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МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ
MSC 30С62, 35J55 DOI: 10.14529/mmpl50302
QUANTITATIVE ESTIMATES ON JACOBIANS FOR HYBRID INVERSE PROBLEMS
G. Alessandrini, Department of Mathematics and Geosciences, University of Trieste, Trieste, Italy, [email protected],
V. Nesi, Department of Mathematics, Sapienza University of Rome, Rome, Italy, nesi@mat .uniromal. it
We consider ст-harmonie mappings, that is mappings U whose components щ solve a divergence structure elliptic equation div(aVMj) = 0, for i = l,...,n. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.
Keywords: elliptic equations; Beltrami operators; hybrid inverse problems; composite materials.
In memoria di Alfredo.
Introduction
The appearance of coupled physics methods has provoked a sharp change of perspective in inverse boundary problems. The simultaneous use of different physical modalities to interrogate, through exterior measurements, a body whose interior parameters are unknown has enabled to single out interior functional which carry useful, and possibly stable, information on the parameters of interest. Such methods are also known under the name of "hybrid inverse problems". Notable examples are the coupling of Magnetic Resonance with Electrical Impedance Tomography [1], Ultrasound and Electrical Impedance Tomography [2], Magnetic Resonance and Elastography [3]. To fix ideas, let us focus on Ultrasound Modulated Electrical Impedance Tomography. In EIT the goal is to determine the, possibly anisotropic, electrical conductivity a = {aij} of a body tt by repeated boundary measurements of voltage u\qq and current distribution aVu ■ v with u solving the elliptic PDE
div(aVu) = 0, in tt. (1)
As is well known [4], the stability is very weak and, in fact, in the anisotropic case, also non-uniqueness occurs [5]. By combining electrical measurements with ultrasound measurements it is possibile to focus on a tiny spot near any point x E tt and it has been shown by Ammari et al. [2] that one can detect the localized energy
H(x) = aVu ■ Vu(x). (2)
If one repeats the experiments with different boundary voltages, it is possible to extract the functional
Hij (x) = aVui ■ Vuj (x). (3)
Вестник ЮУрГУ. Серия «Математическое моделирование 25
where ui,...un is an array of different solutions to (1). In Monard and Bal [6,7], it is shown how, from such functional, one may obtain the conductivity a in a satisfactory stable fashion. The crucial point, however, is to be able to set up an array of boundary data ..., 0n and corresponding solutions ul,.. .un in such a way that the functional Hj are non degenerate.
In other words, calling U : Q ^ Rn, the mapping U = (ul,... ,un), which we shall designate V-harmonic" mapping, it is required that the Jacobian determinant det DU does not vanish. And, furthermore, for the purpose of stability, a quantitative lower bound would be needed.
This is the main question that we wish to address in this note, which essentially stays behind all coupled physics problems mentioned above, and other inverse problems as well. The same issue showed up, for instance, in the field of groundwater transmissivity detection [8].
This kind of questions also arises in the branch of the homogenization theory which studies effective properties of composite materials. We give a brief outline here.
Indeed, the positivity of Jacobians of injective a-harmonic mappings has attracted attention in several applications. In two dimensions, the first application of this positivity has been given in [9]. The long standing problem of improving the so-called Hashin -Shtrikman bounds [10] for the effective conductivity of composite materials was addressed in that paper. The method used was based on ideas of Mural and Tartar [11] and Tartar [12], a reference not easy to find. We refer to [13] for a more complete treatment. The bottom line is as follows. The question of interest, in the simplest not yet known at that time, case is the following. Three numbers 0 < al < a2 < a3, representing the conductivity of three isotropic materials, called the phases, three "volume fractions" pl,p2 and p3, summing up to 1 and representing the area proportions of the phases and a 2 x 2
3
matrix A, parametrizing the affine boundary data, are given. Assume that a = ^^ Xi(x)ai
i=l
where xi represents the characteristic function of the set where a is equal to ai times the
identity matrix and -r^ f xi(x)dx = pi,i = 1, 2, 3. Then one aims to determine a bound 1 1 n
from below for the following quantity
F(A) = inf inf / Trace[(DUo(x) + A)Ta(x)(DUo(x) + A)]dx. (4)
Xl,X2,X3 Uo£Wi'2(n) |Q| J
n
The overall problem is non linear and actually is linked with the notion of quasi-convexity.
