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428 K. Laidoune, G. Metafune, D. Pallara and A. Rhandi

Proof. We take ω(y)=e

δ|y|

, W

(y)=W

(y)=e

γ|y|

for δ<γ<(βΛ)

−1

c and

use Theorem 3.3 with a = t and a − a

= b

− b = b − a =

. The thesis then

follows using Proposition 2.7. 

Example.

(i) The above corollary applies with any γ<(βΛ)

−1

c and without any restriction

on V ≥ 0when

lim sup

|x|→∞

|x|

1−β



F (x) ·

|x|



< −c, 0 <c<∞,

for some β>2and|F (x)|≤c

|x|

β−ε

for some ε, c

> 0. This is obvious

if V = 0 and, in the general case, it follows by observing that the kernel

p is pointwise dominated, by the maximum principle, by the corresponding

kernel of the operator with V =0.

(ii) Let us consider the Schr¨odinger operator Δ − a

|x|

with a>0,s>2. Then

Corollary 3.4 applies with β =1+

and any δ<

s+2

. This yields

0 <p(x, y, t) ≤ c

exp



−

s+2

s−2



exp



−δ|y|

s+2



:= c(t)φ(y).

Using the symmetry of p and the semigroup law (see [9, Example 3.13]), we

obtain

p(x, y, t) ≤ c

exp



−

s+2

s−2



exp



−δ|x|

s+2



exp



−δ|y|

s+2



This estimate was obtained in [9, Example 3.13].

(iii) Let us generalize the previous situation to the case of the operators

A =Δ−|x|

|x|

·D −|x|

with r>1. We distinguish three cases.

(a) If s<2r,thenβ = r +1andδ can be any positive number less than

r+1

. Therefore

0 <p(x, y, t) ≤ c

exp



−

r+1

r−1



exp



−δ|y|

r+1



(b) If s =2r,thenβ = r + 1 as before but now δ must be less than

√

2(r+1)

and δ<

s+2

. Then we get, as in (ii)

0 <p(x, y, t) ≤ c

exp



−

s+2

s−2



exp



−δ|y|

s+2



:= c(t)φ(y). (3.9)

In this case one can also obtain estimates with respect to x proceeding

as in (ii). We consider the formal adjoint A

∗

=Δ+|x|

|x|

· D +(N +

r − 1)|x|

r−1

−|x|

. The associated minimal semigroup has the kernel

∗

(x, y, t)=p(y,x,t) which satisﬁes (3.9), by the same argument as

above. This yields p(t, x, y) ≤ c(t)φ(x) and, proceeding as in (ii),

p(x, y, t) ≤ c

exp



−

s+2

s−2



exp



−δ|x|

s+2



exp



−δ|y|

s+2



Estimates on Transition Kernels 429

Under conditions similar to those of Corollary 3.4, the estimate of p can be

improved with respect to the time variable, loosing the exponential decay in y.

Corollary 3.5. Assume that

lim sup

|x|→∞

|x|

1−β



F (x) ·

|x|

−

|x|

2α

V (x)



< 0, (3.10)

for some α>0 and β>2.If|F(x)|+



V (x) ≤ c(1+|x|

)

and ω(x):=(1+|x|

)

with 0 <kγ

+ γ

≤ α, γ

≥

β−2

and k>N+2, then there exists a constant

C>0 such that

0 <p(x, y, t) ≤

(1 + |y|

)

−γ

for all x, y ∈ R

, 0 <t≤ 1 where

σ =

β − 2

((k − 2)γ

+ γ

) .

Proof. Observe that W

(x)=(1+|x|

)

is a Lyapunov function for every 0 <r≤

α.Ifζ

(x, t) is the corresponding function deﬁned in (2.1), then Proposition 2.8

yields

(x, t) ≤ c

−2r

β−2

for x ∈ R

and 0 <t≤ 1. We set a = t and a−a

= b

−b = b−a =

where s ≥ 1

will be chosen later and we apply Theorem 3.3 with ω(x)=W

(x)=



1+|x|



and W

(x)=



1+|x|



kγ

+γ

.Thusweobtain

p(x, y, t) ≤ C



−

2(kγ

+γ

)

β−2

+ t

−

2γ

β−2

−s



(1 + |y|

)

−γ

Minimizing over s we get s =

4γ

β−2

and the thesis follows. 

