A STRICTLY CONVEX SOBOLEV FUNCTION WITH
NULL HESSIAN MINORS
ZHUOMIN LIU AND JAN MALÝ
Abstract. Given 1 ≤ p < k ≤ n, we construct a strictly convex
function f ∈ W 2,p ((0, 1)n ) with α-Hölder continuous derivative
for any 0 < α < 1 such that rank ∇2 f < k almost everywhere
in (0, 1)n . In particular, the mapping F = ∇f is an example of a
W 1,p homeomorphism whose differential has rank strictly less than
k almost everywhere in the unit cube. This Sobolev regularity is
sharp in the sense that if g ∈ W 2,p , p ≥ k, and rank ∇2 g < k
a.e., then g cannot be strictly convex on any open portion of the
domain.
1. Introduction
Let f be a smooth function on Rn . Suppose that the Hessian determinant of f vanishes everywhere. Then f has a special behavior: any
point on the space lies on a straight line along which ∇f is constant.
Following [21], we call such a function 1-developable. (See Remark 1
below for a discussion of this terminology.) In particular, the graph of
the function is a ruled surface, this means that it is a union of straight
lines. (This property is much weaker: it requires only the derivative
in direction of the associated line to be constant.) From geometrical
point of view, the reason for developability is that the graph of f has
vanishing Gaussian curvature. We are interested in the question what
the limiting regularity for such a behavior is.
The main result of this paper is an example on the “negative side”
of the problem. However, our motivation comes from the positive sidethe developability of functions with vanishing Hessian determinant or
Hessian minors. These functions have been studied in connection with
isometric immersions. An isometric immersion of co-dimension k is a
mapping from a domain of Rn into Rn+k , 1 ≤ k < n, which preserves
length of any curve that passes through each point in its domain and
angles between any two of them. In particular, most of the rigidity results for isometric immersions, rather than being treated directly, are
obtained from the rigidity of functions with vanishing Hessian determinant. This is because, under the W 2,1 regularity assumption ([24], see
The first author was supported by the ERC CZ grant LL1203 of the Czech Ministry of Education. The second author is supported by the grant GA ČR P201/1508218S of the Czech Science Foundation. The authors would like to thank Robert
Jerrard and Reza Pakzad for many useful discussions.
1
2
ZHUOMIN LIU AND JAN MALÝ
also [21] for the W 2,2 case), an isometric immersion of co-dimension k
satisfies the property that the Hessian of each of its coordinate function
has rank less than or equal to k (or to be consistent with our language
later on, strictly less than k + 1) almost everywhere. Therefore, we can
expect images of such isometric immersions to be developable. Actually, Hartman and Nirenberg [16] proved that C 2 isometric immersions
have their images as developable surfaces. Even stronger result has
been obtained by Pogorelov [31], [30], here the key assumption is that
the image under the gradient map has vanishing Lebesgue measure.
Such a kind of regularity fails dramatically if we assume, say, that our
mapping is a C 1 isometric immersion, see the discussion below. Hence
it is natural to consider the question in the scale of Sobolev spaces.
A first positive results in this direction was established by Kirchheim
[22] who proved that any any W 2,∞ function f in a bounded domain
in R2 with almost everywhere vanishing Hessian determinant must be
1-developable. Following this method, Pakzad [29] improved W 2,∞ to
W 2,2 and applied this result to rigidity of isometric immersions from a
bounded domain in R2 into R3 . For a generalization to mappings from
a bounded domain in Rn into Rn+1 see Liu and Pakzad [25]. Jerrard
and Pakzad [21] later considered also an arbitrary codimension. We
will soon see that their results serve as the positive counter-part of our
example.
Likewise interesting is the negative side, namely flexibility of isometric immersions below a critical regularity assumption. The pioneering
work of Nash [28] and Kuiper [23] established the existence of C 1 isometric immersions of any Riemannian manifold into another manifold
of higher dimension, and in particular, into balls of arbitrary small diameter. The image of such an isometric immersion does not contain
any line segment. Their method was extended further in the framework
of convex integration. Actually, further analysis of the convex integration technique leads to progress in the negative direction for isometric
immersions of Hölder continuous class C 1,α . It was first proved by
Conti, De Lellis and Székelyhidi [9] that the same Nash and Kuiper
type flexibility holds for α < 1/13 and then for α < 1/7 by De Lellis
and Székelyhidi [11]. We mention here that there is no consensus what
the critical exponent α should be.
The examples mentioned above are nowhere twice differentiable and
thus it has no reasonable sense to speak on their Hessian determinant.
Concerning Sobolev examples, they are of totally different nature. The
function f (x) = |x| defined in n-dimensional unit ball breaks the expected regularity at the origin, although its Hessian determinant vanishes a.e. More generally, we could consider 1-homogeneous functions,
i.e. a functions of the form f (x) = x w(x/|x|) where w is nonconstant
and smooth. The regularity of such examples is W 2,p with 1 ≤ p < n.