A
function of DU, with U — Ax G W0l'2(Q). This function turns out to be the minimum of three quadratic functions, as shown by Kohn and Strang in [14]. However, for our
U0
U(x) = U0(x)+ Ax is the a-harmonic mapping with affine boundary data given by U = Ax on 3Q. Optimal lower bound for (4) were found by Kohn and Strang exactly exploiting the connection with the optimal bound for effective conductivity found by Mural and Tartar and, later, by Cherkaev and Lurie [15]. The optimality is restricted to the case when only two isotropic phases are present that is, only two materials are "mixed". For three or more phases, the methods based on compensated compactness gave suboptimal bounds. In this specific context, the compensated compactness method uses simply the constraint that
DU
The classic strategy gives the so-called "Wiener bound", that is the harmonic mean bound. It is obtained considering the test fields B in the class
Bo := {B e L2(tt) : y^y J B(x)dx = A v n
One obtains
F(A) > Fo(A) := inf I Trace[B(x)Ta(x)B(x)]dx
BeBo \il\ J
n
-1
m / 1 /* \
Trace
A I |Q| / a-1(x)dx I A n
(5)
Tartar's ideas, based on compensated compactness, in this simplified context lead to an improved bound (called the "translation bound" by G.W. Milton) obtained by considering the new test field in the class
B1 := {B e L2(Q) : щ J B (x)dx = A, щ J det B(x)dx = det A.
n n
One obtains
F(A) > Fi(A) := Ы ^ f Trace[B(x)Ta(x)B(x)]dx.
n
The computation is more involved than (5). This technique, however, gives an optimal answer in two dimensions, when specialized to the case of two-phase isotropic materials. When one deals with more than two phases this approach is no longer optimal. The results in [16] have the following corollary. Set
B2 := { B e L2(Q) : щ J B(x)dx = A, щ J det B(x)dx = det A det B > 0 , a.e. in Q
nn
One has
F(A) > F2(A) := Ы |Q| J Trace[B(x)Ta(x)B(x)]dx. (6)
n
In fact, in [9], it is proved that F2(A) > F1(A), as soon as one deals with more that two isotropic phases, for suitable choices of the given parameters pi and matrices A. Later, new optimal microgeometries were found for multiphase materials in [17] and, using again the positivity of the Jacobian determinant, it was possible to prove their optimality according to a stricter criterion, see [18]. The key is exactly the universal bound given on the Jacobian
det A det B > 0
this context, it is highly desirable not to have any constraint on the regularity of the interfaces between phases. When a is non-symmetric, applications to composites have been
given, for instance, in the context of the classic Hall effect by Briane and Milton [19,20]. Other applications have considered the problem of determining which electric fields are realizable by Briane, Milton, and Treibergs [21]. On the other hand one would like to have similar improvements in higher dimensions. Briane and Nesi [22] studied the case of laminates of high rank showing that, for these special microgeometries the positivity of the determinant of the "Jacobians" of the corrector matrix holds in any dimension. To explain the result in detail would require too long a digression. However, roughly speaking, one
a
positivity of the Jacobian determinant if one makes assumptions on the "microgeometry". On the other hand, even in the very restricted setting of periodic boundary conditions, particularly adapted to composites, and even under the assumption of dealing with only two isotropic phases, there is no hope to control the sign of the Jacobian determinant of a
explicit example was provided by Briane, Milton and Nesi [23].
We now go back to the precise subject of the present paper. We pose the following problem. Problem 1.
