Everywhere differentiability of absolute minimizers for locally strongly convex Hamiltonian with n ≥ 3
Introduction
Let and suppose that is convex and coercive (i.e., ). In 1960's Aronsson initiated the study of minimization problems for the -functional see [3], [4], [5], [6]. Given any domain , by Aronsson a function is called an absolute minimizer for H in Ω (write for simplicity) if It turns out that the absolute minimizer is the correct notion of minimizers for such -functionals.
The existence of absolute minimizers for given continuous boundary in bounded domains was proved by Aronsson [5] for and Barron-Jensen-Wang [9] for general ; while their uniqueness was built up by Jensen [23] for (see also [2], [8], [13]), and by Jensen-Wang-Yu [24] and Armstrong-Crandal-Julin-Smart [1] for and , respectively, with having empty interior.
Moreover, if is convex and coercive, absolute minimizers coincide with viscosity solutions to the Aronsson equation (a highly degenerate nonlinear elliptic equation) see Jensen [23] for , and Crandall-Wang-Yu [15] and Yu [33] (and also [1], [9], [10], [20], [13]) in general. Here for , for , and for . For the theory of viscosity solution see [14]. In the special case , the Aronsson equation (1.1) is the ∞-Laplace equation and its viscosity solutions are called as ∞-harmonic functions. If but , 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 -regular; indeed ∞-harmonic function in whole is not -regular. Such a function also leads to a well-known conjecture on the - and -regularity with 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 with must be constant functions.
Next, when , Savin [30] established their interior -regularity and then deduced the corresponding Liouville theorem, that is, all ∞-harmonic functions in whole plane with a linear growth at ∞ (that is, for all ) must be linear functions. Later, the interior -regularity for some was proved by Evans-Savin [17] and the boundary -regularity by Wang-Yu [32]. Recently, Koch-Zhang-Zhou [25] proved that for all and all ∞-harmonic functions u in planar domains, which is sharp as ; also that the distributional determinant is a nonnegative Radon measure.
Moreover, when , 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 with a linear growth at ∞, then locally uniformly for some vector e with . But -regularity and the corresponding Liouville theorem of ∞-harmonic functions are completely open.
On the other hand, if is locally strongly convex, Wang-Yu [31] obtained the linear approximation property of absolute minimizer, and when , the -regularity and hence the corresponding Liouville theorem. As usual, is called to be locally strongly convex if for any convex subset U of , there exists a constant depending on U such that .
Recently, under the assumptions that 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 , if and only if all absolute minimizers for H are -regular, and also if and only if the corresponding Liouville theorem holds. In [28], we proved that if is locally strongly convex, then for all for all absolute minimizers u in planar domains, where and when ; and also that the distributional determinant is a nonnegative Radon measure. But, when , the everywhere differentiability, -regularity and the Liouville theorem is not clear.
If and 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 and is locally strongly convex. Let be any domain. If , then u is differentiable everywhere in Ω.
Theorem 1.2 Suppose that and is locally strongly convex. If with a linear growth at ∞, then there exists a unique vector e such that
When , 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 is convex and coercive and is not a constant in any line segment. By [29], if 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 -norm with . Recall that if , then -norm belongs to and is convex, and hence both of the conclusions of Theorem 1.1, Theorem 1.2 hold. If or ∞, the -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 satisfies
- (H1)
there exist such that
- (H2)
.
Remark 1.3 Suppose that is locally strongly convex. One may check by hand that if and only if for any convex subset U of , there exists a constant depending on U such that is convex in U.
Remark 1.4 Suppose that is locally strongly convex. (i) If for some domain , letting be arbitrary subdomain, we have . Next, by [28, Lemma A.8], there exists a which is strongly convex in such that in . Thus . The strong convexity of implies that there exists a such that . Set for . Then satisfies (H1)&(H2). Write for all . We have . Since u and have the same regularity in U, we only need to prove the everywhere differentiability of in U. (ii) If has a linear growth at ∞, then by [29] we have . Let and as above. Then u is linear if and only if is linear. So we only need to prove is linear.