Example. Let us consider again the operators

A =Δ−|x|

|x|

·D −|x|

with r>1. Again we distinguish three cases.

(a) If s +1≤ r,thenβ = r +1 and γ

. It is easily seen that (3.10) holds for

every α>0 and hence

p(x, y, t) ≤ Ct

−(k−2)

r−1

−

2γ

r−1



1+|y|



−γ

for every γ

≥ 0, 0 <t≤ 1, y ∈ R

(b) If r<s+ 1, then (3.10) holds for β = s +2andeveryα>0. So, we have to

distinguish two cases.

(i) If s ≤ 2r,thenγ

and

p(x, y, t) ≤ Ct

−(k−2)

−

2γ



1+|y|



−γ

430 K. Laidoune, G. Metafune, D. Pallara and A. Rhandi

(ii) If s>2r,thenγ

and

p(x, y, t) ≤ Ct

−

k−2

−

2γ



1+|y|



−γ

for every γ

≥ 0, 0 <t≤ 1, y ∈ R

Remark 3.6. The results of this section generalize Theorem 4.1 and its corollaries

in [10] and also the results obtained in [9] in the case of exponential decay but not

for polynomial decay, where the results in [9] are more precise.

4. Regularity properties

In this section we obtain the diﬀerentiability of the transition semigroup T (·)

associated with the transition kernels p in C

) in the case where the coeﬃcients

F and V are of exponential type.

We assume here that a

∈ C

), V ∈ C

)andF ∈ C

). All

results of this section can be proved exactly by the same arguments as in [10,

Section 5 and Section 6].

Theorem 4.1. Suppose that there exist constants β>2,c>0 such that

lim sup

|x|→∞

|x|

1−β



F (x) ·

|x|

−

V (x)

δβ|x|

β−1



< −c.

Assume moreover that

V (x)+|DV (x)| + |F (x)| + |DF(x)| + |D

F (x)|≤c

exp(c

|x|

β−ε

)

for some ε, c

> 0. Then the following estimates hold

(i) 0 <p(x, y, t) ≤ c

exp{c

−

β−2

}exp{−γ|y|

}

(ii) |D

p(x, y, t)|≤c

exp{c

−

β−2

}exp{−γ|y|

}

(iii) |D

p(x, y, t)|≤c

exp{c

−

β−2

}exp{−γ|y|

}

(iv) |∂

p(x, y, t)|≤c

exp{c

−

β−2

}exp{−γ|y|

}

for suitable c

,γ>0 and for all 0 <t≤ T and x, y ∈ R

Remark 4.2. (a) Assuming only that there exist constants β>2,c > 0such

that

lim sup

|x|→∞

|x|

1−β



F (x) ·

|x|

−

V (x)

δβ|x|

β−1



< −c,

and V (x)+|F(x)|≤C exp(|x|

)forsomeC>0andγ<β, the functions

p log

p and p log p are integrable in Q(a, b)andinR

for ﬁxed t ∈ [a, b]

Estimates on Transition Kernels 431

respectively and



Q(a,b)

p(x, y, t)|

p(x, y, t)

dy dt

≤



Q(a,b)

(|F (y)|

+ V

(y))p(x, y, t) dy dt



Q(a,b)

p(x, y, t)log

p(x, y, t) dy dt



p(x, y, t) − p(x, y, t)logp(x, y, t)

t=b

t=a

dy < ∞.

In particular, p

belongs to W

1,0

(Q(a, b)) (see [10, Theorem 5.1]). This im-

plies in particular that p ∈ W

2,1

(Q(a, b)) provided that also DF is of expo-

nential type for some k>N+ 2 (see [10, Theorem 5.2]).

