THE MONGE-AMP ERE SYSTEM CONVEX INTEGRATION IN ARBITRARY DIMENSION AND CODIMENSION MARTA LEWICKA

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THE MONGE-AMP`

ERE SYSTEM: CONVEX INTEGRATION IN

ARBITRARY DIMENSION AND CODIMENSION

MARTA LEWICKA

Abstract. In this paper, we study ﬂexibility of the weak solutions to the Monge-Amp`ere

system (MA) via convex integration. This system of Pdes is an extension of the Monge-Amp`ere

equation in d= 2 dimensions, naturally arising from the prescribed curvature problem, and

closely related to the classical problem of isometric immersions (II).

Our main result achieves density, in the set of subsolutions, of the H¨older regular C1,α so-

lutions to the weak formulation (VK) of (MA), for all α < 1

1+d(d+1)/k where dis an arbitrary

dimension and kis an arbitrary codimension of the problem. This result seems to be opti-

mal, from the technical viewpoint, for the corrugation-based convex integration scheme. In

particular, it covers the codimension interval k∈1, d(d+ 1), so far uncharted even for the

system (II), since the regularity C1,α with any α < 1 proved by K¨allen in [7], strictly requires

a large codimension k≥d(d+ 1). We also reproduce K¨allen’s result in the context of (MA).

At k= 1, our result agrees with the regularity C1,α for (II) with any α < 1

1+d(d+1) , proved

by Conti, Delellis and Szekelyhidi in [2]. Finally, our results extend the initial ﬁndings by the

author and Pakzad in [10] for (MA) and d= 2, k = 1.

As an application of our results for (VK), we derive an energy scaling bound in the quanti-

tative immersability of Riemannian metrics, for nonlinear energy functionals modelled on the

energies of deformations of thin prestrained ﬁlms in the nonlinear elasticity [8].

1. Introduction

This paper concerns regularity of weak solutions to a multi-dimensional version of the Monge-

Amp`ere equation, which arises from the prescribed curvature problem and is closely related to

the problem of isometric immersions and the dimension reduction of thin ﬁlms. Namely, given

a matrix ﬁeld A:ω→Rd×d

sym on a domain ω⊂Rd, we look for vector ﬁelds wand vsuch that:

v:ω→Rk, w :ω→Rd,

2(∇v)T∇v+ sym∇w=Ain ω. (VK)

When d= 2, k = 1, the left hand side above is the von K´arm´an content whose energy measures

the stretching of a thin ﬁlm with midplate ω, subject to the out of plane displacement vand

the in plane displacement w. Taking curl curl of both sides of (VK) yields then the Monge-

Amp`ere equation: det2v=−curl curl Aon ω, which reﬂects matching the leading order term

in the expansion of the Gaussian curvature κof surfaces {(x, ϵv(x)); x∈ω)} ⊂ R3with the

given scalar function f=−curl curl A. We note that one can achieve any suﬃciently regular

fby imposing A=−(∆−1f)Id2. For arbitrary d, k ≥1, the situation is similar: applying a

multi-dimensional operator C2(see the formula in (1.4)) to both sides of (VK), leads to the

system which equates the leading order term in the expansion of the full Riemann curvature

M.L. was partially supported by NSF grant DMS-2006439. AMS classiﬁcation: 35Q74, 53C42, 35J96, 53A35.

arXiv:2210.04363v3 [math.AP] 24 Feb 2024

2 MARTA LEWICKA

tensor of the manifold {(x, ϵv(x)); x∈ω)} ⊂ Rd+kwith the table F=−C2(A) : ω→Rd4:

v:ω→Rk,

Det ∇2v.

=⟨∂i∂jv, ∂j∂tv⟩−⟨∂i∂tv−∂j∂sv⟩ij,st:1...d =Fin ω. (MA)

Again, any Fsatisfying the symmetries and Bianchi’s identities can be achieved this way.

The closely related problem to (VK) and (MA) is the problem of ﬁnding an isometric immersion

uof the given Riemannian metric g:ω→Rd×d

sym,>, into a higher dimensional space Rd+k:

u:ω→Rd+k,

(∇u)T∇u=gin ω. (II)

This problem reduces to (VK) when gathering the leading (lowest order) terms in the ϵ-

expansions, for the family of Riemannian metrics {gϵ= Idd+ 2ϵ2A}ϵ→0and the family of

metrics generated by the immersions {uϵ= (idd+ϵ2w, ϵv)}ϵ→0. When posed in arbitrary di-

mension dbut low codimension k= 1, it has been shown in [3, Theorem 1.1] that any local

subsolution to (II) can be uniformly approximated by a sequence of solutions {un}∞

n=1 of regu-

larity C1,α, for any H¨older exponent α < 1

1+2d∗where d∗=d(d+ 1)/2 is the dimension of Rd×d

sym.