The graphs of 1-homogeneous functions do not contain straight lines,
A STRICTLY CONVEX SOBOLEV FUNCTION WITH NULL HESSIAN MINORS3
but they contain line segments. More precisely, every point except the
origin lies on a ray along which ∇f is constant. Our motivation was
to find whether any such kind of “partial regularity” cannot hold for
W 2,p -solutions of the null-Hessian equation. Indeed, for some partial
differential equations it happens that all singularities are located in a
small closed set (of zero Lebesgue or even Hausdorff measure). Outside
this singular set the solution behaves as a classical one. See e.g. [15]
for information on such a phenomenon.
The 1-homogeneous examples are not strong enough to disprove partial regularity. We present an example which shows that there is no
partial regularity for W 2,p -solutions of the null-Hessian equation (and
null-Hessian minor equations, see below). Our example is a continuously differentiable W 2,p -function, 1 ≤ p < n, whose graph contains no
segment. Indeed, our function f is strictly convex and the mapping
F = ∇f is a Sobolev homeomorphism whose Jacobian determinant
vanishes a.e. Such a homeomorphism has been recently constructed by
Hencl [17]. An example of a bi-Sobolev homeomorphism with vanishing Jacobian minors is due to Černý [7]. Our example improves the
example by Hencl (and partially that of Černý) in the direction that
it is a gradient of a scalar function. Another example of a wild convex
solution f of the null-hessian equationhas been constructed by Alberti
and Ambrosio [1]. Their example, however, is not strictly convex and
its graph does contain uncountably many line segments. For other
related examples see [19], [32], [26], [27].
More generally, instead of the requirement that rank ∇2 f < n a.e.
we may require rank ∇2 f < k a.e., where k is an integer between 2
and n. The price to be paid is that the Sobolev regularity of the
example is weakened to W 2,p with p < k. Our motivation is again
from a positive result of Jerrard and Pakzad [21], who further proved
that any W 2,p function in a bounded domain of Rn with p ≥ k and
rank ∇2 f < k a.e. is weakly (n − k + 1)-developable. For the precise
definition of (weak) m-developability we refer to [21]; we need only to
record a consequence of this property, namely that any point x in the
domain lies on a m-dimensional hyperplane among which ∇f is Hm a.e. equal to a constant (and thus f to an affine function) at least on
a neighborhood of x. In particular, f cannot be strictly convex and its
gradient cannot be a homeomorphism.
The definition of weak developability does not exclude a pathological situation that x belongs to two m-dimensional hyperplanes, along
each of which ∇f is Hm -a.e. constant, but the corresponding values
of ∇f do no meet. Therefore, a more desirable property is (strong)
m developability, where ∇f is just locally constant along hyperplanes
and no exceptional set is allowed. Such a (n − k + 1)-developability can
be obtained if p ≥ min{2k − 2, n} [21]. What exactly happens between
4
ZHUOMIN LIU AND JAN MALÝ
the exponents k and min{2k − 2, n} is not yet known, see [21], Remark
19.
Our main result reads as follows:
Theorem 1. Let k ∈ {2, . . . , n}, 1 ≤ p < k. Then there exists a strictly
convex function f ∈ W 2,p ((0, 1)n ) ∩ C 1 ([0, 1]n ) such that rank ∇2 f < k
almost everywhere in (0, 1)n . In particular, F = ∇f serves as an
example of a W 1,p homeomorphism with rank ∇F < k almost everywhere. In addition, the mapping F may be α-Hölder continuous for
each α ∈ (0, 1).
In particular when taking k = n, we have the following corollary,
Corollary 1. If 1 ≤ p < n, then there exists strictly convex W 2,p solution to the degenerate Monge-Ampère equation
det(∇2 f ) = 0
a.e.
on the n-dimensional unit cube.
The philosophy behind such an example is that, for such a low
Sobolev regularity, the pointwise second derivative does not carry all
the information needed to describe the true Hessian minors. Among
several ways how to associate a more appropriate Jacobian or Hessian
to a function let us mention two most important ones. The formula
det(∇F ) = div(F1 ∇F2 × · · · × ∇Fn )
enables us to define the Jacobian determinant in the sense of distribution if the expression under the divergence makes sense as a locally
integrable function, this has been introduced by Ball [4]. Similar definitions can be used to define the Hessian determinant of f in the sense
of distributions. Although the distributional Jacobian determinant of
∇f is one of these possibilities, there are other ones available under
lower Sobolev regularity, see [5], [18], [13]. There is also a more sophisticated way than integration by parts how to introduce the Jacobian
in the sense of distribution, see [6].
One does not need any additional Sobolev regularity to define the
Hessian minors of a function f as measures if f is convex. For the
definition of Hessian measures see [2], [3], [34]. Also, the n-th order
Hessian measure of f is the Jacobian of the monotone function ∇f in
the sense of Alberti and Ambrosio [1]. The convexity assumption can
be relaxed, see [14], [20].
