X j =1 N X i =1 ν i u ij ! v γ − dA + 32 | ρ | q,K s s − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q + 12 | ρ | q,K s s − β (cid:18) s − N ˆ L s v γ dA (cid:19) q − q − D s (cid:26) ˆ L s (cid:20) s − N N l − + s − N +1 N − l − + (cid:18) − N − N ( N − (cid:19) l − N − (cid:21) ( X, ν ) L N +1 v γ dA (cid:27) . Setting the notations ψ ( s ) := s − N ˆ L s v γ dA and F ( s ) := ˆ L s (cid:20) s − N N l − + s − N +1 N − l − + (cid:18) − N − N ( N − (cid:19) l − N − (cid:21) ( X, ν ) L N +1 v γ dA, we can rewrite (3.24) as(3.25) ψ ′ ( s ) − | ρ | q,K s s − β ( ψ ( s )) q − q − | ρ | q,K s s − β ( ψ ( s )) q − q ≤ − C ˆ L s s − N − ( s − l ) N X i,j =1 u ij + N X j =1 N X i =1 ν i u ij ! v γ − dA − F ′ ( s ) . Now recall that for every h ∈ L ( M ), we have(3.26) ˆ L s h dA = ˆ M U ( s − l ) h dA, where U : R → [0 ,
1] is the Heaviside (Unit Step) function. Moreover, since v γ is continuous at x = 0 and l approximates the geodesic distance in M for | x | small, we have(3.27) ψ ( s ) → ω N v γ (0) , as s → + , see [5, Section 2]). In addition, since M is C and strictly spacelike, we have ( X, ν ) L N +1 = O ( | x | )as | x | →
0. We hence infer that lim s → + ˆ L s ( X, ν ) L N +1 l − N − dA = 0 , lim s → + ˆ L s ( X, ν ) L N +1 s − N l − dA = 0 , lim s → + ˆ L s ( X, ν ) L N +1 s − N +1 l − dA = 0 . (3.28) We now integrate (3.25) from 0 to t ∈ (0 , R ]. Exploiting (3.26)–(3.28), Fubini’s theorem andobserving that | ρ | q,K s ≤ | ρ | q,K R , for any s ∈ (0 , R ], we obtain(3.29) ψ ( t ) − ω N v γ (0) − | ρ | q,K R ˆ t s − β ψ q − q ( s ) ds − | ρ | q,K R ˆ t s − β ψ q − q ( s ) ds ≤ − C ˆ L t S t ( l ) N X i,j =1 u ij + N X j =1 N X i =1 ν i u ij ! v γ − dA − F ( t ) , where S t ( l ) := ˆ t s − N − ( s − l ) U ( s − l ) ds. A direct computation shows that(3.30) S t ( l ) = 1 N ( N − l − N + 12 N l t − N − N − t − N > < l < t . In addition, since N ≥
3, we see that F ( t ) ≥ t ∈ (0 , R ]. We deduce fromthese facts that the right-hand side of (3.29) is non-positive and thus we deduce the estimate(3.31) ψ ( t ) ≤ ω N + 32 | ρ | q,K R ˆ t s − β ψ q − q ( s ) ds + 12 | ρ | q,K R ˆ t s − β ψ q − q ( s ) ds ∀ t ∈ (0 , R ] . Moreover, in view of (3.27), ψ has a continuous extension on [0 , R ] by setting ψ (0) := ω N v γ (0)and since v ≤
1, the inequality (3.31) holds for all t ∈ [0 , R ]. Therefore, applying Lemma 3.1 with T = R , C = ω N , C = | ρ | q,K R , C = | ρ | q,K R , we conclude that(3.32) ψ ( t ) ≤ ω q N + 3 ω − q N | ρ | q,K R q (2 − β ) t − β + | ρ | q,K R q (1 − β ) t − β q ∀ t ∈ [0 , R ] . Now, observe that it easily follows from (3.30) that for 0 < l < R/ S R ( l ) > c ( N ) R − N , where c ( N ) > N only. Then, from (3.29) and (3.32), recalling that F ( R ) ≥ ν i = u i v and dA = vdx , we get ω N v γ (0) ≥ R − N ˆ K R v γ +1 dx + c ( N ) R − N ˆ K R/ v γ − N X i,j =1 u ij + v − N X j =1 N X i =1 u ij u i ! dx − | ρ | q,K R ˆ R s − β ω q N + 3 ω − q N | ρ | q,K R q (2 − β ) s − β + | ρ | q,K R q (1 − β ) s − β q − ds − | ρ | q,K R ˆ R s − β ω q N + 3 ω − q N | ρ | q,K R q (2 − β ) s − β + | ρ | q,K R q (1 − β ) s − β q − ds. (3.33)Up to the translation argument pointed out at the beginning of the proof and noticing that(3.34) v − N X j =1 N X i =1 u ij u i ! = v − N X j =1 (cid:18) ∂∂x j |∇ u | (cid:19) = 14 N X j =1 " ∂∂x j (cid:0) −|∇ u | (cid:1)p − |∇ u | = |∇ v | , we finaly deduce (1.7) from (3.33). (cid:3) SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 17 Haarala’s gradient estimate
In this section, we first give a detailed version of Haarala’s proof of [32, Theorem 3.5], whichis reminiscent of [5, Theorem 3.5], and whose key final ingredient is a Moser iteration technique.Then, under further assumptions, we provide an estimate in Proposition 4.2 on the L q norm ofthe function ν defined by(4.1) ν ( x ) := 1 p − |∇ u ( x ) | x ∈ R N , where u ∈ C ( R N ) is a given strictly spacelike function. As seen in Section 2, the quantity ν has ageometrical interpretation as ν = ν N +1 = v − is the ( N + 1)-th component in L N +1 of the Gaussmap associated to M , see (2.2), and it plays a crucial role to deduce gradient estimates for (1.2)(see [5, 14, 29]). In terms of ν we have the following. Theorem 4.1 (Haarala [32]) . Assume N ≥ , q > N and ρ ∈ C ( R N ) . Let u ∈ C ( R N ) be astrictly spacelike classical solution to (1.2) and let ν be the function defined by (4.1) . There existsa positive constant c depending only on N and q such that for any x ∈ R N , R > it holds, sup B R/ ( x ) ν ≤ c − ˆ B R ( x ) ν q dx ! Nq ( q − N ) + R Nq − N − ˆ B R ( x ) | ρ | q dx ! Nq ( q − N ) − ˆ B R ( x ) ν q dx ! q . Proof of Theorem 4.1.