Can we find Dirichlet data
such that the corresponding solution mapping U = (ul, • • • ,un) is such that det DU is
a
Note that, in this context, it is essential that the choice of the boundary data is aa
a
of regularity shall be needed.
Problem 1 has a different phenomenology depending on the space dimension. When n = 2 the issue is more or less completely understood, whereas when n = 3 or higher, various kinds of pathologies show up. A review of such pathologies and a discussion of the open issues when n > 3 shall be the object of Section 3.
n=2
the Jacobian determinant under essentially minimal regularity assumptions. This is the content of our main Theorem 2 which is the new contribution of this paper to this subject.
We start reviewing the main known results in dimension n = 2. It was proved in Bauman et al [16] that, if a is Holder continuous, Q has Cl'a boundary and $ is a Cl'a diffeomorphism onto the boundary of a convex domain, then det DU > 0 everywhere.
a
of classical results on two dimensional elliptic first order systems with Holder coefficients see, for instance, [24] Appendix and also [25, Proposition 5.1], the result extends as well to the non-symmetric case. On the other hand, the present authors [26], proved that when a is merely and $ is a homeomorphism onto the boundary of a convex domain, then det DU > 0
aU
$ = (0l, ••• ,0n) : d Q ^ R
n
(7)
log det DU G BMO and, subsequently [27], this result was improved to
det DU A
(8)
(9)
that is the class of Muckenhoupt weights [28].
We recall that for purely harmonic mappings, Lewy's Theorem [29], states that for two-dimensional harmonic homeomorphisms, the Jacobian determinant cannot vanish
n=2
diffeomorphisms. However the Jacobian determinant may vanish at boundary points.
It is also worth mentioning that the convexity assumption on the target of the boundary mapping $ is sharp, Choquet [30], Alessandrini and Nesi [31], if one wishes to have a condition expressed merely on the "shape" of the target and not on its parametrization.
a
det DU Q
on a well-known one by Meyers, is illustrated.
In the next Section 1 we shall prove a quantitative version of the result in [16]. The starting
$
viewed as a parametrization of the boundary of the convex target, see Definitions 1, 2, 4. The subsequent step consists on a quantitative lower bound of the modulus of the gradient of a scalar solution to equation (1), Theorem 1. This estimate may be interesting on its own. Finally we state and prove our main result, Theorem 2.
1. The Quantitative Bounds
Let 0 : R ^ R be a T-periodic Cl function. Let u : [0, to) ^ [0, to) be a continuous strictly increasing function such that u(0) = 0.
Definition 1. Given m,M G R, m < M, we say that 0 is quantitatively unimodal if there exists numbers tl < t2 <t3 < t4 <tl + T such that
0(t) = m
0(t) = M
0 (t) > min{u(t - t2),u(t3 - t)},
—0 (t) > min{u(t - t4),u(tl + T - t)}
In the sequel we will refer to the quadruple {T, m, M, u} as to the "character of
0
The concept of unimodality, but not this terminology, first appears in Kneser [32], when he proved Rado's conjecture [33] concerning the case of "purely" harmonic mappings. The terminology "unimodality" was introduced in this context by Leonetti and Nesi [34], following the work of Alessandrini and Magnanini [35]. A different terminology (almost two-to-one functions) has also been used for the same concept, Nachman, Tamasan and Timonov [36].
Let r C R2 be a simple closed curve parametrized by a T-periodic Cl mapping
$ : R ^ R2 (11)
in such a way that $|[0T) is one-to-one.
Definition 2. We say that r is quantitatively convex if for every £ G R2, |£| = 1 the function
0t = $ • £
t G [ti,t2J ,
t G [ts,t4] ,
t G Ma] , t G [t4,ti + T] .
(10)
is quantitatively unimodal and its character of unimodality is given by {T,m,, M,,u} with m,, M, such that M, — m, > D, for a given D > 0.