It is also interesting to place our example in the setting of critical
regularity regarding the developability of convex functions. To associate the k-th order Hessian measure µk (g) to a generic convex function
g we follow the approach of Trudinger and Wang [34], see Definition 1.
If f is the function from our example, then µk (f ) is a singular measure (Theorem 3). On the other hand, Colesanti and Hug [8, Theorem
2] proved that a convex function g is (n−k+1)-developable outside the
A STRICTLY CONVEX SOBOLEV FUNCTION WITH NULL HESSIAN MINORS5
support of µk (g). Therefore, in our example, supp µk (f ) must be the
whole domain of f . The same result from [8] yields the sharpness of
our Sobolev exponent in the setting of convex functions. Indeed, if
g ∈ W 2,k is convex, then µk (g) is obviously absolutely continuous with
respect to the Lebesgue measure. Hence, if in addition, [∇2 g]k = 0
a.e., then µk (g) = 0 and g is (n−k+1)-developable.
Remark 1. The term “developable” was originally used in differential geometry for surfaces which can be flattened without stretching
or compressing; if fact, these are images of Euclidean domains under
“regular” isometric immersions. Then a graph of a smooth function
ϕ : Ω ⊂ R2 → R is developable in this conventional terminology if and
only if its Gaussian map is, roughly speaking, constant on lines, and
this is 1-developability of ϕ. The n-dimensional case with (n−1) developability is known to experts and it is not so straightforward to find
this result in literature. We are grateful to Reza Pakzad for instructions how to combine results from Spivak’s books [33] to verify this
knowledge.
2. Preliminaries
If A is an n × n matrix, kAk will denote the operator norm of A.
Let A be a real symmetric n × n matrix. The eigenvalues of A will
be denoted by λi = λi (A) indexed in nondecreasing order counting
multiplicity, i.e.,
λ1 ≤ λ2 ≤ · · · ≤ λn .
We are interested in the expressions
[A]k = Sk (λ1 , . . . , λn ),
k = 1, . . . , n,
where Sk is the k-th elementary symmetric function on Rn
X
Sk (λ1 , . . . , λn ) =
λi1 · · · λik .
i1 <i2 <···<ik
Pn
In particular, [A]1 = Tr A = i=1 λi and [A]n = det A = λ1 · · · λn .
The characteristic polynomial P(A) = det(λI − A) has then the form
P(A) = λn − [A]1 λn−1 + [A]2 λn−2 − · · · + (−1)n [A]n .
The expressions [A]k are important invariants as the order of eigenvalues may be difficult to watch. Alternatively, we obtain [A]k as the sum
of all k × k principal minors of A, hence, [A]k depends smoothly on A.
Recall that
λ1 = inf{Av · v : v ∈ Rn , kvk = 1},
(2.1)
λn = sup{Av · v : v ∈ Rn , kvk = 1}.
Now, let A be positive (positive semidefinite), so that λ1 ≥ 0. Then
(2.2)
λn = kAk
6
ZHUOMIN LIU AND JAN MALÝ
and
rank A < k ⇐⇒ λ1 = · · · = λn−k+1 = 0 ⇐⇒ λn−k+1 = 0
⇐⇒ λn−k+1 · · · λn = 0.
Since
n
λn−k+1 · · · λn ≤ Sk (λ1 , . . . , λn ) ≤
λn−k+1 · · · λn ,
k
we infer that
rank A < k ⇐⇒ [A]k = 0.
3. Main step
Lemma 1. Let k ∈ {2, . . . , n}, 1 ≤ p < k and ε0 > 0. Let A be
a positively semidefinite symmetric mapping and Q ⊂ Rn be a closed
cubical interval. Then there exist τ ∈ (0, 1) depending only on n, k and
p and a function g ∈ C 2 (Rn ) with support in Q such that such that
|g(x)| + |∇g(x)| ≤ ε0 ,
(3.1)
(3.2)
x ∈ Q,
1
|∇2 g(x)v · v| ≤ |Av · v| + ε0 kvk2 ,
2
k∇2 g(x)kk ≤ [A]k ,
(3.3)
Z
p/k
v ∈ Rn , x ∈ Q,
x ∈ Q,
p/k
[A + ∇2 g(x)]k dx ≤ τ 2 |Q|[A]k ,
(3.4)
Q
and
(3.5)
Z
[A + ∇2 g(x)]k − [A]k dx ≤ ε0 |Q|.
Q
We devote the rest of this section to the proof of Lemma 1. This will
be done in several steps.
3.1. Laminating. Let λi = λi (A). Consider a positively oriented orthonormal basis (v1 , . . . , vn ) of Rn such that each vi is a eigenvector
of A associated with λi . Set m = n − k + 1. Given ω > 0, set
λm
(3.6)
hω (x) = − 2 sin ω(x · vm ) ,
x ∈ Rn .
2ω
Then
∇hω (x) = − λ2ωm cos(ω(x · vm ))vm ,
(3.7)
∇2 hω (x) = λ2m sin(ω(x · vm ))vm ⊗ vm
The function hω is a simplified version of the function g that we are
looking for. It still does not have a compact support, however, some
computation with hω will be useful.