Let F ∈ C (cid:16) B (0) (cid:17) ∩ C ∞ ( B (0)) be defined by F ( y ) := 1 − p − | y | , y ∈ B (0). By direct computation we easily check that(4.2) ∇ F ( y ) = y p − | y | , D F ( y ) = 1 p − | y | I N + 1 p (1 − | y | ) y ⊗ y, for any y = ( y , . . . , y N ) ∈ B (0), where I N is the identity matrix in R N , y ⊗ y is the symmetricmatrix whose ( i, j ) entry is given by y i y j , for i, j = 1 , . . . , N . In particular since u is strictlyspacelike, and recalling that ν is given by (4.1), we have(4.3) D F ( ∇ u ) = ν I N + ν ∇ u ⊗ ∇ u. One then easily checks that D F ( ∇ u ) is positive definite for any x ∈ R N as(4.4) ν | ξ | ≤ ( D F ( ∇ u ) ξ, ξ ) R N ≤ ν | ξ | ∀ ξ ∈ R N . Now, since u is of class C , strictly spacelike and satisfies classically (1.2), we can differentiate(1.2) with respect to the variable x i . Taking into account (4.2), we easily check that − div (cid:0) D F ( ∇ u ) ∇ u i (cid:1) = ρ i in R N , in the classical sense, for any i = 1 , . . . , N . In particular, we have(4.5) ˆ R N ( D F ( ∇ u ) ∇ u i , ∇ ψ ) R N dx = ˆ R N ρ i ψ dx ∀ ψ ∈ C c ( R N ) . Taking ψ = ϕu i as test function in (4.5), where ϕ ∈ C c ( R N ) is such that ϕ ≥
0, and summingover all indices i = 1 , . . . , N , we infer that(4.6) N X i =1 ˆ R N ( D F ( ∇ u ) ∇ u i , ∇ ϕ ) R N u i dx = ˆ R N ϕ ( ∇ u, ∇ ρ ) R N dx − N X i =1 ˆ R N ( D F ( ∇ u ) ∇ u i , ∇ u i ) R N ϕ dx. Now, thanks to (4.4) and since ϕ ≥
0, we deduce that( D F ( ∇ u ) ∇ u i , ∇ u i ) R N ϕ ≥ , for any i = 1 , . . . , N , and therefore (4.6) leads to the inequality(4.7) N X i =1 ˆ R N ( D F ( ∇ u ) ∇ u i , ∇ ϕ ) R N u i dx ≤ ˆ R N ϕ ( ∇ u, ∇ ρ ) R N dx. Observe that ϕ ( ∇ u, ∇ ρ ) R N = ν ( ∇ u, ∇ ( ρϕν − )) R N − ( ∇ u, ∇ ϕ ) R N ρ + ν − ( ∇ u, ∇ ν ) R N ϕρ. Integrating by parts and taking into account (1.2), we obtain(4.8) ˆ R N ϕ ( ∇ u, ∇ ρ ) R N dx = ˆ R N ν − ϕρ dx − ˆ R N ( ∇ u, ∇ ϕ ) R N ρ dx + ˆ R N ν − ( ∇ u, ∇ ν ) R N ϕρ dx. Let A := ν − D F ( ∇ u ). Since ∇ ν = ν P Ni =1 u i ∇ u i , it follows from (4.7) and (4.8) that(4.9) ˆ R N ( A ∇ ν, ∇ ϕ ) R N dx ≤ ˆ R N ν − ϕρ dx − ˆ R N ( ∇ u, ∇ ϕ ) R N ρ dx + ˆ R N ν − ( ∇ u, ∇ ν ) R N ϕρ dx, for any ϕ ∈ C c ( R N ) such that ϕ ≥
0. Let p > N , η ∈ C c ( R N ) and plug ϕ = η ν p − in (4.9). As ∇ ϕ = 2 ην p − ∇ η + ( p − η ν p − ∇ ν, this yields( p − ˆ R N η ν p − ( A ∇ ν, ∇ ν ) R N dx ≤ ˆ R N η ν p − ρ dx − ( p − ˆ R N η ν p − ( ∇ u, ∇ ν ) R N ρ dx − ˆ R N ην p − ( ∇ u, ∇ η ) R N ρ dx − ˆ R N ην p − ( A ∇ ν, ∇ η ) R N dx (4.10)By definition, we have A ( x ) = (1 −|∇ u ( x ) | ) I N + ∇ u ( x ) ⊗∇ u ( x ), x ∈ R N , and therefore A ∇ u = ∇ u .Moreover, in view of (4.4), for any x ∈ R N , it holds(4.11) ν ( x ) − | ξ | ≤ ( A ( x ) ξ, ξ ) R N ≤ | ξ | ∀ ξ ∈ R N . In particular, for any x ∈ R N the map ( ξ, ζ ) ( A ( x ) ξ, ζ ) R N is a inner product on R N , and by theCauchy-Schwarz inequality we have(4.12) | ( Aξ, ζ ) R N | ≤ p ( Aξ, ξ ) R N p ( Aζ, ζ ) R N ∀ ξ, ζ ∈ R N . Replacing ∇ u by A ∇ u , we infer from (4.12) that | ( ∇ u, ∇ ν ) R N | = | ( A ∇ u, ∇ ν ) R N | ≤ p ( A ∇ u, ∇ u ) R N p ( A ∇ ν, ∇ ν ) R N = |∇ u | p ( A ∇ ν, ∇ ν ) R N . Using the same trick to estimate | ( ∇ u, ∇ η ) R N | , exploiting the fact that |∇ u | ≤ | ( A ∇ ν, ∇ η ) R N | , we deduce from (4.10) that( p − ˆ R N η ν p − ( A ∇ ν, ∇ ν ) R N dx ≤ ˆ R N η ν p − ρ dx +( p − ˆ R N η ν p − | ρ | p ( A ∇ ν, ∇ ν ) R N dx +2 ˆ R N | η | ν p − | ρ | p ( A ∇ η, ∇ η ) R N dx +2 ˆ R N | η | ν p − p ( A ∇ ν, ∇ ν ) R N p ( A ∇ η, ∇ η ) R N dx. (4.13) SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 19
We now use Young inequality (3.7) to estimate the last three terms in (4.13). For the reader’sconvenience, and since it will be crucial to keep track of the dependence on the parameter p in theconstants, we give the full details of the estimates. First, applying (3.7) with ǫ = 1 / ( p −
1) we get( p − η ν p − | ρ | p ( A ∇ ν, ∇ ν ) R N ≤ ( p − ( p − η ν p − ρ + p − η ν p − ( A ∇ ν, ∇ ν ) R N , while 2 | η | ν p − | ρ | p ( A ∇ η, ∇ η ) R N = (cid:16)p p − | η | ν p − | ρ | (cid:17) (cid:18) √ p − ν p p ( A ∇ η, ∇ η ) R N (cid:19) ≤ ( p − η ν p − ρ + 1 p − ν p ( A ∇ η, ∇ η ) R N . Similarly, we infer that2 ν p p ( A ∇ η, ∇ η ) R N | η | ν p − p ( A ∇ ν, ∇ ν ) R N ≤ p − ν p ( A ∇ η, ∇ η ) R N + p − η ν p − ( A ∇ ν, ∇ ν ) R N . Summing up, we deduce from (4.13) that( p − ˆ R N η ν p − ( A ∇ ν, ∇ ν ) R N dx ≤ p − ˆ R N ν p ( A ∇ η, ∇ η ) R N dx + (cid:20) ( p − + ( p − p − (cid:21) ˆ R N η ν p − ρ dx. (4.14)From (4.14) and since p > N , N ≥
3, then by elementary algebraic considerations we get(4.15) ˆ R N η ν p − ( A ∇ ν, ∇ ν ) R N dx ≤ p − ˆ R N ν p ( A ∇ η, ∇ η ) R N dx + 5 ˆ R N η ν p − ρ dx. Now, applying Sobolev’s inequality to φ := ην p − ∈ C c ( R N ), taking into account that (cid:12)(cid:12)(cid:12) ∇ ( ην p − ) (cid:12)(cid:12)(cid:12) ≤