Unless other specifying, we always assume that 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 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 of a Hamiltonian , where is a smooth approximation of H and satisfies (H1)&(H2) with the same constants .
- (c)
A uniform approximation to via smooth functions (see Theorem 2.3), which is an appropriate modification of Evans' approximation via -harmonic functions in [16]. The point is that none of -order derivatives of is involved in the linearization of the equation (2.2) for .
- (d)
An integral flatness estimate for (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 and then conclude the uniform approximation of to . The reason to use instead of -harmonic functions is that the linearization of -harmonic equation contains 3-order derivatives of ; 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 and . Since none of -order derivatives of is involved in the equation for , 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 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 satisfies (H1)&(H2). To prove Theorem 1.1, let Ω be any domain of , and . We recall the following linear approximation property of u as established by [29]. Lemma 2.1 For any and any sequence which converges to 0, there exist a subsequence and a vector such that and
Proof of Theorem 2.3
Let H, , u and be as in Section 2. Note that satisfies (H1)&(H2) with the same λ and Λ. Since implies by a standard quasilinear elliptic theory (see [21]), there exists a unique smooth solution to (2.2). Theorem 2.3 (i) follows from the known maximum principle. We also note that by a standard argument, in (that is Theorem 2.3 (iv)) follows from Theorem 2.3 (ii)&(iii), and the
A generalization of Evans-Smart' adjoint method
Let H, , u and be as in Section 2. For convenience, we write as H, and as u, as below. Let be the linearized operator given in (3.1), and be its dual operator, that is, for any . Observe that
Fix a smooth domain . For each point , we consider the adjoint problem where denotes the Dirac measure at . Equivalently,
Proof of Theorem 2.4
Let and in this section. Let H, , u and be as in Section 2. For convenience, we write as H, and as u, as below.
Note that the condition (2.4) and Theorem 2.3 imply that Moreover, let and be given in Theorem 4.1. The condition (2.5) implies that Lemma 4.8 (ii) holds, that is The proof of Theorem 2.4 is then divided into 3 steps.
Step 1. We first show that 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 “ 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
References (33)
- et al.
An asymptotic sharp Sobolev regularity for planar infinity harmonic functions
J. Math. Pures Appl. (9)
(2019) - et al.
-regularity of the Aronsson equation in
Ann. Inst. Henri Poincaré, Anal. Non Linéaire
(2008) - et al.
Convexity criteria and uniqueness of absolutely minimizing functions
Arch. Ration. Mech. Anal.
(2011) - et al.
An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions
Calc. Var. Partial Differ. Equ.
(2010) Minimization problems for the functional
Ark. Mat.
(1965)Minimization problems for the functional . II
Ark. Mat.
(1966)Extension of functions satisfying Lipschitz conditions
Ark. Mat.
(1967)Minimization problems for the functional . III
Ark. Mat.
(1969)On certain singular solutions of the partial differential equation
Manuscr. Math.
(1984)- et al.
Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term
Commun. Partial Differ. Equ.
(2001)
The Euler equation and absolute minimizers of functionals
Arch. Ration. Mech. Anal.
An efficient derivation of the Aronsson equation
Arch. Ration. Mech. Anal.
A remark on infinity harmonic functions
Proceedings of the USA–Chile Workshop on Nonlinear Analysis
Electron. J. Differ. Equ. Conf.
Optimal Lipschitz extensions and the infinity Laplacian
Calc. Var. Partial Differ. Equ.
Uniqueness of ∞-harmonic functions and the eikonal equation
Commun. Partial Differ. Equ.
User's guide to viscosity solutions of second order partial differential equations
Bull. Am. Math. Soc. (N.S.)
Cited by (2)
Regularity of absolute minimizers for continuous convex Hamiltonians
2021, Journal of Differential EquationsA QUANTITATIVE SOBOLEV REGULARITY FOR ABSOLUTE MINIMIZERS INVOLVING HAMILTONIAN H(p) IN PLANE
2022, SIAM Journal on Mathematical Analysis