A STRICTLY CONVEX SOBOLEV FUNCTION WITH NULL HESSIAN MINORS7
3.2. Homogenization. We claim that there exist τ ∈ (0, 1), depending only on n, k and p, and a constant κ such that
p/k
κ = w*-lim[A + ∇2 hω ]k
(3.8)
ω→∞
(the weak* convergence in L1 (Rn )∗ ) and
p/k
0 < κ ≤ τ 4 [A]k .
(3.9)
Moreover,
w*-lim[A + ∇2 hω ]k = [A]k .
(3.10)
ω→∞
Consider a linear isometry P such that
P(e1 ) = vm
and write
B(y) = A + ∇2 h1 (Py)
where h1 is the function defined in (3.6) with ω = 1. Hence,
B(y) = A +
λm
2
sin y1 vm ⊗ vm .
Then the function
p/k
u(y) = [B(y)]k
is 2π-periodic in the variables y1 . By [10, Theorem 2.1.5], the weak*
limit of y 7→ u(ωy) as ω → ∞ is a constant κ identified as
Z
Z
1
1
p/k
κ=
u(y) dy =
[B(y)]k dy.
(2π) (−π,π)×(0,1)n−1
(2π) (−π,π)×(0,1)n−1
Moreover, we observe that vm is an eigenvector of B(y) corresponding
to the eigenvalue λm (1 + 21 sin y1 ) and that, for i 6= m, vi remain to
be eigenvectors of B(y) corresponding to the eigenvalues λi . Thus, the
complete list of eigenvalues of B(y) is
(3.11)
(λ1 , . . . , λm−1 , λm (1 +
1
sin y1 ), λm+1 , . . . , λn ),
2
however, they are no more sorted in nondecreasing order here. Nevertheless, they can be used to compute the invariants of B(y), namely,
p/k
u(y) = [B(y)]k
= Sk (λ1 , . . . , λm−1 , λm (1 +
1
sin y1 ), λm+1 , . . . , λn )p/k .
2
Since the function u does not depend on y2 , . . . , yn , it is enough to
average over y1 . Also, it does not matter whether we average over
y1 ∈ (−π, π) or y1 ∈ (−π/2, π/2) if the expression depends only on
8
ZHUOMIN LIU AND JAN MALÝ
sin y1 . Using the change of variables t = sin y1 we compute
Z
1
p/k
κ=
[B(y)]k dy
2π (−π,π)×(0,1)n−1
Z
1
p/k
=
[B(y)]k dy1
π (−π/2,π/2)×{0}n−1
Z 1
1
Sk (λ1 , . . . , λm−1 , λm (1 + sin y1 ), λm+1 , . . . , λn )p/k dy1
=
π
2
(−π/2,π/2)
=
1
π
Z
1
−1
p/k
Sk λ1 , . . . , λm−1 , λm (1 + 21 t), λm+1 , . . . , λn
√
dt.
1 − t2
The numerator of the last integrand has the form
([A]k (1 + at))p/k ,
where
P
a=
i1 <···<ik
m∈{i1 ,...,ik }
P
1
λ λ
2 i1 i2
λi1 λi2 · · · λik
i1 <···<ik
1
≥ a0 :=
2
· · · λik
≥
λ λ
· · · λn
1
P m m+1
2
i1 <···<ik λi1 λi2 · · · λik
.
n
k
(Here we have used the order λ1 ≤ · · · ≤ λn , namely, if i1 < · · · < ik ,
then λi1 λi2 · · · λik ≤ λm λm+1 · · · λn .) Since the function
s 7→ (1 + st)p/k + (1 − st)p/k ,
s ≥ 0,
is decreasing, we infer that
1
κ≤
π
=
Z
1
p/k
p/k
Z
[A]k (1 + at)
1 1 [A]k (1 + a0 t)
√
√
dt ≤
dt
π −1
1 − t2
1 − t2
−1
p/k
τ 4 [A]k
with
τ :=
1 Z
π
1
−1
This verifies (3.8) with (3.9).
(1 + a0 t)p/k 1/4
√
dt
< 1.
1 − t2
A STRICTLY CONVEX SOBOLEV FUNCTION WITH NULL HESSIAN MINORS9
A similar argument shows
Z
Z
1
1
[B(y)]k dy =
[B(y)]k dy1
2π (−π,π)×(0,1)n−1
π (−π/2,π/2)×{0}n−1
Z 1
1
Sk (λ1 , . . . , λm−1 , λm (1 + sin y1 ), λm+1 , . . . , λn ) dy1
=
π
2
(−π/2,π/2)
=
=
1
π
Z
1
π
Z
1
−1
1
−1
Sk λ1 , . . . , λm−1 , λm (1 + 21 t), λm+1 , . . . , λn
√
dt
1 − t2
1 + at
√
dt[A]k = [A]k
1 − t2
which establishes (3.10).