12 ( p − η ν p − |∇ ν | + 2 ν p − |∇ η | , and exploiting (4.11) we deduce that(4.16) (cid:12)(cid:12)(cid:12) ην p − (cid:12)(cid:12)(cid:12) ∗ ≤ c ( p − ˆ R N η ν p − ( A ∇ ν, ∇ ν ) R N dx + 2 c ˆ R N ν p − |∇ η | dx, where c is the Sobolev constant for the embedding of D , ( R N ) ֒ → L ∗ ( R N ), in particular c depends only on N . Finally, combining (4.16) with (4.15), exploiting again (4.11) and taking intoaccount that ν ≥
1, we infer that(4.17) (cid:12)(cid:12)(cid:12) ην p − (cid:12)(cid:12)(cid:12) ∗ ≤ c ˆ R N ν p |∇ η | dx + 3 c ( p − ˆ R N η ν p − ρ dx. Take k ∈ N . Set R k := ( + k ) R , and B k := B R k ( x ). Clearly, by definition, we have B = B R ( x ), B k +1 ⊂ B k and we easily check that R k R k +1 = k +22 k +1 , m N ( B k ) m N ( B k +1 ) ≤ C , for some positive constantdepending only on N , where m N ( · ) denotes the Lebesgue measure in R N . Let η k ∈ C c ( R N ) besuch that 0 ≤ η k ≤ η k ≡ B k +1 , supp ( η k ) ⊂ B k and(4.18) |∇ η k | ≤ R k − R k +1 = 2 k +2 R − . Taking η = η k in (4.17), the estimate (4.18) leads to(4.19) − ˆ B k +1 ν ( p − NN − dx ! N − N ≤ k +2 c − ˆ B k ν p dx + c ( p − R − ˆ B k ν p − ρ dx. where c is a positive constant depending only on N .Now, fix once for all a number q > N and let p ≥ q . Using H¨older’s inequality to estimate theintegrals in the right-hand side of (4.19) we get that(4.20) − ˆ B k ν p dx ≤ (cid:18) − ˆ B k ν q dx (cid:19) q (cid:18) − ˆ B k ν ( p − qq − dx (cid:19) q − q − ˆ B k ν p − ρ dx ≤ (cid:18) − ˆ B k | ρ | q dx (cid:19) q (cid:18) − ˆ B k ν ( p − qq − dx (cid:19) q − q . Then from (4.19), (4.20), and observing that | B R ( x ) || B k | ≤ | B R ( x ) || B R/ ( x ) | ≤ N for all k ∈ N (which is usedto bound the first integral means in the right-hand sides of (4.20) with the same integral meanson the whole B R ( x )), then, after elementary computations (taking into account that 4 k ≥ p − ≥ − ˆ B k +1 ν ( p − NN − dx ! N − N ( p − ≤ kp − g p − ( p − p − (cid:18) − ˆ B k ν ( p − qq − dx (cid:19) ( q − p − q , where we have set g := 4 Nq c − ˆ B R ( x ) ν q dx ! q + c Nq R − ˆ B R ( x ) | ρ | q dx ! q . In particular notice that g is independent of p and k . Now, setting α := N ( q − N − q we define p k := α k ( q −
2) + 2 , Φ k := (cid:18) − ˆ B k ν α k q dx (cid:19) αkq , k ∈ N . Clearly p = q , in addition, since q > N , we check that α > p k ≥ q for any k ∈ N , p k → + ∞ , as k → + ∞ . Taking p = p k in (4.21) then by construction, and after easy algebraicmanipulations, we infer that for any k ∈ N Φ k +1 ≤ q − kα − k (( q − √ g ) q − α − k Φ k . Then, arguing by induction and passing to the limit as k → + ∞ we deduce that(4.22) lim k → + ∞ Φ k ≤ q − P ∞ j =1 jα − j (( q − √ g ) q − P ∞ j =0 α − j Φ . Now, since α = N ( q − N − q > q − ∞ X j =0 α − j = Nq − N , q − ∞ X j =1 jα − j = qN ( N − q − N ) . Finally, as lim k → + ∞ Φ k = | ν | ∞ ,B R/ ( x ) , Φ = (cid:16) − ´ B R ( x ) ν q dx (cid:17) q , from (4.22), (4.23), recalling thedefinition of g and after elementary computations we deduce thatsup B R/ ( x ) ν ≤ c − ˆ B R ( x ) ν q dx ! Nq ( q − N ) + R Nq − N − ˆ B R ( x ) | ρ | q dx ! Nq ( q − N ) − ˆ B R ( x ) ν q dx ! q , for some positive constant c depending only on N and q . The proof is complete. (cid:3) SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 21
When ρ is in L q ( R N ) with q > N , if the solution u is globally bounded and there exists ν > ν − ν ) + has compact support, then an L q -estimate for ( ν − ν ) + holds. This is thecontent of the next result which is [32, Theorem 3.6] revisited. Proposition 4.2.
Assume N ≥ , q > N , ρ ∈ L q ( R N ) ∩ C ( R N ) , and let u ∈ C ( R N ) ∩ L ∞ ( R N ) be a strictly spacelike classical solution to (1.2) . Assume that there exists a number ν > suchthat ( ν − ν ) + has compact support. There exists a positive constant c depending only on q and ν such that (4.24) | ( ν − ν ) + | q ≤ c | u | ∞ | ρ | q . Proof.
Since u is a classical solution to (1.2), a standard integration by parts gives(4.25) ˆ R N ν ( ∇ u, ∇ ϕ ) R N dx = ˆ R N ϕρ dx ∀ ϕ ∈ C c ( R N ) . In particular, thanks to the assumptions and by density we infer that (4.25) holds true also forany ϕ ∈ W ,p (Ω), where Ω is a given smooth bounded domain of R N and p ≥
1. Hence, since weare assuming that ( ν − ν ) + has compact support, by taking ϕ = uν − ( ν − ν ) q + in (4.25) we get ˆ R N ( ν − ν ) q + |∇ u | dx = ˆ R N uν − ( ν − ν ) q + ( ∇ u, ∇ ν ) R N − q ˆ R N u ( ν − ν ) q − ( ∇ u, ∇ ν ) R N dx + ˆ R N uν − ( ν − ν ) q + ρ dx. (4.26)From (4.26), recalling A = ν − D F ( ∇ u ) and the fact that A ∇ u = ∇ u , using the ellipticity boundsin (4.12), and observing that ν − ( ν − ν ) + ≤
1, we obtain(4.27) ˆ R N ( ν − ν ) q + |∇ u | dx ≤ q ˆ R N | u | ( ν − ν ) q − |∇ u | p ( A ∇ ν, ∇ ν ) R N dx + ˆ R N | u | ( ν − ν ) q − | ρ | dx. Now, observe that saying ν ≥ ν is equivalent to |∇ u | ≥ γ ν := 1 − ν . As ν > γ ν ∈ (0 ,