In the sequel we will refer to the triple {T, D, u} as to the "character of convexity" of r.
Remark 1. If r is quantitatively convex then it is convex, that is, it is the boundary of a convex set G. In fact each tangent line to r turns out to be a support line for G. The following Lemma provides a sufficient condition for quantitative convexity. Roughly speaking, it says that if r is an appropriately parametrized C2 simple closed curve with strictly positive curvature, then it is quantitatively convex in the sense of Definition 2, and the character of convexity can be computed in terms of the parametrization. Here, for the sake of simplicity, we have chosen the arc-length parametrization, because the main purpose of this Lemma is to provide a variety of examples, but we emphasize that in general, the character of convexity does depend on the parametrization of the curve and not only on its image.
We convene to denote by J the matrix representing the counterclockwise rotation of 90 degrees
' 0 -1
' = 0 -)■
Lemma 1. Let T be such that $ G C2 and:
i) |$'| = 1,
ii) 0 <k < $" • JT$' < K , ^
then r is quantitatively convex with character {|r|, K, 2K t}■
Proof Condition i) of Lemma 1, implies that $'(t) = els(t\ 0 < t < T and we may assume that s(0) = 0. Without loss of generality we assume that $ is orientation preserving. Then, by condition ii) of Lemma 1, one has 0 < k < s'(t) < K. Picking, w.l.o.g., £ = e2,
fa(t) — fa(0) = ^ sin(s(T))dr. 0
The function s(t) ranges ^^^r ^te whole interval [0, pickin g tn such th at s(tn) = n, we have
M, — m■ = I ^W1 ds(T) > 1! sin(s)ds = K , 00
fa,(t) = sin(s(t)), t
s(t) = J s'(t)dr > Kt,
0
fa,(t) > 2Kt, 0 < s(t) < n. Thus we may pick D = K and u(t) = 2K t ,t > 0.
_a
3Q Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 25-41
t
We shall consider Q a bounded simply connected domain in R2 with Cl'a boundary. In order to make precise the quantitative character of such regularity we introduce the following definition.
Definition 3. A domain Q C R2 is said to be of class Cwith constants p0,M0, positive and Holder exponent a E (0,1], if for a ny P E dQ, there exist a rigid change of coordinates such that P = 0 and we have
Ü n Bpo(0) = {ж E Bpo(0) : Х2 > ФХ)}, (13)
where ф : [—p0,p0] ^ R2 is a C1a function satisfying
ф(0) = ф'(0) = 0 (14)
and also
и //и i+a \Ф'(х) — Ф'(Х)\ (i-pcpol) + Ро\\ф \\l-([-P0,P0]) + Po sup - _ a- < Mopo . (15)
x,x'Е[-Р0,Р0] \Х Х \ x=x'
Definition 4. Given a C 1,a(dQ; R) function 0, we shall say that it is quantitatively unimodal, if considering the arclength parametrization of dQ, x = x(s), 0 < s < T = IdQ|, the periodic extension of the function [0,T] 9 s ^ 0(x(s)) is quantitatively unimodal with character {T,m, M,u}. For such a function 0, we introduce the following closed arcs, possibly collapsing to a single point:
Tmin = {x G dQ : 0 = m} , , ,
rmax = {x G dQ : 0 = M} . {
Accordingly, a mapping $ G Cl'a(dQ; R2) shall be said quantitatively convex with character {T, D, u} if the periodic extension of $(x(s)) fulfils the conditions of Definition 2.
Let us consider a = {aj}i,j=l,2 a, not necessarily symmetric, matrix of coefficients aj : Q ^ R satisfying the ellipticity condition
^(x)C • С > K-1\C\2 , for every С E R2 (J-l(x)£ • С > K-1\С\2 , for every С E R2
(17)
K
Wij(x) — (Jij(x')\ < E\x — x'\a , Ух,х' E Ü , (18)
for given a 0 < a < 1 and E > 0.
