Everywhere differentiability of absolute minimizers for locally strongly convex Hamiltonian H(p)C1,1(Rn) with n ≥ 3

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Abstract

Suppose that n3 and H(p)C1,1(Rn) is a locally strongly convex Hamiltonian. We obtain the everywhere differentiability of all absolute minimizers for H in any domain of Rn.

Introduction

Let n2 and suppose that HC0(Rn) is convex and coercive (i.e., liminfpH(p)=). In 1960's Aronsson initiated the study of minimization problems for the L-functionalFH(u,Ω)=esssupxΩH(Du(x))for any domain ΩRn and function uWloc1,(Ω); see [3], [4], [5], [6]. Given any domain ΩRn, by Aronsson a function uWloc1,(Ω) is called an absolute minimizer for H in Ω (write uAMH(Ω) for simplicity) ifFH(u,V)FH(v,V)whenever VΩvWloc1,(V)C(V) and u=v on V. It turns out that the absolute minimizer is the correct notion of minimizers for such L-functionals.

The existence of absolute minimizers for given continuous boundary in bounded domains was proved by Aronsson [5] for 12|p|2 and Barron-Jensen-Wang [9] for general H(p)C0(Rn); while their uniqueness was built up by Jensen [23] for 12|p|2 (see also [2], [8], [13]), and by Jensen-Wang-Yu [24] and Armstrong-Crandal-Julin-Smart [1] for H(p)C2(Rn) and H(p)C0(Rn), respectively, with H1(minH) having empty interior.

Moreover, if HC1(Rn) is convex and coercive, absolute minimizers coincide with viscosity solutions to the Aronsson equation (a highly degenerate nonlinear elliptic equation)AH(u):=i,j=1nHpi(Du)Hpj(Du)uxixj=0inΩ, see Jensen [23] for H(p)=12|p|2, and Crandall-Wang-Yu [15] and Yu [33] (and also [1], [9], [10], [20], [13]) in general. Here Hpi=Hpi for HC1(Rn), uxi=uxi for uC1(Rn), and uxixj=2uxixj for uC2(Rn). For the theory of viscosity solution see [14]. In the special case H(p)=12|p|2, the Aronsson equation (1.1) is the ∞-Laplace equationΔu:=i,j=1nuxiuxjuxixj=0inΩ and its viscosity solutions are called as ∞-harmonic functions. If HC0(Rn) but C1(Rn), we refer to [13], [1] for further discussions and related problems on the Euler-Lagrange equation for absolute minimizers.

The regularity of absolute minimizer is then the main issue in this field.

By Aronsson [7], ∞-harmonic functions are not necessarily C2-regular; indeed ∞-harmonic function x14/3x24/3 in whole Rn is not C2-regular. Such a function also leads to a well-known conjecture on the C1,1/3- and Wloc2,t-regularity with 1t<3/2 of ∞-harmonic functions. A seminar step towards this is made by Crandall-Evans [11], who obtained their linear approximation property. They [12] also proved that all bounded ∞-harmonic functions in whole Rn with n2 must be constant functions.

Next, when n=2, Savin [30] established their interior C1-regularity and then deduced the corresponding Liouville theorem, that is, all ∞-harmonic functions in whole plane with a linear growth at ∞ (that is, |u(x)|C(1+|x|) for all xR2) must be linear functions. Later, the interior C1,α-regularity for some 0<α1/3 was proved by Evans-Savin [17] and the boundary C1-regularity by Wang-Yu [32]. Recently, Koch-Zhang-Zhou [25] proved that |Du|αWloc1,2 for all α>0 and all ∞-harmonic functions u in planar domains, which is sharp as α0; also that the distributional determinant detD2udx is a nonnegative Radon measure.

Moreover, when n3, Evans-Smart [18], [19] obtained their everywhere differentiability; Miao-Wang-Zhou [26] and Hong-Zhao [22] independently observed an asymptotic Liouville property, that is, if u is a ∞-harmonic function in whole Rn with a linear growth at ∞, then limR1Ru(Rx)=ex locally uniformly for some vector e with |e|=DuL(Rn). But C1,C1,α-regularity and the corresponding Liouville theorem of ∞-harmonic functions are completely open.

On the other hand, if HC2(Rn) is locally strongly convex, Wang-Yu [31] obtained the linear approximation property of absolute minimizer, and when n=2, the C1-regularity and hence the corresponding Liouville theorem. As usual, HC0(Rn) is called to be locally strongly convex if for any convex subset U of Rn, there exists a constant λ>0 depending on U such that H(p)λ2|p|2 is convex in U.