1) so that Young inequality (3.7) implies | u | ( ν − ν ) q − |∇ u | p ( A ∇ ν, ∇ ν ) R N ≤ | u | ( ν − ν ) q − ( A ∇ ν, ∇ ν ) R N + 14 ( ν − ν ) q + |∇ u | , | u | ( ν − ν ) q − | ρ | ≤ γ ν | u | ( ν − ν ) q − | ρ | + γ ν ν − ν ) q + . (4.28)Then, from (4.27), (4.28), regrouping the terms, taking into account that |∇ u | ≥ γ ν and recallingthat u ∈ L ∞ ( R N ), we deduce that ˆ R N ( ν − ν ) q + dx ≤ q | u | ∞ γ ν ˆ R N ( ν − ν ) q − ( A ∇ ν, ∇ ν ) R N dx + 2 | u | ∞ γ ν ˆ R N ( ν − ν ) q − ρ dx. (4.29)In order to estimate the first integral in the right-hand side of (4.29), since ∇ ν = ν P Ni =1 u i ∇ u i ,we deduce from (4.7) and the definition of A that(4.30) ˆ R N ( A ∇ ν, ∇ ϕ ) R N dx ≤ ˆ R N ϕ ( ∇ u, ∇ ρ ) R N dx ∀ ϕ ∈ C c ( R N ) . As pointed out at the beginning of the proof, by a density argument and since ( ν − ν ) + hascompact support, we can take ϕ = ( ν − ν ) q − as test function in (4.30). Then, arguing as in(4.14) (with η ∈ C c ( R N ), η ≡ ν − ν ) + ) we get(4.31) ˆ R N ( ν − ν ) q − ( A ∇ ν, ∇ ν ) R N dx ≤ ˆ R N ( ν − ν ) q − ρ dx. Plugging now (4.31) into (4.29) and taking into account that γ ν ∈ (0 , ˆ R N ( ν − ν ) q + dx ≤ q | u | ∞ γ ν ˆ R N ( ν − ν ) q − ρ dx. Finally, it follows from H¨older’s inequality that(4.33) ˆ R N ( ν − ν ) q − ρ dx ≤ (cid:18) ˆ R N ( ν − ν ) q + dx (cid:19) q − q (cid:18) ˆ R N | ρ | q dx (cid:19) q , and thus from (4.32) and (4.33), we readily deduce the claimed inequality (4.24) with c = q qγ ν . (cid:3) Gradient estimates for the Born-Infeld equation
In this section we provide new gradient estimates for strictly spacelike weak solutions of theBorn-Infeld equation ( BI ) with ρ ∈ L qloc ( R N ) ∩ L m ( R N ) (or ρ ∈ L q ( R N ) ∩ L m ( R N )), q > N , m ∈ [1 , ∗ ], where we recall that 2 ∗ denotes the conjugate H¨older exponent of the critical Sobolevexponent. One could have considered ρ ∈ L qloc ( R N ) ∩ X ∗ with q > N arguing merely with littlechanges, but for simplicity we choose ρ ∈ L m ( R N ) with m ∈ [1 , ∗ ]. We begin with a couple ofpreliminary technical results about the imbedding properties of the convex set X defined in (1.6). Lemma 5.1.
Let N ≥ and m ∈ (1 , ∗ ] . There exists a positive constant c depending only on N and m such that (5.1) | φ | m ′ ≤ c |∇ φ | ( N +1) m − NmN ∀ φ ∈ X , where m ′ = mm − is the conjugate exponent of m .Proof. Take φ ∈ X . Since |∇ φ | ≤ R N , we have |∇ φ | ∈ L k ( R N ) for all k ≥
2. Therefore, bySobolev’s inequality, we infer that φ ∈ L k ∗ ( R N ) for k ∈ [2 , N [ and(5.2) | φ | k ∗ ≤ c |∇ φ | k ≤ c |∇ φ | /k , where c = c ( N, k ) is the Sobolev constant for the embedding of D ,k ( R N ) ֒ → L k ∗ ( R N ). Observethat given m ∈ ]1 , ∗ ], it is always possible to find k ∈ [2 , N [ such that k ∗ = m ′ , i.e. k = mN ( N +1) m − N .Using (5.2), we then deduce (5.1) with c ( N, m ) := c (cid:16) N, mN ( N +1) m − N (cid:17) . (cid:3) Lemma 5.2.
Given N ≥ and s > N , there exist two positive constants c , c , both dependingonly on N and s , such that (5.3) | φ | ∞ ≤ c (cid:18) c |∇ φ | N + sNs + |∇ φ | s (cid:19) ∀ φ ∈ X . Proof.
By Morrey-Sobolev’s inequality we know that there exists a positive constant c dependingonly on N and s , such that(5.4) | φ | ∞ ≤ c k φ k W ,s ( R N ) , ∀ φ ∈ W ,s ( R N ) , SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 23 where k φ k W ,s ( R N ) = | φ | s + |∇ φ | s is the standard norm in W ,s ( R N ). Now, on the one hand wehave, as in (5.2), |∇ φ | s ≤ |∇ φ | /s , ∀ φ ∈ X and on the other hand, since s > N we can always find k ∈ ] N , N [ such that k ∗ = s , namely k = NsN + s , to estimate | φ | s = | φ | k ∗ by help of (5.2). Then (5.4) yields | φ | ∞ ≤ c ( c |∇ φ | N + sNs + |∇ φ | /s ) , ∀ φ ∈ X , where c = c ( N, NsN + s ) is the Sobolev constant for the embedding of D , NsN + s ( R N ) ֒ → L s ( R N ), andthis leads to (5.3) with c ( N, s ) := c ( N, NsN + s ). (cid:3) We now state and prove the first gradient estimate of this section (namely Proposition 1.2 inSection 1). We begin by studying the case ρ ∈ L m ( R N ) with m ∈ (1 , ∗ ]. Proposition 5.3.
Assume N ≥ , q > N and ρ ∈ L m ( R N ) ∩ C ( R N ) with m ∈ (1 , ∗ ] . Let u ∈ X ∩ C ( R N ) be a strictly spacelike weak solution of ( BI ) . There exist γ ∈ (0 , N ) dependingonly on N , c > depending only on N and m , such that for any R > , x ∈ R N , (5.5) (1 − |∇ u | ) γ ( x ) ≥ ω N ω N + cR − N | ρ | NmN − m m γ +1 − P (cid:16) | ρ | q,K R ( x ) R q − Nq (cid:17) , where P ( k ) := (cid:18) (cid:19) q − q ω − q N q − N k + (cid:18) (cid:19) q − q − q ω − q N ( q − N ) q − ( q − k q − + 2 q − ω − N ( q − N ) q − (cid:18)
32 + 1 q − N (cid:19) k q + 2 q − q ω − q N q − N k + (cid:18) (cid:19) q − q − q ω − q N ( q − N ) q (2 q − k q − , for k ≥ . (5.6) Proof.