3.3. Definition of g. Choose ε > 0 which may depend on A and will
be specified later. Consider a smaller concentric cubical subinterval Q0
of Q and a smooth cut-off function η with support in Q such that
(3.12)
0 ≤ η ≤ 1,
η = 1 on Q0 .
Set
gω = ηhω .
Then
(3.13)
∇gω = η∇hω + hω ∇η,
∇2 gω = η∇2 hω + ∇η ⊗ ∇hω + ∇hω ⊗ ∇η + hω ∇2 η.
Fix x ∈ Q and let t = η(x) sin(ω(x · vm )), then t ∈ [−1, 1] and from
(3.7),
tm λm
A + η(x)∇2 hω (x) = A +
vm ⊗ vm ,
2
Thus, vi are eigenvectors of both η(x)∇2 hω (x) and A + η(x)∇2 hω (x).
Denote
(
1
tλm , i = m,
(3.14)
νi = 2
0,
i 6= m.
Then νi are eigenvalues of η(x)∇hω (x) corresponding to vi (not sorted
in the natural order) and thus λi + νi are the corresponding eigenvalues
of A + η(x)∇hω (x). It then easily follows that
k
(3.15)
[A + η(x)∇hω (x)]k ≤ 32 [A]k .
We specify the choice of Q0 such that
p/k
3 p
(3.16)
[A]k |Q \ Q0 | ≤ ε|Q|
2
and fix the function η satisfying (3.12). It remains to specify the frequency ω.
10
ZHUOMIN LIU AND JAN MALÝ
3.4. The choice of ω. We test the weak* convergence in (3.8) by the
characteristic function of Q0 and find ω0 , thanks to (3.9), such that
Z
p/k
p/k
(3.17)
[A + ∇2 hω (x)]k dx ≤ τ 3 |Q0 |[A]k ,
ω > ω0 .
Q0
Since the function
p/k
M 7→ [M]k
is locally uniformly continuous, there exists δ > 0 such that
1
δ < min{ε0 , λm }
2
(3.18)
and for each M, M0 ∈ Rn×n
sym with kMk ≤ 2kAk we have
(3.19)
p/k
p/k
kM0 − Mk < δ =⇒ |[M0 ]k − [M]k | < ε.
Now, by (3.6) and (3.7) we have
kAk
,
ω2
kAk
|∇hω | ≤ C2
,
ω
|hω | ≤ C2
(3.20)
so by (3.13) there exists ω1 ≥ ω0 such that
(3.21)
|gω (x)| + |∇gω (x)| < ε0 ,
k∇2 gω (x) − η(x)∇2 hω (x)k < δ,
ω > ω1 , x ∈ Q.
Set h = hω and g = gω with ω > ω1 . Then (3.1) is satisfied.
3.5. Property (3.2). Since we have chosen ω, from now on we write
h and g for hω and gω , respectively. Fix x ∈ Q and v ∈ Rn . We may
asume that kvk = 1. A spectral decomposition argument and (3.7)
show that
(3.22)
1
|∇2 h(x)v · v| ≤ |Av · v|.
2
Then by (3.18) and (3.21) we have
|∇2 g(x)v · v| ≤ |η(x)∇2 h(x)v · vk + k∇2 g(x) − η(x)∇2 h(x)k
1
1
≤ kAvk + δ ≤ kAvk + ε0 ,
2
2
which establishes (3.2).
A STRICTLY CONVEX SOBOLEV FUNCTION WITH NULL HESSIAN MINORS
11
3.6. Property (3.3). We use (2.2), (3.7) and (3.21) to observe that
kη(x)∇2 h(x)k ≤ 21 λm ,
k∇2 g(x)k ≤ kη(x)∇2 h(x)k + δ
(3.23)
≤ 21 λm + 12 λm ≤ λm .
Hence,
k∇2 g(x)kk ≤ λkm ≤ λm · · · λn ≤ [A]k ,
which yields (3.3).
3.7. Property (3.4). Now, using (3.15), (3.19), (3.17) and (3.16) we
estimate
Z
Z
p/k
p/k
2
[A + ∇ g(x)]k dx ≤
[A + η∇2 h(x)]k dx
Q
Q\Q0
Z
p/k
p/k
2
2
[A + ∇ g(x)]k − [A + η∇ h(x)]k
dx
+
Q\Q0
Z
p/k
+
[A + ∇2 h(x)]k dx
Q0
≤
3 p
2
p/k
[A]k
p/k
|Q \ Q0 | + ε|Q \ Q0 | + τ 3 |Q0 |[A]k
p/k
≤ (2ε + τ 3 [A]k )|Q|.
Under a suitable choice of ε we deduce
Z
p/k
p/k
[A + ∇2 g(x)]k dx ≤ τ 2 [A]k |Q|,
Q
which concludes the proof of (3.4).