On the other hand, as showed in [7], regularity C1,α with any α < 1 can be achieved in suf-

ﬁciently high codimension k; however even for the local result this argument strictly requires

k≥2d∗, whereas it yields no outcome for k < 2d∗. These two results, albait both relying

on convex integration, use diﬀerent constructions of the cascade of perturbations: Kuiper’s

corrugations in [3] and Nash’s spirals in [7], and one cannot be deduced from the other.

To our knowledge, there has been no result for the codimension interval k∈(1,2d∗) interpolat-

ing the regularity in [3] and [7], or even improving the exponent 1

1+2d∗without the requirement

k≥2d∗. The purpose of our paper is to achieve precisely this goal, for the system (VK).

Our main result states that any C1-regular pair (v, w) which is a subsolution of (VK), can

be uniformly approximated by a sequence of solutions {(vn, wn)}∞

n=1 of regularity C1,α for any

H¨older exponent α < 1

1+2d∗/k , in case of arbitrary dand k. Our proof only uses corrugations,

extending the construction in [3] in an optimal manner. Clearly, the obtained critical regularity

exponent 1/2 at k= 2d∗is inferior to the exponent 1 from a version of the same construction

as in [7], that we also demonstrate in our paper. We expect that the superposition of both

techniques should yield a tighter interpolation, which is the subject of the ongoing research.

We state our results and oﬀer further discussion in the subsections below.

1.1. Convex integration by corrugations, arbitrary d and k. The following theorem is

our main result. We refer to it as ﬂexibility of (VK) up to C1,1

1+2d∗/k :

Theorem 1.1. Let ω⊂Rdbe an open, bounded domain. Given two vector ﬁelds v∈ C1(¯ω, Rk),

w∈ C1(¯ω, Rd)and a matrix ﬁeld A∈ C0,β (¯ω, Rd×d

sym), assume that:

D=A−1

2(∇v)T∇v+ sym∇wsatisﬁes D> c Iddon ¯ω,

for some c > 0, in the sense of matrix inequalities. Fix ϵ > 0and let:

0< α < min nβ

2,1

1 + d(d+ 1)/k o.

THE MONGE-AMP`

ERE SYSTEM: CONVEX INTEGRATION IN ARBITRARY DIMENSION AND CODIMENSION3

Then, there exists ˜v∈ C1,α(¯ω, Rk)and ˜w∈ C1,α(¯ω, Rd)such that the following holds:

∥˜v−v∥0≤ϵ, ∥˜w−w∥0≤ϵ, (1.1)1

A−1

2(∇˜v)T∇˜v+ sym∇˜w= 0 in ¯ω. (1.1)2

This result generalizes [10, Theorem 1.1], where we proved ﬂexibility for (VK) up to C1,1

7in

dimensions d= 2, k= 1. In that special case, motivated by theory of elasticity, the left hand

side of (VK) represents the von K´arm´an stretching content 1

2∇v⊗ ∇v+ sym∇wwritten in

terms of the scalar out-of-plane displacement vand the in-plane displacement wof the middle

plate ωof a thin ﬁlm. The case d= 2 is special and ﬂexibility (in codimension 1) of (VK)

holds up to C1,1

5as shown in [2, Theorem 1.1] using the conformal equivalence of 2-dimensional

metrics to the Euclidean metric. In our follow-up extension [9] of the present work, we likewise

show that any k≥1 allow for ﬂexibility up to C1,1

1+4/k when d= 2. After the completion

of our both works, we learned of the very recent preprint [1] in which ﬂexibility of (VK) for

d= 2, k = 1 has been futher improved to hold up to C1,1

The main new technical ingredient allowing for the ﬂexibility range stated in Theorem 1.1, is

the following “stage”-type construction in the convex integration algorithm for (VK):

Theorem 1.2. Let the vector ﬁelds v∈ C2(¯ω, Rk),w∈ C2(¯ω, Rd)and the matrix ﬁeld A∈

C0,β(¯ω, Rd×d

sym)be given on an open, bounded domain ω⊂Rd. Assume that:

D=A−1

2(∇v)T∇v+ sym∇wsatisﬁes 0<∥D∥0≤1.

Fix two constants M, σ such that:

M≥max{∥v∥2,∥w∥2,1}and σ≥1.