By Lemma 5.2 we know that u ∈ L ∞ ( R N ) since u ∈ X (see also [12, Lemma 2.1]). Also, itis easily seen that u is a classical solution of ( BI ). Then from Theorem 1.1 and Lemma 2.8-(i),we infer that there exists a positive constant γ ∈ (0 , N ) depending only on N such that for anyfixed x ∈ R N , R >
0, one has ω N v γ ( x ) ≥ R − N ˆ K R ( x ) v γ +1 dx − | ρ | q,K R ( x ) ˆ R s − β ω q N + 3 ω − q N | ρ | q,K R ( x ) q (2 − β ) s − β + | ρ | q,K R ( x ) q (1 − β ) s − β q − ds − | ρ | q,K R ( x ) ˆ R s − β ω q N + 3 ω − q N | ρ | q,K R ( x ) q (2 − β ) s − β + | ρ | q,K R ( x ) q (1 − β ) s − β q − ds, where v = p − |∇ u | and β = Nq . Notice that we discarded the second integral in the right-handside of (1.7) because it is non-negative.Using repeatedly the elementary convexity inequality(5.7) ( a + b ) α ≤ α − ( a α + b α ) ∀ a, b > , ∀ α ≥ , and taking into account that 2 − β = 2 q − Nq , we deduce from elementary computations that ω N v γ ( x ) ≥ R − N ˆ K R ( x ) v γ +1 dx − (cid:18) (cid:19) q − q ω − q N q − N | ρ | q,K R ( x ) R − β − (cid:18) (cid:19) q − q − q ω − q N ( q − N ) q − ( q − | ρ | q − q,K R ( x ) R (2 − β )( q − −
32 2 q − ( q − N ) q − | ρ | qq,K R ( x ) R q (1 − β ) − q − q ω − q N q − N | ρ | q,K R ( x ) R − β − (cid:18) (cid:19) q − q − q ω − q N ( q − N ) q (2 q − | ρ | q − q,K R ( x ) R (1 − β )(2 q − − q − ( q − N ) q | ρ | qq,K R ( x ) R q (1 − β ) . (5.8)For the first integral in the right-hand side of (5.8), we claim that there exists a positive constant c = c ( N, m ) depending only on N , m such that(5.9) R − N ˆ K R ( x ) v γ +1 dx ≥ ω γ +2 N (cid:18) ω N + cR − N | ρ | NmN − m m (cid:19) γ +1 . Since ρ ∈ L m ( R N ) ⊂ X ∗ , [12, Proposition 2.6] implies that u coincides with the unique minimizerof the energy I ρ in X . Hence, by the minimality of I ρ ( u ), and since for 0 ∈ X one has I ρ (0) = 0,we deduce that I ρ ( u ) ≤
0. Using this, the elementary inequality t ≤ − √ − t , for all t ∈ [0 , |∇ u | ˆ R N (cid:16) − p − |∇ u | (cid:17) dx h ρ, u i ≤ | ρ | m | u | m ′ ≤ c | ρ | m |∇ u | ( N +1) m − NmN where m ′ = mm − is the conjugate exponent of m and c = c ( N, m ) is a positive constant dependingonly on N , m . Then, the inequality (5.10) implies that(5.11) |∇ u | ≤ (2 c | ρ | m ) mN N − m ) . Now, taking u ∈ X in the weak formulation of ( BI ), we get(5.12) ˆ R N |∇ u | p − |∇ u | dx = ˆ R N ρu dx. Since ρ ∈ L m ( R N ) and thanks to H¨older’s inequality, Lemma 5.1 and (5.11), we deduce that(5.13) (cid:12)(cid:12)(cid:12)(cid:12) ˆ R N ρu dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ρ | m | u | m ′ ≤ ( N +1) m − NN − m ( c | ρ | m ) m ∗ , where m ∗ = NmN − m , and thus(5.14) ˆ R N |∇ u | p − |∇ u | dx ≤ c | ρ | m ∗ m , with c := 2 ( N +1) m − NN − m c m ∗ (which is a positive constant depending only on N , m ). Starting fromthe identity ˆ B R ( x ) p − |∇ u | dx = ˆ B R ( x ) p − |∇ u | dx + ˆ B R ( x ) |∇ u | p − |∇ u | dx, SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 25 and using the fact that p − |∇ u | ≤
1, we deduce from (5.14) that(5.15) ˆ B R ( x ) p − |∇ u | dx ≤ ω N R N + c | ρ | m ∗ m . Finally, since u is strictly spacelike, we can write, recalling that v = p − |∇ u | > R N , ω N R N ≤ ˆ B R ( x ) v γ +1 dx ! γ +2 ˆ B R ( x ) v − dx ! γ +1 γ +2 ≤ ˆ B R ( x ) v γ +1 dx ! γ +2 (cid:16) ω N R N + c | ρ | m ∗ m (cid:17) γ +1 γ +2 , (5.16)where we used H¨older’s inequality and (5.15). Since, B R ( x ) ⊂ K R ( x ) by definition, the claim(5.9) now follows from (5.16).At last, plugging (5.9) inside (5.8) leads to the estimate v γ ( x ) ≥ ω γ +1 N (cid:18) ω N + cR − N | ρ | NmN − m m (cid:19) γ +1 − (cid:18) (cid:19) q − q ω − q N q − N | ρ | q,K R ( x ) R − β ) − (cid:18) (cid:19) q − q − q ω − q N ( q − N ) q − ( q − | ρ | q − q,K R ( x ) R (2 − β )( q − − q − ω − N ( q − N ) q − (cid:18)
32 + 1 q − N (cid:19) | ρ | qq,K R ( x ) R q (1 − β ) − q − q ω − q N q − N | ρ | q,K R ( x ) R − β − (cid:18) (cid:19) q − q − q ω − q N ( q − N ) q (2 q − | ρ | q − q,K R ( x ) R (1 − β )(2 q − . (5.17)As 1 − β = q − Nq , the estimate (5.17) yields (5.5) with P defined by (5.6). (cid:3) If we assume furthermore that ρ ∈ L q ( R N ) in the statement of Proposition 5.3, then it is possibleto deduce a global bound independent of the radius and the base point. More precisely we canabsorb them in a free parameter k >
0. This is the content of the next result.
Proposition 5.4.
Let N ≥ , ρ ∈ L q ( R N ) ∩ L m ( R N ) ∩ C ( R N ) , with q > N , m ∈ (1 , ∗ ] . Assumelet u ∈ X ∩ C ( R N ) is a strictly spacelike weak solution of ( BI ) . There exist two positive constants γ ∈ (0 , N ) depending only on N , c depending only on N and m , such that for any k > one has (5.18) inf R N (1 − |∇ u | ) γ ≥ ω γ +1 N (cid:18) ω N + c | ρ | NmN − m m | ρ | Nqq − N q k − Nqq − N (cid:19) γ +1 − P ( k ) , where P ( k ) is given by (5.6) Proof.
The proof is identical to that of Proposition 5.3 up to (5.17), with the only caution thathere | ρ | q,K R ( x ) is replaced by | ρ | q . Set k := R − β | ρ | q and observe that R − N = k − Nqq − N | ρ | Nqq − N q .Since R > k > v γ ( x ) ≥ ω γ +1 N (cid:18) ω N + c | ρ | NmN − m m | ρ | Nqq − N q k − Nqq − N (cid:19) γ +1 − P ( k ) , where P ( k ) is given by (5.6). Finally, since the constants γ , c , as well as P ( k ) are independentof x ∈ R N (in particular the right-hand side of (5.19) does not depend on x ) then we readilyobtain (5.18). (cid:3) Remark 5.5.
We point out that (5.18) is invariant under the transformation u ˜ u t , ρ ¯ ρ t ,where ˜ u t ( x ) := tu (cid:0) xt (cid:1) , ¯ ρ t ( x ) := t ρ (cid:0) xt (cid:1) and t is a given positive number. Such a transformationis naturally associated to Problem ( BI ) in the sense that u ∈ X if and only if ˜ u t ∈ X and u is asolution to ( BI ) if and only if ˜ u t is a solution of the same problem with datum ¯ ρ t . Remark 5.6.
A crucial property used in the proofs of Proposition 5.3, Proposition 5.4 (see (5.10) , (5.13) ) and in that of Lemma 5.1 is the continuous embedding of the space X ֒ → L p ∗ ( R N ) , for all p ∈ [2 , N ) . This reflects on the fact that for u ∈ X , ρ ∈ L m ( R N ) , m ≥ , the duality pairing (5.20) h ρ, u i = ˆ R N ρu dx makes sense only for m ∈ [1 , ∗ ] . We now analyze the case m = 1. The counterpart of Proposition 5.3 is the following Proposition 5.7.