3.8. Property (3.5). The same argument using (3.15), (3.19), (3.17)
and (3.16) gives,
Z
Z
2
[A + ∇2 g(x)]k − [A]k dx ≤ [A + η∇ h(x)]k dx
Q
Q\Q0
Z
+ [A + ∇2 g(x)]k − [A + η∇2 h(x)]k dx
Q\Q0
Z
2
+ [A + ∇ h(x)]k − [A]k dx
Q0
k
≤ 23 [A]k |Q \ Q0 | + ε|Q \ Q0 | + ε|Q0 |
≤ 3ε|Q|.
Upon choosing a possibly larger ε so that it also satisfies ε < ε0 /3, we
deduce (3.5).
Remark 2. There is a possible modification of Lemma 1 which has
some applications which we want to present in a subsequent paper. If
we drop the assumption that A is positively semidefinite (but still, A
12
ZHUOMIN LIU AND JAN MALÝ
is symmetric), the assertion, with the exception of (3.2), holds if [·]k
replaced by
J·Kk := Sk (|λ1 |, . . . , |λn |).
For the purpose of the proof we need to sort the eigenvalues according
to their absolute values. Then all computations remain the same; in
the construction of h we do not change the sign of the only varying
eigenvalue. The error estimates in the construction of g also do not
rely on the sign of the eigenvalues.
4. Final construction
4.1. The plan of the construction. Let Q0 = [0, 1]n . We start with
f0 (x) := 21 |x|2 .
Let εj > 0 with εj → 0 be given, we claim that we can use induction
to construct a sequence (gj )j of C 2 mappings on Rn such that, for
j = 1, 2, . . . , gj has a support in Q0 , and the functions gj and
fj := f0 + g1 + · · · + gj ,
j = 1, 2 . . . .
satisfy
|gj (x)| + |∇gj (x)| ≤ εj ,
(4.1)
(4.2)
x ∈ Q0 ,
∇2 fj (x) is positive definite for every x ∈ Q0 ,
k∇2 gj (x)kk ≤ 2[∇2 fj−1 (x)]k ,
(4.3)
x ∈ Q0 ,
and
Z
(4.4)
[∇
2
p/k
fj (x)]k
Q0
Z
dx ≤ τ
p/k
[∇2 fj−1 (x)]k dx.
Q0
Furthermore, we need the property that for each x ∈ Q0 and j ∈ N
there exists an interval Qj (x) containing x such that
(4.5)
diam Qj (x) < εj ,
fj = fj+1 = . . . and ∇fj = ∇fj+1 = . . .
on ∂Qj (x).
If we succeed in constructing the sequence fj , we use (4.1) to establish
existence of the limit function
f = lim fj .
j→∞
A STRICTLY CONVEX SOBOLEV FUNCTION WITH NULL HESSIAN MINORS
13
4.2. Details of the construction. Throughout we use uniform con-
tinuity of ∇2 fj−1 on Q0 . By the induction hypothesis, λ1 ( ∇2 fj−1 (x)
is bounded away from zero on Q0 . Let
Λj = inf{λ1 ∇2 fj−1 (x) : x ∈ Q0 },
Kj = sup{k∇2 fj−1 k : x ∈ Q0 }.
n×n
are locally uniformly continuous,
Since continuous functions on Rsym
there exists δj > 0 such that
(4.6)
1
δj < Λj
4
and for each M0 , M ∈ Rn×n
sym we have
)
(
|[M0 ]k − [M]k | < 21 Λkj , and
kMk ≤ 2Kj ,
(4.7)
=⇒
p/k
p/k
kM0 − Mk < δj
|[M0 ]k −[M]k | < 1 (τ −τ 2 )Λpj .
2
Then we find βj > 0 such that for each x, y ∈ Q0 we have
|y − x| < βj =⇒ k∇2 fj−1 (y) − ∇2 fj−1 (x)k < δj .
Find mj ∈ N such that mj is a multiple of mj−1 (if j ≥ 2) and
√
n
(4.8)
< min 21 εj , βj .
mj
Consider the partition
h
hz − 1 z i
z1 − 1 z1 i
1
1
n
,
× ··· ×
,
: z ∈ {0, . . . , mj } .
Qj :=
mj
mj
mj
mj
For each Q ∈ Qj let xQ be the center of Q and set
AQ := ∇2 fj−1 (xQ ).
We use Lemma 1 to find a function gQ ∈ C 2 (Rn ) with support in Q such
that g = gQ satisfies (3.1)–(3.4) with ε0 = min{εj , δj } and A = AQ .
Set
gj (x) := gQ (x),
for x ∈ Q ∈ Qj .
Then (4.1) follows directly from (3.1).
4.3. Property (4.2). If x ∈ Q ∈ Qj , then
|x − xQ | ≤ diam Q ≤ βj
and thus
(4.9)
k∇2 fj−1 (x) − AQ k < δj .