(b) From Theorem 3.3 and (a) (cf. [10, Theorem 5.3]) one can observe that the

assumption a

∈ C

) is not needed for (i) and (ii).

As a consequence we obtain the diﬀerentiability of T (·)inC

Theorem 4.3. Under the assumptions of Theorem 4.1, the transition semigroup

T (·) is diﬀerentiable on C

) for t>0.

Example. Let a ∈ R. From Theorem 4.3 we deduce that the operator

A =Δ−|x|

x · D − a

|x|

with r>0ands ≥ 0 generates a diﬀerentiable semigroup in C

). This result

is known for a = 0 (see [13, Proposition 4.4]).

References

[1] A. Aibeche, K. Laidoune, A. Rhandi, Time dependent Lyapunov functions for some

Kolmogorov semigroups perturbed by unbounded potentials, Archiv Math. 94 (2010),

565–577.

[2] V.I. Bogachev, N.V. Krylov, M. R¨ockner, On regularity of transition probabilities and

invariant measures of singular diﬀusions under minimal conditions, Comm. Partial

Diﬀ. Eq., 26 (2000), 2037–2080.

[3] V.I. Bogachev, M. R¨ockner, S.V Shaposhnikov, Global regularity and estimates of

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Appl.52(2008) 209–236.

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Studies in Mathematics 12, Amer. Math. Soc., 1996.

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weighted Sobolev spaces, J. Funct. Anal.183 (2001), 1–41.

432 K. Laidoune, G. Metafune, D. Pallara and A. Rhandi

[7] O.A. Ladyz’enskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equa-

tions of Parabolic Type, Amer. Math. Soc., 1968.

[8] L. Lorenzi, M. Bertoldi, Analytical Methods for Markov Semigroups, Chapman-Hall

/CRC 2007.

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J. evol. equ. 6 (2006), 433–457.

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Singular Diﬀusions, Theory Probab. Appl. 54 (2010), 68–96.

[11] G. Metafune, D. Pallara, M. Wacker, Feller semigroups on R

, Semigroup Forum

65 (2002), 159–205.

[12] G. Metafune, D. Pallara, M. Wacker, Compactness properties of Feller semigroups,

Studia Math. 153 (2002), 179–206.

[13] G. Metafune, E. Priola, Some classes of non-analytic Markov semigroups,J.Math.

Anal. Appl. 294 (2004), 596–613.

[14] C. Spina, Kernel estimates for a class of Kolmogorov semigroups,Arch.Math.91

(2008), 265–279.

Karima Laidoune

D´epartement de Math´ematiques

Facult´e des Sciences

Universit´e Ferhat Abbes S´etif

Route de Scipion

19000 S´etif, Algeria

e-mail: klaidoune@hotmail.fr

Giorgio Metafune and Diego Pallara

Dipartimento di Matematica “Ennio De Giorgi”

Universit`a di Lecce

C.P.193

I-73100, Lecce, Italy

e-mail: giorgio.metafune@unisalento.it

diego.pallara@unisalento.it

Abdelaziz Rhandi

Dipartimento di Ingegneria dell’Informazione

e Matematica Applicata

Universit`a degli Studi di Salerno

Via Ponte Don Melillo

I-84084 Fisciano (Sa), Italy

e-mail: rhandi@diima.unisa.it

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 433–445

 2011 Springer Basel AG

Metric-induced Morphogenesis

and Non-Euclidean Elasticity:

Scaling Laws and Thin Film Models

Marta Lewicka

Abstract. The purpose of this paper is to report on recent developments con-

cerning the analysis and the rigorous derivation of thin ﬁlm models for struc-

tures exhibiting residual stress at free equilibria. This phenomenon has been

observed in diﬀerent contexts: growing leaves, torn plastic sheets and speciﬁ-

cally engineered polymer gels. The study of wavy patterns in these contexts

suggest that the sheet endeavors to reach a non-attainable equilibrium and

hence assumes a non-zero stress rest conﬁguration.