We shall consider the Wl'2(Q) solution u to the Dirichlet problem
{
div((Vw) = 0 in Ü , u = ф on д Ü .
(19)
We recall that, in view of the classical regularity theory, и in fact belongs to C(П), for some в < a and its norm is dominated by the C^"-norm of ф, modulo a constant which only depends on p0, M0, K and with p0, M0 as in Definition 3.
Lemma 2. Let fa : dQ ^ R be quantitatively unimodal with character {|dQ|,m, M,u} and assume that
\£fa(x(s)) — £fa(x(s'))\ < EIs — s'la , Ws, s' e [0, |dQ|]. (20)
Then there exist k,8 only depending on the character of unimodality (see Definitions 1, 4) and on a, E, such that if
x e Q and dist(x, rmin U rmax) < 5 , (21)
then
|Vu(x)| > k. (22)
Proof. Up to a Cl'a diffeomorphism, with constants only depending on p0,M0 and ^Q|, we may assume that Q = B^0).
It is well known that in such new coordinates u solves a new Dirichlet problem of type (19)
with a new matrix of coefficients and new boundary data that, however, satisfy analogous
assumptions with constants and parameters only depending on the same a-priori data. For
the sake of not to overburn the notation we stick to the one of (19).
By the C 13 regularity of u, if dist(x, Tmax) < then u(x) > M — Cn with C > 0 only
depending on the a-priori data.
Let us pick n such that
M—m
M — Cn > -.
1 ~ 2
Hence, by Harnack's inequality [37],
u(x) — m > C Mp1 > 0, for every x e B1_n(0).
C
and Gilbarg [38, Lemma 7], which applies to equations in divergence form, and Holder continuous a, we obtain
|Vu(x)| > Ko > 0 , yx e rmin , (23)
K0
By C1il3 regularity we have
|Vu(x)| > k0 — C513 Wx e Q such that dist(x, rmin) < 5. Picking 5 such that C53 < Kr, we obtain
|Vu(x)| > f > 0 , if dist(x, rmin) < 5. (24)
A symmetrical result applies in the neighborhood of Tmax.
□
Lemma 3. Under the same assumptions as in Lemma 2, there exists r > 0 such that
V^x)^ L > 0 , Wx e Q , dist(x, dQ) < r. (25)
Here L and r are positive and only depend on the a-priori data.
Proof. If we pick x E dQ, and write x = x(s) such that dist(x, rmin U rmax) > 8, we have
|Vw(x(s) • x'(s)|
d
-ФШ
> w(8).
By C13 regularity
| > — Cr13 , Wx E Q such that dist(x,d Q) <r.
Picking r such that
Cr3 = ^ min{ft, w(i)} ,
the thesis follows.
□
Theorem 1. Let Q be a simply connected domain, C1,a-regular with constants {p0,M0} (see Definition 3). Let fa : dQ ^ R be quantitatively unimodal with given character {ldQI,m, M,u}(see Definitions 1, 4) an(i let it satisfy the Holder condition (20). Let ° = {aij (x) }i,j=1,2 satisfy the ellipticity condition (17) and the Holder bound (18). Let u E W 1,2(Q) be the solution of the Dirichlet problem (19).
Then there exists C > 0, only depending on the a-priori data as above, such that
|Vu(x)| > C > 0 , for every x E Q . (26)
Remark 2. Under stronger regularity assumptions, in particular assuming that a is Lipschitz continuous, a similar result was proven already in [40, Theorem 3.2].