Let N ≥ , q > N and ρ ∈ L ( R N ) ∩ C ( R N ) . Assume u ∈ X ∩ C ( R N ) is astrictly spacelike weak solution of ( BI ) . For any given s > N there exist three positive constants γ , c , c with γ ∈ (0 , N ) depending only on N , c , c depending only on N and s , such that forany x ∈ R N , R > one has (1 − |∇ u | ) γ ( x ) ≥ ω γ +1 N (cid:18) ω N + c R − N | ρ | ss − (cid:19) γ +1 − P (cid:16) | ρ | q,K R ( x ) R q − Nq (cid:17) if k u k X ≤ , ω γ +1 N ω N + c R − N | ρ | NsNs − ( N + s )1 ! γ +1 − P (cid:16) | ρ | q,K R ( x ) R q − Nq (cid:17) if k u k X > , where ω N is the volume of the unit ball in R N and P = P ( k ) is given by (5.6) .Proof. In this proof, any c i , i ∈ N denotes a positive constant depending only on N and s . Theargument is similar to the one used in the proof of Proposition 5.3, but some adjustments areneeded for Claim 5.9. To this end, fixing s > N , arguing as in (5.10) and exploiting Lemma 5.2,it follows that(5.21) 12 |∇ u | ≤ | ρ | | u | ∞ ≤ c | ρ | (cid:18) c |∇ u | N + sNs + |∇ u | s (cid:19) . Assume that k u k X ≤ |∇ u | ≤ N + sNs > s we get(5.22) |∇ u | ≤ c | ρ | s s − . Combining (5.3), (5.22) we have(5.23) | u | ∞ ≤ c | ρ | s − . From (5.23) we deduce that(5.24) (cid:12)(cid:12)(cid:12)(cid:12) ˆ R N ρu dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ρ | | u | ∞ ≤ c | ρ | ss − , and thus, arguing as in (5.12)–(5.16), we infer that for any x ∈ R N , R >
0, it holds that R − N ˆ K R ( x ) v γ +1 dx ≥ ω γ +2 N (cid:16) ω N + c R − N | ρ | ss − (cid:17) γ +1 . SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 27
This gives the counterpart of Claim 5.9 and this completes the proof of (5.27) when k u k X ≤ k u k X > N + sNs > s weinfer that(5.25) |∇ u | ≤ c | ρ | Ns Ns − N − s ) . Hence, as in the previous case, from (5.3) and (5.25) one concludes that(5.26) | u | ∞ ≤ c | ρ | N + sNs − N − s , and arguing as in (5.24) and (5.12)–(5.16), it follows that R − N ˆ K R ( x ) v γ +1 dx ≥ ω γ +2 N (cid:18) ω N + c R − N | ρ | NsNs − N − s (cid:19) γ +1 , for any x ∈ R N , R > (cid:3) Finally, the counterpart of Proposition 5.4 for m = 1 goes as follows. Proposition 5.8.
Let N ≥ , ρ ∈ L q ( R N ) ∩ L ( R N ) ∩ C ( R N ) , with q > N . Assume u ∈X ∩ C ( R N ) is a strictly spacelike weak solution of ( BI ) . For any given s > N there exist threepositive constants γ , c , c with γ ∈ (0 , N ) depending only on N , c , c depending only on N and s , such that for any k > one has (5.27) inf R N (1 − |∇ u | ) γ ≥ ω γ +1 N ω N + c | ρ | ss − | ρ | Nqq − Nq k − Nqq − N ! γ +1 − P ( k ) if k u k X ≤ , ω γ +1 N ω N + c | ρ | NsNs − ( N + s )1 | ρ | Nqq − Nq k − Nqq − N ! γ +1 − P ( k ) if k u k X > , where ω N is the volume of the unit ball in R N and P ( k ) is given by (5.6) .Proof. The proof is identical to that of Proposition 5.4 with the necessary adjustments pointedout in the proof of Proposition 5.7. (cid:3)
Remark 5.9.
We observe that, differently from (5.18) , the condition in (5.27) is not invariantunder the natural transformation u ˜ u t , ρ ¯ ρ t described in Remark 5.5. More precisely, forany t > one has that k ˜ u t k X = t N k ˜ u k X , | ¯ ρ t | ss − | ¯ ρ t | Nqq − N q = t N − ss − | ρ | ss − | ρ | Nqq − N q and | ¯ ρ t | NsNs − ( N + s ) | ¯ ρ t | Nqq − N q = t N Ns − ( N + s ) | ρ | NsNs − ( N + s ) | ρ | Nqq − N q . W ,qloc regularity of the minimizer of the Born-Infeld energy In this final section, we prove Theorem 1.4.
Proof of Theorem 1.4.
Let N ≥ ρ ∈ L q ( R N ) ∩ L m ( R N ), with q > N , m ∈ [1 , ∗ ] and let u ρ ∈ X be the unique minimizer of I ρ . Let ( ρ n ) n ⊂ C ∞ c ( R N ) be a standard sequence of mollifications of ρ and consider the sequence ( u n ) n ⊂ X , where u n is the unique minimizer of I ρ n for any n ∈ N .From [12, Theorem 1.5, Remark 3.4] we know that for any n ∈ N the function u n belongs to C ∞ ( R N ), it is strictly spacelike in R N , and it is a weak (and also classical) solution to ( BI ) withdatum ρ n . Moreover, since ρ n → ρ in L m ( R N ), as n → + ∞ , then from [12, Theorem 5.3, Corollary n → + ∞ we have that ( u n ) n converges to u ρ weaklyin X and uniformly in R N .We fix once for all two positive numbers s > N and R >
0. We divide the proof in successivesteps and even if this is tedious, we keep track of the constants and on the parameters on whichthey depend.