Then, there exist ˜v∈ C2(¯ω, Rk)and ˜w∈ C2(¯ω, Rd)such that, denoting:

D=A−1

2(∇˜v)T∇˜v+ sym∇˜w,

the following holds:

∥˜v−v∥1≤C∥D∥1/2

0,∥˜w−w∥1≤C∥D∥1/2

0(1 + ∥∇v∥0),(1.2)1

∥∇2˜v∥0≤CMσd∗/k,∥∇2˜w∥0≤CM σd∗/k(1 + ∥∇v∥0),(1.2)2

∥˜

D∥0≤C∥A∥0,β

Mβ∥D∥β/2

0+∥D∥0

σ,(1.2)3

where d∗=d(d+ 1)/2and where the constants Cdepend only on d, k and ω.

We brieﬂy outline how our construction diﬀers from [10] and [3]. There, a stage consisted of

precisely d∗“steps”, each cancelling one of the rank-one “primitive” deﬁcits in the decompo-

sition of D. The initially chosen frequency of perturbation was multiplied by a factor σat

each step, leading to the increase of the second derivative by σd∗and thus to the exponent d∗

replacing d∗/k in (1.2)2, while the remaining error in Dwas of order 1/σ, leading to (1.2)3.

Presently, we ﬁrst observe that ksuch deﬁcits may be cancelled at once, by using klinearly

independent codimensions. Further, when all the ﬁrst order primitive deﬁcits are cancelled, one

may proceed to cancelling the second order deﬁcits obtained as the one-dimensional decompo-

sitions of the error between the original and the decreased D; the corresponding frequencies

must be then increased by the factor σ1/2, precisely due to the decrease of Dby the factor

4 MARTA LEWICKA

1/σ. One may inductively proceed in this fashion, cancelling even higher order deﬁcits, and

adding k-tuples of single codimension perturbations, for a total of N=lcm(k, d∗) steps. The

frequencies get increased by the factor of σover each multiple of k, leading to the increase of

the second derivatives by σ, and by the factor of σ1/2over each multiple of d∗, where the deﬁcit

decreases by the factor of 1/σ. In the ﬁnal count, the total increase of the second derivatives

has the factor σN/k, while the decrease of the deﬁcit has the factor 1/σN/d∗. The relative

change of order is thus (N/k)/(N/d∗) = d∗/k, as stated in Theorem 1.2.

We point out that for this scheme to work, it is essential to use the optimal “step”-type

construction in which the chosen one-dimensional primitive deﬁcit is cancelled at the expense

of introducing least error possible. Our previous deﬁnition from [10] would not work for this

purpose, and we need to superpose three corrugations rather than two.

1.2. Convex integration by spirals, k ≥2d∗.For a large codimension, one can reach ﬂexi-

bility of (VK) up to C1,1, motivated by a similar result for (II) in [7]:

Theorem 1.3. In the context of Theorem 1.1, assume that the codimension ksatisﬁes k≥

2d∗=d(d+ 1). Then, the same result is valid for any exponent in the range:

0< α < min nβ

2,1o.

The “stage” construction allowing for ﬂexibility as above, is the counterpart of Theorem 1.4:

Theorem 1.4. Let ω⊂Rdand kbe as in Theorem 1.3. Fix an exponent δ > 0. Then, there

exists σ0>1depending only on ωand δ, such that we have the following. Given v∈ C2(¯ω, Rk),

w∈ C2(¯ω, Rd),A∈ C0,β(¯ω, Rd×d

sym)and given two constants M, σ wih the properties:

D=A−1

2(∇v)T∇v+ sym∇wsatisﬁes 0<∥D∥0≤1,

M≥max{∥v∥2,∥w∥2,1}, σ ≥σ0,

there exist ˜v∈ C2(¯ω, Rk),˜w∈ C2(¯ω, Rd)such that, denoting:

D=A−1

2(∇˜v)T∇˜v+ sym∇˜w,

the following bounds are valid, with constants Cdepending only on d, k, ω and δ:

∥˜v−v∥1≤C∥D∥1/2

0,∥˜w−w∥1≤C∥D∥1/2

0(1 + ∥∇v∥0),(1.3)1

∥∇2˜v∥0≤CMσδ,∥∇2˜w∥0≤CMσδ(1 + ∥∇v∥0),(1.3)2

∥˜

D∥0≤C∥A∥0,β

Mβ∥D∥β/2

0+∥D∥0

σ.(1.3)3

An outline of this construction, based on the approach in [7], is as follows. Firstly, each rank-one

“primitive” deﬁcit is cancelled using two codimensions, via spiral-like perturbations of the ﬁelds

v, w, rather than via one-dimensional corrugations. This allows for a better order in the second

order deﬁcit. Since we now have 2d∗codimensions available, we may rank-one decompose the

new deﬁcit as well and cancel it right away by adjusting the original perturbations. Proceeding

this way, it is possible to cancel arbitrarily high order of deﬁcits, keeping the frequency at a

chosen value σwhile assuring that Dis decreased by the factor 1/σN, for arbitrarily large N.