Proof. As is well-known, there exists U E W1,2(Q), called the stream function associated u
VU = JaVu everywhere in Q , J = ^ 1 o^ ^ . ^^
Using complex notation z = x1 + ix2, f = u + iu, the system (27) can be rewritten as
fz = vfz + vfz in Q , (28)
where, the so called complex dilatations ¡i, v are given by
. _ о"22-о"11~ЧО-12+о"21) v _ 1-deta+i(ai2-a2l) fOQ)
Г = 1+Tr ст+det a ' V = 1+Tr ст+det a ' ^ J>
Q"22-o"11-i(o"12+o"21) v _ 1-deta+i(a12-a21)
1+Tr a+det a '
and satisfy the following ellipticity condition
K - 1
П + lvl< KTT, (30)
and, being a Holder continuous, also fi and v satisfy an analogous Holder bound. In [16], it is proven that f is a Cl'3 diffeomorphism of Q ontо f (Q). The lower bound obtained in Lemma 3, implies that, setting
Qr = {x E Q : dist(x, дQ) > r} ,
f : Q\Qr ^ C, is a bilipschitz homeomorphism with constants only depending on the a-priori data. We have identified C with R2 in the canonical way.
Hence f (Q) is also a Clil3 domain with constants controlled by the a-priori data. Note that also |d(f (Q))| is controlled.
Let us denote g = f-l(w), w G C. A straightforward calculation gives
gw = -v(g)gw - v(g)gw. (31)
g
and Holder continuity, with constants only depending on the a-priori data. By standard interior regularity estimates, gw is bounded in f (Qr). Using (31), we have
which can be rewritten as
which in turn implies
gw|2-gw|2<c2 in f(Qr), (32)
fw ? -fw? > C-2 in Qr
\vu > C-1 in Qr. (33)
Hence, in combination with Lemma 3, the thesis follows.
□
Theorem 2. Let Q and a be as in Theorem 1. Let $ = (0l; 02) : dQ ^ R2 be quantitatively convex, see Definitions 2, 4, with character {^Q^ D,u}. Let U = (ul;u2) G Wl'2(Q; R2) solve
div(aVui) = 0 in Q ,
{
ui = 0i on dQ. ^^^
C>0
U : Q ^ U(Q) c R2
is a Cdiffeomorphism and
det DU > C2 > 0 in Q. (35)
Remark 3. A similar result, under slightly more restrictive hypotheses, has been recently proved by G.S. Alberti [39]. In fact the approach in [39] is based on estimates in [40,41]
a
a
U
Trace(DUT DU) 2 det DU '
have been recently studied in [42, Theorems 3.1, 3.4].
U
bound (35) remains to be proven. For any £ G R2, |£|2 = 1, we may apply Theorem 1 to
u^ = U • £
| Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 25-41
and obtain
\(DU )f| = \Vu^ \> C> 0 in H. (36)
Or equivalently
\DUTDU£ • = \Vu^\2 > C2 > 0 in H, (37)
that is the eigenvalues A^x) and A2(x) of the symmetric matrix DUT(x)DU(x) are uniformly bounded from below:
Aj(x) > C2 > 0 , i = 1, 2 , V x G H.
Therefore
(det DU)2 = A1(x)A2(x) > C4 > 0 , everywhere in H. (38)
Since U is sense preserving, one has det DU > C2 > 0 everywhere in H
□
Remark 4. Theorem 2, has some feature in common with the results in [31]. In the latter paper the authors consider harmonic mappings which are extensions of given Dirichlet
U
H
determinant on the boundary, so implicitly imposing constraints on the parametrization of the boundary of the image. One may wonder whether an assumption just on the shape of the target may suffice. This is not the case even in the purely harmonic case. Indeed one may exhibit a sequence Un of sense preserving, injective, harmonic mappings of the unit disk onto itself, fixing Un(0) = 0, such that det DUn(0) ^ 0 as n ^ The convergence holds uniformly on compact subsets of the unit disk. The limit harmonic mapping in not univalent. See [43, Section 4.1].