Step 1: there exists a positive constant C depending only on N , m , s and | ρ | m such that(6.1) | u n | ∞ ≤ C ∀ n ∈ N . We first notice that Step 1 holds true, as u n → u ρ uniformly in R N . Nevertheless, for the sakeof completeness, we give a direct proof of (6.1). Indeed, if m ∈ (1 , ∗ ] then thanks to (5.11) andsince | ρ n | m ≤ | ρ | m , which is a standard property of the mollified sequence, we get(6.2) |∇ u n | ≤ c | ρ | mN N − m ) m , for some positive constant c depending only on N and m . Fixing s > N , then from Lemma 5.2and (6.2) we readily infer that(6.3) | u n | ∞ ≤ c | ρ | m ( N + s )( N − m ) s m + c | ρ | mN ( N − m ) s m , where c , c are positive constants depending only on N , s , and we are done. Similarly, when m = 1, then, fixing s > N , taking into account (5.22)–(5.23) (if |∇ u n | ≤ |∇ u n | > | u n | ∞ ≤ max (cid:26) c | ρ | s − ; c | ρ | N + sNs − N − s (cid:27) , where c , c are positive constants depending only on N and s . At the end, from (6.3) (or (6.4) if m = 1) we obtain (6.1), with a positive constant C depending only on s , | ρ | m , N and m . Step 2: there exists a constant δ ∈ [0 ,
1) depending only on R , N , m , | ρ | m (and s if m = 1)such that for any n ∈ N there exists a bounded open neighborhood Λ n of supp ( ρ n ) ( supp ( ρ n ) isthe support of ρ n ) such that(6.5) |∇ u n | ∞ , Λ ∁ n ≤ δ. Let R be the positive number chosen at the beginning of the proof and denote by K nR ( x ) theprojection of the Lorentz ball associated to u n , centred at x ∈ R N and of radius R . Thanksto Lemma 2.8-(i) and (6.1), for any x ∈ R N , for any n ∈ N we have K nR ( x ) ⊂ B R ′ ( x ), with R ′ = q R + 4 C . Hence, denoting by dist ( x, supp ( ρ n )) the Euclidean distance between x ∈ R N and supp ( ρ n ), and setting Λ n := { x ∈ R N ; dist ( x, supp ( ρ n )) < R ′ } , we infer that K nR ( x ) ∩ supp ( ρ n ) = ∅ for any x ∈ Λ ∁ n , for all n ∈ N . In particular, | ρ n | q,K nR ( x ) = 0and, if m ∈ (1 , ∗ ], then, by Proposition 5.3 and taking into account that | ρ n | m ≤ | ρ | m , we deducethat(6.6) |∇ u n ( x ) | ≤ − ω γ +1) γ N (cid:18) ω N + c R − N | ρ | NmN − m m (cid:19) γ +1) γ ∀ x ∈ Λ ∁ n , ∀ n ∈ N , SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 29 where γ ∈ (0 , N ) depends on N only, and c depends on N , m only. Similarly, when m = 1, usingProposition 5.7 we deduce that for any x ∈ Λ ∁ n , for any n ∈ N it holds(6.7) |∇ u n ( x ) | ≤ − min ω γ +1) γ N (cid:16) ω N + c R − N | ρ | ss − (cid:17) γ +1) γ , ω γ +1) γ N (cid:18) ω N + c R − N | ρ | NsNs − ( N + s ) (cid:19) γ +1) γ . for some positive constants c , c depending only on N and s , and γ is as in the previous case.Finally, Step 2 follows immediately from (6.6) (or (6.7) if m = 1), with δ ∈ [0 ,
1) dependingonly on N , m , | ρ | m and R (or on N , s , | ρ | and R , if m = 1). Step 3: there exists a constant θ ∈ [0 ,
1) independent of n such that(6.8) |∇ u n | ∞ ≤ θ ∀ n ∈ N . Let us set ν n ( x ) := 1 p − |∇ u n ( x ) | , x ∈ R N , and observe that proving Step 3 is equivalent to finding a constant ¯ ν ≥ n suchthat sup R N ν n ≤ ¯ ν ∀ n ∈ N . To this end, let δ ∈ [0 ,
1) be the number given by Step 2 and fix a number ν > √ − δ ≥
1. Thenby definition and Step 2 it follows that ( ν n − ν ) + has compact support for all n ∈ N , where ( · ) + stands for the positive part of a function. This means that the assumptions of Proposition 4.2 aresatisfied by u n , for all n ∈ N , with a uniform value of ν independent of n ∈ N . Hence, by (4.24),(6.1), and recalling that | ρ n | q ≤ | ρ | q , we deduce that(6.9) | ( ν n − ν ) + | q ≤ c | ρ | q ∀ n ∈ N , for some positive constant c depending only on ν , s , N , q , m , | ρ | m . In particular, since ν n ≤ ( ν n − ν ) + + ν in R N ,then, exploiting the elementary inequalities (5.7), (3.15) and (6.9) we infer that for any x ∈ R N ,for any R > | ν n | q,B R ( x ) ≤ q − q (cid:18) c | ρ | q + ν ω q N R Nq (cid:19) ∀ n ∈ N . Then(6.10) − ˆ B R ( x ) ν qn dx ! q ≤ q − q c − ˆ B R ( x ) | ρ | q dx ! q + ν ∀ n ∈ N . Finally, thanks to Theorem 4.1, we get the estimate(6.11)sup B R/ ( x ) ν n ≤ c − ˆ B R ( x ) ν qn dx ! Nq ( q − N ) + R Nq − N − ˆ B R ( x ) | ρ | q dx ! Nq ( q − N ) − ˆ B R ( x ) ν qn dx ! q , where c is a positive constant depending only on N and q and x ∈ R N is arbitrary. Hence,combining (6.10), (6.11), we infer that(6.12) sup B R/ ( x ) ν n ≤ ν ∀ n ∈ N , for some constant ν ≥ ν , s , N , q , m , | ρ | m , | ρ | q and R . Therefore, as ¯ ν isindependent of x and x ∈ R N is arbitrary, then (6.12) yieldssup R N ν n ≤ ν ∀ n ∈ N . As pointed out before, this is equivalent to (6.8) and he proof of Step 3 is complete.
Step 4: there exists a number α ∈ (0 ,
1) independent of n such that for any bounded domainΩ ⊂ R N there exists a positive constant C independent of n , such that(6.13) k u n k C ,α (Ω) ≤ C ∀ n ∈ N . Let τ ∈ (0 ,
1) and let η τ ∈ C ∞ ([0 , + ∞ )) be such that rη τ ∈ C ∞ ([0 , + ∞ )), r η τ ( r ) r isincreasing and satisfying η τ ( r ) r = r for r < − τ, − τ for r > − τ. Let us consider the function a τ : R N → R N defined by(6.14) a τ ( z ) := z p − η τ ( | z | ) | z | . Clearly a τ ∈ C ( R N , R N ) and by a straightforward computation we check that a τ satisfies thegrowth and ellipticity conditions of [37, (1.2)], with p = 2, s = 0, with ellipticity constant equal toone and for some constant L > τ (see [13, Proof of Theorem 1.6, Step 6] forthe details). Namely, denoting by ∂a τ the Jacobian matrix of a τ , then exploiting the definition of a τ and the properties of η τ we see that(6.15) | a τ ( z ) | + | ∂a τ ( z ) || z | ≤ L | z | , ∀ z ∈ R N , for some constant L > τ , and(6.16) ( ∂a τ ( z ) λ, λ ) ≥ | λ | , ∀ z ∈ R N , λ ∈ R N . Now, fixing τ ∈ (0 , − θ ), where θ ∈ [0 ,
1) is the number given by Step 3, it follows byconstruction that η τ ( |∇ u n | ) |∇ u n | = |∇ u n | for all n ∈ N . Hence ( u n ) n is a sequence of smoothsolutions to − div ( a τ ( ∇ u n )) = ρ n in R N . and thus, as a τ depends only on the gradient variable z , we can apply [37, Theorem 1.4] with ω ≡ ω ) and s = 0. To this end, let α M ∈ (0 ,