THE MONGE-AMP`

ERE SYSTEM: CONVEX INTEGRATION IN ARBITRARY DIMENSION AND CODIMENSION5

1.3. The Monge-Amp´ere system. We now proceed to interpreting Theorem 1.1 in the

context of the Monge-Amp´ere system. We introduce this new system of partial diﬀerential

equations as the strong formulation of (VK). Recall that for a matrix ﬁeld A= [Aij ]i,j=1...2:

R2→R2×2, the scalar ﬁeld curl curlA is deﬁned by taking the curl operator on each row of

A, and then applying another curl on thus formed two-dimensional vector ﬁeld:

curl curlA= curl∂1A12 −∂2A11, ∂1A22 −∂2A21

=∂1∂1A22 −∂1∂2A21 −∂1∂2A12 +∂2∂2A11.

It is well known that the kernel of curl curl when restricted to R2×2

sym matrix ﬁelds, consists

precisely of symmetric gradients. We will be concerned with the following generalization of

curl curl, serving the same characterisation in higher dimensions:

Deﬁnition 1.5. Given a d-dimensional square matrix ﬁeld A= [Aij ]i,j=1...d :ω→Rd×don a

domain ω⊂Rd, we deﬁne C2(A) : ω→Rd4by:

C2(A)ij,st =∂i∂sAjt +∂j∂tAis −∂i∂tAjs −∂j∂sAit for all i, j, s, t = 1 . . . d. (1.4)

It can be checked that the components of the Riemann curvature tensor of a family of metrics

Idd+ϵA on ω, are given, to the leading order, by the components of C2(A):

Riem(Idd+ϵA)ij,st =−ϵ

2C2(A)ij,st +O(ϵ2) for all i, j, s, t = 1 . . . d. (1.5)

For dimension d= 2, the above formula yields the linearization of the Gaussian curvature:

κ(Id2+ϵA) = −ϵ

2curl curlA+O(ϵ2). We have the following:

Lemma 1.6. Let ω⊂Rdbe an open, bounded, contractible domain with Lipschitz boundary.

Given a symmetric matrix ﬁeld A∈L2(ω, Rd×d

sym), the following conditions are equivalent:

(i) A= sym∇wfor some w∈H1(ω, Rd),

(ii) C2(A)=0in the sense of distributions on ω.

Observe that for A= (∇v)T∇vgiven through a vector ﬁeld v:ω→Rk, there holds:

C2(∇v)T∇vij,st = 2⟨∂i∂tv, ∂j∂sv⟩ − 2⟨∂i∂sv, ∂j∂tv⟩.

When d= 2 and k= 1, the above reduces to the familiar formula: curl curl(∇v⊗ ∇v) =

−2 det ∇2v. Following this motivation, we introduce:

Deﬁnition 1.7. For v:ω→Rkdeﬁned on a domain ω⊂Rd, we set Det ∇2v:ω→Rd4in:

Det ∇2vij,st =⟨∂i∂sv, ∂j∂tv⟩−⟨∂i∂tv, ∂j∂sv⟩for all i, j, s, t = 1 . . . d. (1.6)

Given F:ω→Rd4, we call the following system of Pdes, the Mong´e-Amper´e system:

Det ∇2v=Fon ω.

Lemma 1.6 can be restated in this context as follows. Given a matrix ﬁeld A:ω→Rd×d

sym on a

domain ω⊂Rd, the problem (VK) is equivalent to (disregarding the regularity questions):

v:ω→Rk,

Det ∇2v=−C2(A),(MA)

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THEMONGE-AMP`ERESYSTEM:CONVEXINTEGRATIONINARBITRARYDIMENSIONANDCODIMENSIONMARTALEWICKAAbstract.Inthispaper,westudyflexibilityoftheweaksolutionstotheMonge-Amp`eresystem(MA)viaconvexintegration.ThissystemofPdesisanextensionoftheMonge-Amp`ereequationind=2dimensions,naturallyarisingfromtheprescribedcurv...

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THE MONGE-AMP ERE SYSTEM CONVEX INTEGRATION IN ARBITRARY DIMENSION AND CODIMENSION MARTA LEWICKA

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