14
ZHUOMIN LIU AND JAN MALÝ
Choose v ∈ Rn , kvk = 1. By (3.2), (4.9) and (4.6) we have
∇2 fj (x)v · v = (∇2 fj−1 (x)v + ∇2 gj (x)v) · v−
≥ AQ v · v − |∇2 gj (x)v · v| − k∇2 fj−1 (x) − AQ k
1
≥ AQ v · v − AQ v · v − 2δj
2
≥ Λj − 2δj > 0.
This verifies (4.2).
4.4. Property (4.3). Let x ∈ Q ∈ Qj . By (4.9) and (4.7),
[AQ ]k ≤ [∇2 fj−1 (x)]k + |[∇2 fj−1 (x)]k − [AQ ]k |
1
≤ [∇2 fj−1 (x)]k + Λkj .
2
Since
Λkj ≤ [AQ ]k ,
we obtain
1
[AQ ]k ≤ [∇2 fj−1 (x)]k + [AQ ]k
2
and thus
[AQ ]k ≤ 2[∇2 fj−1 (x)]k .
Using (3.3) we infer that
k∇2 gj (x)kk ≤ [AQ ]k ≤ 2[∇2 fj−1 (x)]k .
4.5. Property (4.4). If x ∈ Q ∈ Qj , then by (3.23)
k∇2 gj (x)k ≤ λn (AQ ) ≤ Kj ,
thus
k∇2 fj (x)k ≤ 2Kj .
Notice that
p/k
Λpj ≤ [AQ ]k .
By (4.7) and (4.9),
p/k
p/k
p/k
|[∇2 fj−1 (x)]k − [AQ ]k | ≤ 12 (τ − τ 2 )[AQ ]k , and thus,
p/k
p/k
p/k
|[∇2 fj (x)]k − [AQ + ∇2 gj (x)]k | < 12 (τ − τ 2 )[AQ ]k .
Integrating with respect to x ∈ Q we obtain
Z
Z
p/k
p/k
2
[∇ fj (x)]k dx ≤ [AQ +∇2 gj (x)]k dx
Q
Q
(4.10)
p/k
+ 21 (τ −τ 2 )[AQ ]k |Q|
A STRICTLY CONVEX SOBOLEV FUNCTION WITH NULL HESSIAN MINORS
15
and
Z
p/k
p/k
[∇2 fj−1 (x)]k dx ≥ [AQ ]k |Q|
Q
(4.11)
p/k
− 12 (τ −τ 2 )[AQ ]k |Q|
p/k
≥ 12 (1 + τ )[AQ ]k |Q|.
From (4.10), (3.4) and (4.11) we obtain
Z
Z
p/k
p/k
2
[∇ fj (x)]k dx ≤ [AQ +∇2 gj (x)]k dx
Q
≤
Q
p/k
+ 21 (τ −τ 2 )[AQ ]k |Q|
p/k
p/k
τ 2 [AQ ]k |Q| + 12 (τ −τ 2 )[AQ ]k
p/k
= 12 τ (1+τ )[AQ ]k |Q|
Z
|Q|
p/k
[∇2 fj−1 (x)]k dx.
≤τ
Q
Summing over Q ∈ Qj we conclude (4.4).
4.6. Property (4.5). Given x ∈ Q0 , we define Qj (x) as any Q ∈
Qj with
x ∈ Q. This is obviously an interval of diameter estimated
√
2 n
by mj ≤ εj , see (4.8). Since both gj and ∇gj vanishes on ∂Qj , we
have ∇fj = ∇fj−1 on ∂Qj . Since each mi+1 is a multiple of mi , i.e.,
the boundaries of the partitioning cubical intervals form an increasing
sequence (in terms of inclusion), it is clear from the construction that
(4.5) holds.
5. Sobolev estimate
From (4.4) we obtain
Z
Z
p/k
p/k
j
2
[∇2 f0 (x)]k dx = τ j → 0.
(5.1)
[∇ fj (x)]k dx ≤ τ
Q0
Q0
By (4.3),
Z
2
p
p/k
Z
p/k
[∇2 fj−1 (x)]k dx ≤ 2p/k τ j−1 ,
|∇ gj (x)| dx ≤ 2
(5.2)
Q0
Q0
P
and thus j gj converges in W02,p (Q0 ) (and the sum can be identified
as f − f0 ). It follows that
f = lim fj = f0 +
j→∞
∞
X
j=1
gj ∈ W 2,p (Q0 ).
16
ZHUOMIN LIU AND JAN MALÝ
6. The rank property
Since [A]k ≤ nk kAkk holds for any A ∈ Rn×n
sym , from (5.1) and (5.2)
we infer that
Z
Z
p/k
p/k
2
[∇ f (x)]k dx = lim
[∇2 fj−1 (x)]k dx = 0
j→∞
Q0
Q0
(Alternatively, we could use Fatou’s lemma). Therefore, [∇2 f ]k = 0
a.e., which means that rank ∇2 f < k a.e.