Recently, under the assumptions that HC0(Rn) is convex and coercive, it was shown by Peng-Wang-Zhou [29] that H is not a constant in any line segment if and only if all absolute minimizers for H have the linear approximation property; moreover, when n=2, if and only if all absolute minimizers for H are C1-regular, and also if and only if the corresponding Liouville theorem holds. In [28], we proved that if HC2(R2) is locally strongly convex, then H(Du)αWloc1,2 for all α>12τH for all absolute minimizers u in planar domains, where 0<τH12 and τH=1/2 when HC2(R2); and also that the distributional determinant detD2udx is a nonnegative Radon measure. But, when n3, the everywhere differentiability, C1,C1,α-regularity and the Liouville theorem is not clear.

If n3 and HC1,1(Rn) is locally strongly convex, this paper aims to prove the following everywhere differentiability (Theorem 1.1 below) and asymptotic Liouville property (Theorem 1.2 below) of absolute minimizers.

Theorem 1.1

Suppose that n3 and HC1,1(Rn) is locally strongly convex. Let ΩRn be any domain. If uAMH(Ω), then u is differentiable everywhere in Ω.

Theorem 1.2

Suppose that n3 and HC1,1(Rn) is locally strongly convex. If uAMH(Rn) with a linear growth at, then there exists a unique vector e such thatH(e)=H(Du)L(Rn)andlimR1Ru(Rx)=exlocally uniformly in Rn.

When n3, it is unclear to us whether the assumption for H in Theorem 1.1, Theorem 1.2 can be relaxed to the weaker (and also necessary in some sense) assumption that HC0(Rn) is convex and coercive and is not a constant in any line segment. By [29], if HC0(Rn) is convex and coercive, and is constant in some line-segment, then both of Theorem 1.1, Theorem 1.2 are not necessarily true.

In particular, it would be interesting to prove the everywhere differentiability of absolute minimizer for lα-norm with 1<α<2. Recall that if 2<α<, then lα-norm belongs to C2(Rn) and is convex, and hence both of the conclusions of Theorem 1.1, Theorem 1.2 hold. If α=1 or ∞, the lα-norm will be constant in some line-segment.

By Remark 1.3, Remark 1.4 below, we only need to prove Theorem 1.1, Theorem 1.2 when HC0(Rn) satisfies

  • (H1)

    there exist 0<λΛ< such thatboth of H(p)λ2|p|2 and Λ2|p|2H(p) are convex in Rn.

  • (H2)

    H(0)=minpRnH(p)=0.

Remark 1.3

Suppose that HC0(Rn) is locally strongly convex. One may check by hand that HC1,1(Rn) if and only if for any convex subset U of Rn, there exists a constant Λ>0 depending on U such that Λ2|p|2H(p) is convex in U.

Remark 1.4

Suppose that HC1,1(Rn) is locally strongly convex.

(i) If uAMH(Ω) for some domain ΩRn, letting UΩ be arbitrary subdomain, we have k=DuL(U)<. Next, by [28, Lemma A.8], there exists a H˜C1,1(Rn) which is strongly convex in Rn such that H˜=H in B(0,k+1). Thus uAMH˜(U). The strong convexity of H˜ implies that there exists a p0Rn such that minpRnH˜(p)=H(p0). Set H¯(p)=H˜(p+p0)H˜(p0) for pRn. Then H¯ satisfies (H1)&(H2). Write u¯(x)=u(x)p0x for all xU. We have u¯AMH¯(U). Since u and u¯ have the same regularity in U, we only need to prove the everywhere differentiability of u¯ in U.

(ii) If uAMH(Rn) has a linear growth at ∞, then by [29] we have k:=DuL(Rn)<. Let u¯ and H¯ as above. Then u is linear if and only if u¯ is linear. So we only need to prove u¯ is linear.

Unless other specifying, we always assume that HC0(Rn) satisfies (H1)&(H2) below. Note that the geometric&variational approach used in dimension 2 (see Savin [30] and also [29], [31]) is not enough to prove Theorem 1.1, Theorem 1.2, since it includes a key planar topological argument. Moreover, since HC0(Rn) does not have Hilbert structure necessarily, it is not clear whether one can prove Theorem 1.1 by using the idea of Evans-Smart [18]—a PDE approach based on maximal principle (see also Remark 2.6 (ii)). But, in Section 2, we are able to prove Theorem 1.1, Theorem 1.2 by borrowing some idea of Evans-Smart [19]—a PDE approach based on an adjoint argument, and using the following crucial ingredients:

  • (a)

    The linear approximation property of any given absolute minimizer u for H as obtained in Fa-Wang-Zhou [29] and Wang-Yu [31] (see Lemmas 2.1&2.5).

  • (b)

    A stability result in [28] (see Lemma 2.2) which allows to approximate u via absolute minimizers uγ of a Hamiltonian Hγ, where Hγ is a smooth approximation of H and satisfies (H1)&(H2) with the same constants λ,Λ.

  • (c)

    A uniform approximation to uγ via smooth functions uγ,ϵ (see Theorem 2.3), which is an appropriate modification of Evans' approximation via e1ϵHγ-harmonic functions in [16]. The point is that none of k3-order derivatives of Hγ is involved in the linearization of the equation (2.2) for uγ,ϵ.