1) be theuniversal maximal regularity exponent associated to the homogeneous equation − div ( a τ ( ∇ u )) = 0 . In view of (6.15), (6.16) it follows that α M depends only on N , τ (see [37, Section 1.1, (1.21)]). Inparticular α M is independent of n ∈ N . Now, fix α ∈ (0 , min { α M , − Nq } ) and a bounded domainΩ ⊂ R N . Let Ω ⊂ R N be a bounded domain such that Ω ⊂⊂ Ω . We deduce from [37, Theorem1.4] that, for any ball B r ( x ) ⊂ Ω and for any x, y ∈ B r/ ( x ),(6.17) |∇ u n ( x ) −∇ u n ( y ) | ≤ c (cid:2) I ρ n − α ( x, r ) + I ρ n − α ( y, r ) (cid:3) | x − y | α + c − ˆ B r ( x ) |∇ u n | dx · (cid:18) | x − y | r (cid:19) α , SHARP GRADIENT ESTIMATE FOR THE PRESCRIBED MEAN CURVATURE EQUATION 31 where c is a positive constant depending only on N , L , the ellipticity constant of a τ (which isequal to one, see (6.16)), diam (Ω ), and where(6.18) I ρ n − α ( x, r ) = ˆ r ´ B t ( x ) | ρ n ( ξ ) | dξt N − (1 − α ) dtt is the truncated Riesz potential (see [37, Section 1.1]). Since | ρ n | q ≤ | ρ | q , applying H¨older’sinequality, we obtain for any x ∈ B r/ ( x ), ˆ B t ( x ) | ρ n ( ξ ) | dξ ≤ ( ω N t ) N ( q − q | ρ n | q,B t ( x ) ≤ c t N ( q − q | ρ | q , where c is a positive constant depending only on N , q , and we deduce from the definition that(6.19) I ρ n − α ( x, r ) ≤ c | ρ | q ˆ r t − α − Nq dt ∀ n ∈ N , for any x ∈ B r/ ( x ). Our choice of α implies α < − Nq so that the integral in the right-handside of (6.19) is finite, with a value depending only on N , q , α and r . This proves that for any x ∈ B r/ ( x )(6.20) I ρ n − α ( x, r ) ≤ c | ρ | q ∀ n ∈ N , where c is a positive constant depending only on N , q , α and r .At the end, from (6.17), (6.20), taking into account that |∇ u n | ≤ u n is uniformlybounded (as proved in Step 1), we conclude that for any ball B r ( x ) ⊂ Ω , one has(6.21) k u n k C ,α ( B r/ ( x )) ≤ c ∀ ∈ N , where c is a positive constant depending only on N , L , α , q , diam (Ω ), | ρ | q , r . Finally, sinceby construction Ω ⊂⊂ Ω then (6.21) and a standard finite subcovering argument readily imply(6.13), with a constant independent of n ∈ N . Step 5: for any bounded domain Ω ′ ⊂ R N there exists a positive constant C independent of n such that(6.22) k u n k W ,q (Ω ′ ) ≤ C ∀ n ∈ N . Indeed, fix two bounded domains Ω ′ ⊂⊂ Ω ⊂ R N . Observe that as u n is a smooth strictly spacelikeclassical solution to ( BI ), we can write the equation in non-divergence form. This yields(6.23) (1 − |∇ u n | )∆ u n + N X i,j =1 ( u n ) i ( u n ) j ( u n ) ij = − (1 − |∇ u n | ) ρ n . In particular, setting a nij := δ ij (1 − |∇ u n | ) + ( u n ) i ( u n ) j , and f n := − (1 − |∇ u n | ) ρ n , we see that u = u n is a strong solution in Ω of the equation N X i,j =1 a nij ( x ) u ij = f n . Clearly, by construction, one has that | f n | q ≤ | ρ n | q ≤ | ρ | q and | a nij ( x ) | ≤ i, j = 1 , . . . , N ,for any x ∈ Ω, n ∈ N . Moreover, if θ ∈ [0 ,
1) is the constant independent of n given by Step 3,then one can check that a nij ( x ) ξ i ξ j ≥ (1 − θ ) | ξ | ∀ ξ ∈ R N ∀ x ∈ Ω . In addition, by construction, thanks to Step 4 and exploiting the elementary properties of H¨olderianfunctions (namely, if g , g ∈ C ,α (Ω) then g g ∈ C ,α (Ω)) we infer that k a nij k C ,α (Ω) ≤ c , forsome constant c > n , where α ∈ (0 ,
1) is the number given by Step 4. Inparticular, this means that we have a uniform control on the moduli of continuity of the coefficients a nij in Ω, with respect to n ∈ N . Therefore, by standard elliptic regularity theory (see [31, Theorem9.11]) we deduce that(6.24) k u n k W ,q (Ω ′ ) ≤ c ( | u n | q, Ω + | f n | q, Ω ) , where c is a positive constant depending only on N , q , θ , Ω ′ , Ω, and the moduli of continuity ofthe coefficients a nij in Ω ′ . Now, from the proof of [31, Theorem 9.11] and since k a nij k C ,α (Ω) ≤ c ,we deduce that c is independent of n ∈ N .At the end, from (6.24), thanks to Step 1 and since | f n | q ≤ | ρ | q we readily obtain (6.22) with aconstant independent of n ∈ N . The proof of Step 5 is complete. Conclusion: given a smooth bounded domain Ω ′ ⊂ R N , from Step 5 it follows that ( u n ) n isa bounded sequence in W ,q (Ω ′ ) and thus, up to a subsequence, u n ⇀ ¯ u in W ,q (Ω ′ ), for some¯ u ∈ W ,q (Ω ′ ), and as q > N , by the Rellich-Kondrachov’s theorem, up to a further subsequence,it follows that u n → ¯ u in C (Ω ′ ). On the other hand, as u n → u ρ in compact subsets of R N (actually u n → u ρ uniformly in R N , as pointed out at the beginning of the proof) it follows that u ρ = ¯ u in Ω ′ , and thus u ρ ∈ W ,q (Ω ′ ). In addition, since u n → u ρ in C (Ω ′ ) then it also followsthat |∇ u ρ | ≤ θ , where θ ∈ [0 ,
1) is the number given by Step 3, which is independent of n and Ω ′ .In particular, from the arbitrariness of Ω ′ we infer that u ρ belongs to W ,qloc ( R N ) and that u ρ isstrictly spacelike in R N . To conclude the proof it remains to show that u ρ is the weak solution to( BI ). To this end let us fix ϕ ∈ C ∞ c ( R N ). Then, up to a subsequence, as ∇ u n → ∇ u ρ uniformlyin compact subsets of R N , and since √ −|∇ u n | ≤ √ − θ , where θ ∈ [0 ,
1) is the constant given byStep 3, we conclude that(6.25) lim n → + ∞ ˆ R N ∇ u n · ∇ ϕ p − |∇ u n | dx = ˆ R N ∇ u ρ · ∇ ϕ p − |∇ u ρ | dx. On the other hand, as u n is a weak solution to ( BI ) with datum ρ n , then, passing to the limit as n → + ∞ in the definition of weak solution (see (1.10)) and since ρ n → ρ in L m ( R N ), we have(6.26) lim n → + ∞ ˆ R N ∇ u n · ∇ ϕ p − |∇ u n | dx = lim n → + ∞ ˆ R N ρ n ϕ dx = ˆ R N ρϕ dx. Hence, equating (6.25) and (6.26) we get that u ρ satisfies the definition of weak solution for ( BI ),for any test function ϕ ∈ C ∞ c ( R N ). Finally, as |∇ u ρ | ≤ θ < R N then arguing by density weinfer that the same conclusion holds true for any test function ϕ ∈ X . Therefore u ρ is a weaksolution to ( BI ) and this completes the proof of Theorem 1.4. (cid:3) References [1] A.L. Albujer, L.J. Al´ıas,
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Entire Spacelike Hypersurfaces of Constant Mean Curvature in Minkowski Space , Invent. math , 39–56 (1982). 2 (Denis Bonheure) D´epartement de math´ematique, Universit´e Libre de Bruxelles, Campus de la Plaine- CP214 boulevard du Triomphe, 1050 Bruxelles, Belgium
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