Mathematics Subject Classiﬁcation (2000). 74K20, 74B20.

Keywords. Non-Euclidean plates, nonlinear elasticity, Gamma convergence,

calculus of variations.

1. Elastic energy of a growing tissue and

non-Euclidean elasticity

This paper concerns elastic structures which exhibit non-zero strain at free equi-

libria. Many growing tissues (leaves, ﬂowers or marine invertebrates) attain com-

plicated conﬁgurations during their free growth. Recent work has focused on some

of the related questions by using variants of thin plate theory [1, 5, 4, 23]. How-

ever, the theories used are not all identical and some of them arbitrarily ignore

certain terms and boundary conditions without prior justiﬁcation. This suggests

that it might be useful to rigorously derive an asymptotic theory for the shape of

a residually strained thin lamina to clarify the role of the assumptions used while

shedding light on the errors associated with the use of the approximate theory

that results. Recently, such rigorous derivations were carried out [8, 17, 19, 21] in

the context of standard nonlinear elasticity for thin plates and shells.

The purpose of this paper is to present these results in a concise manner,

departing from the 3d incompatible elasticity theory conjectured to explain the

434 M. Lewicka

mechanism for the spontaneous formation of non-Euclidean metrics. Namely, recall

that a smooth Riemannian metric on a simply connected domain can be realized

as the pull-back metric of an orientation preserving deformation if and only if

the associated Riemann curvature tensor vanishes identically. When this condi-

tion fails, one seeks a deformation yielding the closest metric realization. It is

conjectured that the same principle plays a role in the developmental processes

of naturally growing tissues, where the process of growth provides a mechanism

for the spontaneous formation of non-Euclidean metrics and consequently leads to

complicated morphogenesis of the thin ﬁlm exhibiting waves, ruﬄes and non-zero

residual stress.

Below, we set up a variational model describing this phenomenon by introduc-

ing the non-Euclidean version of the nonlinear elasticity functional, and establish

its Γ-convergence under a proper scaling. Heuristically, a sequence of function-

als F

is said to Γ-converge to a limit functional F if the minimizers of F

,if

converging, have a minimizer of F as a limit.

Consider a sequence of thin 3d ﬁlms Ω

=Ω× (−h/2,h/2), viewed as the

reference conﬁgurations of thin elastic tissues. Here, Ω ⊂ R

is an open, bounded

and simply connected set which we refer to as the mid-plate of thin ﬁlms under

consideration. Each Ω

is now assumed to undergo a growth process, described

instantaneously by a (given) smooth tensor:

=[a

]:Ω

−→ R

3×3

such that det a

(x) > 0.

According to the formalism in [25], the multiplicative decomposition

∇u = Fa

(1.1)

is postulated for the gradient of any deformation u :Ω

−→ R

. The tensor

F = ∇u(a

)

−1

corresponds to the elastic part of u, and accounts for the reorga-

nization of Ω

in response to the growth tensor a

. The validity of decomposition

(1.1) into an elastic and inelastic part requires that it is possible to separate out a

reference conﬁguration, and thus this formalism is most relevant for the descrip-

tion of processes such as plasticity, swelling and shrinkage in thin ﬁlms, or plant

morphogenesis.

The elastic energy of u depends now only on F:

(u)=



W (F )dx =



W (∇u(a

)

−1

)dx, ∀u ∈ W

1,2

(Ω

, R

(1.2)

We remark that although our results are valid for thin laminae that might be

residually strained by a variety of means, we only consider the one-way coupling

of growth to shape and ignore the feedback from shape back to growth (plasticity,

swelling, shrinkage etc.). However, it seems fairly easy to include this coupling

once the basic coupling mechanisms are known.