  • (d)

    An integral flatness estimate for uγ,ϵ (see Theorem 2.4).

Theorem 2.3 will be proved in Section 3. The novelty in the proof of Theorem 2.3 is that we use viscosity solutions to certain Hamilton-Jacobi equation as barrier functions to get a boundary regularity of uγ,ϵ and then conclude the uniform approximation of uγ,ϵ to uγ. The reason to use uγ,ϵ instead of e1ϵHγ-harmonic functions is that the linearization of e1ϵHγ-harmonic equation contains 3-order derivatives of Hγ; see Remark 2.6 (i) for details.

Theorem 2.4 will be proved in Section 5. To this end, we generalize in Section 4 the adjoint arguments of [19] to Hamiltonian Hγ and uγ,ϵ. Since none of k3-order derivatives of Hγ is involved in the equation for uγ,ϵ, all key estimates in Theorem 2.3 and Section 4 rely only on λ and Λ. This is indeed important to get Theorem 2.4. Moreover, since HC0(Rn) does not have Hilbert structure in general, some new ideas are needed to get Theorem 2.4 in Section 5; in particular, the test function used in the proof of flatness estimates in [19] is not enough to us, as another novelty we find a suitable test function and build up some related estimates.

Section snippets

Proofs of Theorems 1.1&1.2

Considering Remark 1.3, we always assume that HC0(Rn) satisfies (H1)&(H2). To prove Theorem 1.1, let Ω be any domain of Rn, and uAMH(Ω). We recall the following linear approximation property of u as established by [29].

Lemma 2.1

For any xΩ and any sequence {rj}jN which converges to 0, there exist a subsequence {rjk}kN and a vector e{rjk}kN such thatlimksupyB(0,1)|u(x+rjky)u(x)rjke{rjk}kNy|=0 andH(e{rjk}kN)=limr0H(Du)L(B(x,r)).

For each xΩ, denote by Du(x) the collection of all

Proof of Theorem 2.3

Let H, Hγ, u uγ and uγ,ϵ be as in Section 2. Note that Hγ satisfies (H1)&(H2) with the same λ and Λ. Since |Hpγ(p)|2Λ2|p|2 impliesϵ|ξ|2[Hpiγ(p)Hpjγ(p)+ϵδij]ξiξjΛ2(|p|2+1)|ξ|2ξRn, by a standard quasilinear elliptic theory (see [21]), there exists a unique smooth solution uγ,ϵC(U)C0(U) to (2.2). Theorem 2.3 (i) follows from the known maximum principle. We also note that by a standard argument, uγ,ϵuγ in C0(U) (that is Theorem 2.3 (iv)) follows from Theorem 2.3 (ii)&(iii), and the

A generalization of Evans-Smart' adjoint method

Let H, Hγ, u uγ and uγ,ϵ be as in Section 2. For convenience, we write Hγ as H, and uγ as u, uγ,ϵ as uϵ below. Let Lϵ be the linearized operator given in (3.1), and Lϵ be its dual operator, that is,Lϵ(v):=[Hpi(Duϵ)Hpj(Duϵ)v]xixj+2[Hpipl(Duϵ)uxixjϵHpj(Duϵ)v]xlϵΔv for any vC(U). Observe thatRnLϵ(v)(x)w(x)dx=Rnv(x)Lϵ(w)(x)dxv,wCc(U).

Fix a smooth domain VU. For each point x0V, we consider the adjoint problemLϵ(v)=δx0inV;v=0onV, where δx0 denotes the Dirac measure at x0. Equivalently,

Proof of Theorem 2.4

Let U=B(0,3) and V=B(0,2) in this section. Let H, Hγ, u uγ and uγ,ϵ be as in Section 2. For convenience, we write Hγ as H, and uγ as u, uγ,ϵ as uϵ below.

Note that the condition (2.4) and Theorem 2.3 imply thatsupU|uϵ|4 and supV|Duϵ|C(λ,Λ). Moreover, let Lϵ and Θϵ be given in Theorem 4.1. The condition (2.5) implies that Lemma 4.8 (ii) holds, that isVΘϵdxC(λ,Λ). The proof of Theorem 2.4 is then divided into 3 steps.

Step 1. We first show thatV[H(en)H(Duϵ)]+ΘϵdxC(λ,Λ)[δ+1ϵeμϵδ]. Here and

Acknowledgement

The authors would like to thank the referee for his valuable comments and suggestions, in particular, Remark 1.3, which allows us to rewrite the original assumption on H as an equivalent but simple one “HC1,1(Rn) is locally strongly convex” in the revision. Q. Miao (the corresponding author) would like to thank the supports of National Natural Science Foundation of China (No. 12001041). Peng Fa (the first author) and Yuan Zhou would like to thank the supports of National Natural Science

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