7. Strict convexity
For each j = 1, 2, . . . we know that ∇2 fj is positive definite everywhere in Q0 , so that fj is strictly convex. Then f as a uniform limit
of convex function is convex as well, and we are only left to show that
the convexity is strict. Let x, y ∈ Q be distinct. By (4.5) there exist
j ∈ N and closed intervals Qj (x) and Qj (y) such that
x ∈ Qj (x),
y ∈ Qj (y),
Qj (x) ∩ Qj (y) = ∅
and
(7.1) fj = fj+1 = . . . and ∇fj = ∇fj+1 = . . . on ∂Qj (x) ∪ ∂Qj (y).
There exist 0 ≤ a < b ≤ 1 such that
x̄ := x + a(y − x) ∈ ∂Qj (x),
ȳ := x + b(y − x) ∈ ∂Qj (y).
Since the function
s 7→ fj (x + s(y − x))
is strictly convex on [0, 1], we have
∇fj (x̄)(y − x) < ∇fj (ȳ)(y − x),
but then by (7.1)
∇fi (x̄)(y−x) = ∇fj (x̄)(y−x) < ∇fj (ȳ)(y−x) = ∇fi (ȳ)(y−x),
i ≥ j.
Since (4.1) implies that ∇fi → ∇f uniformly, it follows that
∇f (x̄)(y − x) < ∇f (ȳ)(y − x).
Therefore, by convexity of f ,
∇f (x)(y − x) ≤ ∇f (x̄)(y − x) < ∇f (ȳ)(y − x) ≤ ∇f (y)(y − x).
Since x and y were arbitrary, we conclude that f is strictly convex.
A STRICTLY CONVEX SOBOLEV FUNCTION WITH NULL HESSIAN MINORS
17
8. Modulus of continuity
Theorem 2. Let ψ : [0, 1] → R be a continuous increasing function
such that ψ(0) = 0. Suppose that
ψ(s) ≥ 2s,
(8.1)
x ∈ [0, 1],
and
ψ(s)
= ∞.
s→0 s
Then the choice of εj can be arranged in such a way that the resulting
function F = ∇f satisfies
(8.2)
lim
|F (y) − F (x)| ≤ ψ(|y − x|),
x, y ∈ Q0 .
Proof. By (8.1) and recall the definition of initial function f0 ,
1
|∇f0 (y) − ∇f0 (x)| ≤ |y − x| ≤ ψ(|y − x|),
2
We will prove
(8.3)
|∇gj (y) − ∇gj (x)| ≤ 2−j−1 ψ(|y − x|),
x, y ∈ Q0 .
j = 1, 2, . . .
for a suitable choice of εj . This will give the result.
Let
1/k
σj = sup{[∇2 fj−1 (x)]k : x ∈ Q0 }
.
By (4.3) we have
k∇2 gj (x)k ≤ 2σj ,
x ∈ Q0 .
We use (8.2) to find rj > 0 such that
ψ(s)
≥ 2k+2 σj ,
s
0 < s < rj
and arrange
εj ≤ 2−k−2 ψ(rj ).
Thus,
|y − x| ≤ rj =⇒ |∇gj (y) − ∇gj (x)| ≤ 2σj |y − x| ≤ 2−k−1 ψ(|y − x|)
and, by (4.1),
|y − x| ≥ rj =⇒ |∇gj (y) − ∇gj (x)| ≤ 2εj ≤ 2−k−1 ψ(rj )
≤ 2−k−1 ψ(|y − x|).
This verifies (8.3) and concludes the proof of Theorem 2.
Corollary 2. The function f can be constructed in such a way that
∇f is α-Hölder continuous for each α ∈ (0, 1).
18
ZHUOMIN LIU AND JAN MALÝ
9. The function ∇f as a homeomorphism
Since the function F = ∇f is continuous on Q0 , in order to verify
that F is a homeomorphism it is enough to show that F is one to one.
However, this is a simple and well known consequence of the fact that
f is strictly convex. Indeed, if x, y ∈ Q0 are distinct, then the strict
convexity of on the segment connecting x and y gives
∇f (x)(y − x) < ∇f (y)(y − x),
but this implies that ∇f (x) 6= ∇f (y).
10. The Hessian measure
Definition 1 (Hessian measure). Let f be a convex function on an
open interval Q ⊂ Rn . The k-th Hessian measure µk (f ) is defined
via approximation. If fj is a sequence of smooth convex functions
converging to f locally uniformly (such a sequence always exists), we
define µk (f ) as the limit of [∇2 fj ]k in the sense of distributions. Such a
limit makes sense and does not depend on the approximating sequence,
see [34].
By Colesanti and Hug [8, Theorem 4], the absolutely continuous part
of µk (f ) is [∇2 f ]k , where ∇2 f is computed from the second derivative
a.e. in the sense of Alexandrov [12, Section 6.4, Theorem 1]. This
coincides a.e. with the Sobolev second derivative, see [35, Theorem
3.4.2]. Hence we obtain the following consequence.
Theorem 3. If f is as in Theorem 1, then µk (f ) is a singular measure.
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Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00 Prague 8, Czech Republic
E-mail address: [email protected]; [email protected]