In (1.2), the energy density W : R

3×3

−→ R

is a nonlinear function, as-

sumed to be C

in a neighborhood of SO(3) and assumed to satisfy the following

Non-Euclidean Plate Theories 435

conditions of normalization, frame indiﬀerence and nondegeneracy:

∃c>0 ∀F ∈ R

3×3

∀R ∈ SO(3) W (R)=0,W(RF )=W(F ),

W (F ) ≥ c dist

(F, SO(3)).

(1.3)

The reason for using a nonlinear elasticity model (rather than the more familiar

linear elasticity) is that, as our analysis shows, the resulting deformations u

when

h → 0, are expected to be of order O(1), even though their gradients are locally

O(h) close to rigid rotations.

We now compare the above approach with the ‘target metric’ formalism [5,

20]. On each Ω

one assumes to be given a smooth Riemannian metric g

=[g

A deformation u of Ω

is then an orientation preserving realization of g

,when

(∇u)

∇u = g

and det∇u>0,

or equivalently, by the polar decomposition theorem,

∇u(x) ∈F

(x)=





(x); R ∈ SO(3)



a.e. in Ω

. (1.4)

It is hence instructive to study the following energy, bounding from below I

(u):

dist

(u)=



dist

(∇u(x), F

(x)) dx ∀u ∈ W

1,2

(Ω

, R

). (1.5)

The functional

dist

measures the average pointwise deviation of the deformation u

from being an orientation preserving realization of g

.Notethat

dist

is comparable

in magnitude with I

,forW =dist

(·,SO(3)). Also, observe that the intrinsic

metric of the material is transformed by a

to the target metric g

=(a

)

and, for isotropic W , only the symmetric positive deﬁnite part of a

given by



plays a role in determining the deformed shape.

2. The residual stress and a result on its scaling

Note that one could deﬁne the energy as the diﬀerence between the pull-back

metric of a deformation u and the given metric: I

str

(u)=



|(∇u)

∇u − g

dx.

However, such ‘stretching’ functional is not appropriate from the variational point

of view, because there always exists u ∈ W

1,∞

such that I

str

(u) = 0. Further,

if the Riemann curvature tensor R

associated to g

does not vanish identically,

say R

ijkl

(x) =0,thenu has a ‘folding structure’ [9]; it cannot be orientation

preserving (or reversing) in any open neighborhood of x.

As proved in [20], the functionals I

below and

dist

have strictly positive

inﬁma for non-ﬂat g

, which points to the existence of non-zero stress at free

equilibria (in the absence of external forces or boundary conditions):

Theorem 2.1. For each ﬁxed h, the following two conditions are equivalent:

(i) The Riemann curvature tensor R

ijkl

≡ 0,

(ii) inf



dist

(u); u ∈ W

1,2

(Ω

, R

)



> 0.

436 M. Lewicka

Several interesting questions further arise in the study of the proposed energy

functionals. A ﬁrst one is to determine the scaling of the inﬁmum energy in terms

of the vanishing thickness h → 0. Another is to ﬁnd the limiting zero-thickness

theories under the obtained scaling laws.

In [20], we considered the case where g

is given by a tangential Riemannian

metric [g

αβ

] on Ω, and is independent of the thickness variable:

= g(x



⎡

⎣

αβ



)

⎤

⎦

∀x



∈ Ω,x

∈ (−h/2,h/2). (2.1)

The above particular choice of the metric is motivated by the results of [14]. The

experiment presented therein consisted in fabricating programmed ﬂat disks of

gels having a non-constant monomer concentration which induces a ‘diﬀerential

shrinking factor’. A disk was then activated in a temperature raised above a critical

threshold, wherein the gel shrunk with a factor proportional to its concentration.

This process deﬁned a new target metric on the disk, of the form (2.1) and radially

symmetric. Consequently, the metric induced a 3d conﬁguration in the initially

planar plate; one of the most remarkable features of this deformation is the onset

of some transversal oscillations (wavy patterns), which broke the radial symmetry.