2. Discontinuous Coefficients. An Example
We elaborate on a well-known example by Meyers [44]. See also Leonetti and Nesi [34] for an application in a related context. For a fixed a > 0 we consider the symmetric matrix of coefficients
/ a-1x2+ax2, (a-1-a)xi x2 \ ' 2 , 2 . 2 , 2 '
a(x)
+ x 2 x 2 I X 2
(a 1-a)xix2 axf+a 1x2 \ xi+x2 x2+x2 /
(39)
Is is a straightforward matter to check that its entries belong to L^ and that a has eigenvalues a and a-1. Therefore a satisfies the uniform ellipticity condition (17) with ellipticity constant
K = max{a, a-1} ,
and a is discontinuous at (0,0) (and only at (0, 0), when a = 1). Let us denote
u1(x) = \x\a-lx1, u2(x) = \x\a-1x2.
A direct calculation shows that Hi G W 1'2(B1(0)), i =1, 2 and that they solve the Dirichlet problem
div(aVui) = 0 in B1(0)
ui = xi on dB1(0).
{
Note also that f = u1 + iu2 is a quasiconformal map ping of 51(0) onto itself and it solves the Beltrami equation
f = a — 1 z
Jz , 7 n Jz .
a + 1 z
Setting U = (u1 ,u2), we compute
det DU = Ifz|2 —Ifz|2 = a|z|2(a_1).
Therefore det DU vanishes at (0, 0) when a > 1, whereas, when a E (0,1), it diverges as z 0.
3. Mappings in Higher Dimensions. Examples and Open Problems
det DU
methods which are intrinsically two-dimensional (the Beltrami equation). Only part of the result can be extended to higher dimensions.
For instance, with minor adaptations of the method developed in the Section 1, one can argue as follows.
Consider Q C Rn, a bounded domain diffeomorphic to a ball of class Cl'a and with constants p0, M0 defined with the obvious slight adaptations of Definition 3. Let a = {aij }i,j=1,2 be the matrix of coefficients and let it satisfy uniform ellipticity with constant K as in (17) and Holder continuity like in (18).
Let G C Rn be a convex body whose boundary T is C2 and having at each point principal curvatures bounded from below by k > 0.
Let $ = (fa1,fa2, ■ ■ ■ , fan) : dQ ^ T be an orientation preserving diffeomorphism such that $-1 are C1,a with const ant Let U = (u1 ,u2, ■■■ ,un) E W 1,2(Q; Rn) be the weak solution to
div(aVui) = 0 in Q ,
ui = fai on Q ,
i = 1, 2 ■ ■ ■ ,n.
Then, by the same arguments used in Section 1, we obtain.
Theorem 3. Under the above stated assumptions, there exists p > 0 and Q > 0 such that U is a diffeomorphism of Q\Qp onto a neighborhood of T, within G and we have
det DU > Q in Q\Qp.
We omit the proof.
When n > 3, there is no chance, under the kind of hypotheses stated above, to obtain
det DU
that have been produced in a wide time span. A first illuminating example goes back to Wood [45] and has the amazing feature of being totally explicit. Wood displayed the following harmonic polynomial mapping from R3 onto R3:
U(x1,x2,x3) = (u1; u2, u3) = (x3 — 3x1x2 + x2x3,x2 — 3x1x3, x3)
U det DU = 0
{x1 = 0}
Later Melas [46] provided an example of a three dimensional harmonic homeomorphism U : 51(0) — B1(0) such that det DU(0) = 0. Subsequently, Laugesen [47], showed that there exists homeomorphisms $ : dB 1(0) — dB 1(0) which are arbitrarily close to the
U = ( u1 , u2 , ■ ■ ■ , un)
Am = 0 U = Ф i = 1, 2- •
in on
Bi(0), dBi(0)
, n.
is such that det DU changes its sign somewhere inside B1(0). Such examples are especially
n > 3
a
mapping is invertible at a topological level (because it may reverse orientation!) and not only as a differentiate mapping.
One may wonder whether changing the topology of the boundary data may help. In the periodic case, obviously the harmonic functions are linear and this might have left the hope that, for variable coefficients, the periodic case may be better that the generic Dirichlet problem. However this is not the case. In [23], it was proved that, in dimension three, one a
periodic arrangement with a smooth interface, but such that the corresponding solution U
sign in the interior of the (unit) cube of periodicity.
If, from the above examples, it seems that few chances are left of finding a universal
a
aU goal would be to find a way to control, in term of the Dirichlet data, the set of points where the Jacobian may degenerate and possibly evaluate the vanishing rate at such points of degeneration.
This appears as a completely open problem, not at all easy as the following example by Jin and Kazdan [48] shows. Let a E C^ (R; R) and set
with
a(x)
a(x3) = 0 a(x3) e (0,ao)
1
а(хз) 0
for for
а(хз) 1 0
хз < 0, x3 > 0
0 0
b(x3)
ao e (0, 1)
(40)
b(x3) = tz
i
1-a2 (хз)
for x3 e R .
We set
U (x) = (x1,x2, —x1x2 + fa(x3))
fa
(bfa')' — 2a = 0 , x3 E R , fa(x3) = 0 , x3 < 0 .
(41)
{
fa > 0 x3 > 0
det DU = | fa
ф' > 0 , for x3 > 0 ф' = 0 , for x3 < 0 .
a
property of unique continuation (whereas this is the case for \DU\2 = Trace(DUTDU)).
det DU
Remark 5. The above example has some striking features. First of all note also that, letting a0 \ 0, we can make a as close as we want to the identity matrix. Moreover U converges, uniformly on each compact subset of R3, to the harmonic polynomial mapping Uo(x) = (x1,x2, -x1x2).
We conclude by noticing that a limiting case of the above construction yields an
a
As before we pose
a(x) =
a(x3) = 0 a(x3) = ao
fr(xs)
l—a2(x3)
1 a(x3) 0
a(x3) 1 0
0 0 b(x3)
for x3 < 0 ,
for x3 > 0 with a0 G (0,1) for x3 G R.
(42)
That is a is piecewise constant, namely
a(x)
100 010 0 0 1
when x3 < 0 and a(x)
1 ao ao 1 00
0 0
x3 > 0
l—a0
Again we pose
U(x) = (xi,x2, -xix2 + 0(x3))
(43)
where now 0 is given by
!
0(x3) = a0(1 — a2) x2 , x3 > 0 , 0(x3) = 0 , x3 < 0 .
We obtain that U is a a-harmonic mapping with C1,1 regularity and, analogously to the previous example, it satifies
det DU
{
2a0(1 — a2) x3 > 0 , for x3 > 0 ,
0 , x3 < 0 .
Acknowledgements. G. Alessandrini was supported by FEA2012 "Problemi Inversi", Universita degli Studi di Trieste, V. Nesi was supported by PEIN Project 20102011 "Calcolo delle Variazioni".
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Received January 9, 2015
УДК 517.9 DOI: 10.14529/mmpl50302
КОЛИЧЕСТВЕННЫЕ ОЦЕНКИ ЯКОБИАНА ДЛЯ ГИБРИДНОЙ ОБРАТНОЙ ЗАДАЧИ
Д. Алессандрини, В. Нези
Рассматриваются ст-гармонические отображения, то есть отображения U с компонентами щ, являющимися решениями эллиптического уравнения div(<rVuj) = 0, для i = 1,... ,n. Исследуется вопрос нахождения таких условий Дирихле, при которых Якобиан отделен от нуля. Результаты такого рода необходимы при решении так называемых гибридных обратных задач, а также в теории усреднения границ для эффективных свойств композиционных материалов.
Ключевые слова: эллиптические уравнения; операторы Бельтрами; гибридные обратные задачи; композитные материалы.
Джованни Алессандрини, профессор, факультет математики и наук о Земле, Уни-вврситбт Триеста (г. Триест, Италия), [email protected].
Винченцо Нези, профессор, факультет математики, Римский университет Ла Са-пиенца (г. Рим, Италия), [email protected].
Поступила в редакцию 9 января 2015 г.