Following our point of view, note that if [g

αβ

] in (2.1) has non-zero Gaussian

curvature κ

αβ

]

,theneachR

≡ 0. In [20], we observed the following:

Theorem 2.2. [g

αβ

] has an isometric immersion y ∈ W

2,2

(Ω, R

) if and only if

−2

inf

dist

≤ C

(for a uniform constant C).Also,κ

αβ

]

≡ 0 if and only if, with a uniform positive

constant c:

−2

inf

dist

≥ c>0.

The existence (or lack thereof) of local or global isometric immersions of a

given 2d Riemannian manifold into R

is a longstanding problem in diﬀerential ge-

ometry, its main feature being ﬁnding the optimal regularity. By a classical result

of Kuiper [15], a C

isometric embedding into R

can be obtained by means of con-

vex integration (see also [9]). This regularity is far from W

2,2

, where information

about the second derivatives is also available. On the other hand, a smooth isom-

etry exists for some special cases, e.g., for smooth metrics with uniformly positive

or negative Gaussian curvatures on bounded domains in R

(see [11], Theorems

9.0.1 and 10.0.2). Counterexamples to such theories are largely unexplored. By

[13], there exists an analytic metric [g

αβ

] with nonnegative Gaussian curvature on

a 2d sphere, with no C

isometric embedding. However such metric always admits

a C

1,1

embedding (see [10] and [12]). For a related example see also [24].

Non-Euclidean Plate Theories 437

3. The prestrained Kirchhoﬀ model

Consider now a class of more general 3d non-Euclidean elasticity functionals:

(u)=



W (x



, ∇u(x)) dx, (3.1)

where the inhomogeneous stored energy density W :Ω×R

n×n

−→ R

satisﬁes the

conditions given below of frame invariance, normalization, growth and regularity,

as in (1.3): with respect to the energy well F

given in (1.4), relative to g

= g as

in (2.1). Note that F

(x)=F(x



) is independent of h and of x

(i) W (x



,RF)=W(x



,F) for all R ∈ SO(3),

(ii) W (x



√

g(x



)) = 0,

(iii) W (x



,F) ≥ c dist

(F, F(x



)), with some uniform constant c>0,

(iv) W has regularity C

in some neighborhood of the set {(x



,F); x



∈ Ω,

F ∈F(x



)}.

Properties (i)–(iii) are assumed to hold for all x ∈ ΩandallF ∈ R

3×3

The following two results provide a description of the limiting behavior of

the energies

as h → 0. Namely, we prove that any sequence of deformations

with

) ≤ Ch

,convergestoaW

2,2

regular isometric immersion y of the

metric [g

αβ

]. Conversely, every y with these properties can be recovered as a limit

of u

whose energy scales like h

. The Γ-limit [3] of the energies is a curvature

functional on the space of all W

2,2

realizations y of [g

αβ

]inR

−→ I

(y)whereI

(y)=





)





αβ

]

−1

(∇y)

∇"n





. (3.2)

Here "n is the unit normal to the image surface y(Ω), while



) are the follow-

ing quadratic forms, nondegenerate and positive deﬁnite on the symmetric 2 × 2

tensors:



)(F )=∇

W (x



, ·)

√



)

(F, F),



)(F

2×2

)=min{



)(

F );

2×2

= F

2×2

We use the following notational convention: for a matrix F ,itsn × m principle

minor is denoted by F

n×m

and the superscript

refers to the transpose of a matrix

or an operator.

Theorem 3.1. Assume that a given sequence of deformations u

∈ W

1,2

(Ω

, R

)

satisﬁes

) ≤ Ch

, (3.3)

where C>0 is a uniform constant. Then, for some sequence of constants c

∈ R

the following holds for the renormalized deformations y



)=u



,hx

) −

∈ W

1,2

(Ω

, R

(i) y

converge, up to a subsequence, in W

1,2

(Ω

, R

) to y(x



)=y(x



) and

y ∈ W

2,2

(Ω, R