1. INTRODUCTION. u yy. u yx

Analisi funzionale. Ð On weak Hessian determinants.Nota di LUIGI D'ONOFRIO, FLAVIAGIANNETTIe LUIGIGRECO, presentata (*) dal Socio C. Sbordone.

ABSTRACT. Ð We consider and study several weak formulations of the Hessian determinant, arising by formal integration by parts. Our main concern are their continuity properties. We also compare them with the Hessian measure.

KEY WORDS: Hessian determinant; Schwartz distributions; Hessian measure.

RIASSUNTO. ÐSui determinanti hessiani deboli.Consideriamo ed esaminiamo varie formulazioni deboli del determinante hessiano, definite come distribuzioni di Schwartz mediante integrazione per parti, principalmente riguardo alle loro proprietaÁ di continuitaÁ. Confrontiamo inoltre tali formulazioni deboli con la misura hessiana.

1. INTRODUCTION

In this paper we discuss several weak formulations of the Hessian determinant which still attract a great interest, seee.g.[3-5, 17, 10, 12]. We begin with the case of functions of two variables. LetVR2be an open set andube a function defined onVwith second order derivatives. We denote byHuthe Hessian determinant

HuˆdetD2_uˆ uxx uxy

uyx uyy

:

Ifuis sufficiently smooth, the following identities can be easily checked: Huˆ(uxuyy)x (uxuxy)yˆ(uyuxx)y (uyuxy)x

ˆ1

2(u uxx)yy‡ 1

2(u uyy)xx (u uxy)xy

ˆ(uxuy)xy 1₂(u2x)yy 1₂(u2y)xx:

On the other hand, each of the above expressions can be used to define a Schwartz distribution onV, providedubelongs to an appropriate Sobolev space. The first one we consider is the distribution of order zero defined by the rule

H0u[W]ˆ

Z

V

Hu(x;y)W(x;y)dx dy

for all test functionsW2D(V). Of course, the natural assumption for definingH0uis u2W_loc2;2(V), as it implies that the Hessian determinantHuis locally integrable onV.

Other distributions will result integrating formally by parts. Precisely, we define H1u[W]ˆ Z V (uxuxyWy uxuyyWx)dx dy ˆ Z V (uyuxyWx uyuxxWy)dx dy H2u[W]ˆ1₂ Z V (u uxxWyy‡u uyyWxx 2u uxyWxy)dx dy H 2u[W]ˆ1₂ Z V (2uxuyWxy ux2Wyy u2yWxx)dx dy for allW2D(V).

In general,H1uis a distribution of order one. It is well defined foru2W2; 4 3

loc. Indeed, by Sobolev imbeddingux;uy2L4loc, thusuxuxy anduxuyy are locally integrable and the first integral in the definition of H1uconverges. The same argument shows that the

second integral also converges for u2W2;43

loc. We remark that the two integrals give the same distribution. Therefore, we can also write

H1u[W]ˆ 1₂

Z

V

W_x(uxuyy uyuxy)‡Wy(uxxuy uxyux)dx dy:

REMARK1.1. The definition ofH₁uis related to the well-known concept of weak (or distributional) Jacobian introduced by Ball [2]. Actually, H1u is precisely the weak

Jacobian of the gradient mapDu.

To justify the definitions ofH2uandH2uwe again use Sobolev imbeddings: W_loc2;2W2;43

locWloc2;1Weloc2;1 C

0 W_loc1;2

…1:1†

where we denoted byWe_loc2;1the space ofW_loc1;1-functions whose second order distributional derivatives are Radon measures onV. Therefore, clearlyH2umakes sense foru2Wloc2;1, or even foru2We2;1

loc (with obvious interpretation ofuxx,uyy anduxyRadon measures). Surprisingly, H₂u makes sense for u2W_loc1;2 and does not require second order derivatives.

Thus, we are imposing weaker and weaker conditions onu. It will be clear that, if we are in a position to define two of the above expressions, they yield the same distribution. For example, ifu2We_loc2;1, we can define bothH2uandH2u, and we have

H2u[W]ˆH2u[W] …1:2†

1.1. Weak Hessian inRn.

To deal with functions of several variables, we will use the formalism of differential forms which is quite effective. LetVRnbe an open subset and letu2C1_{(V). The}

Hessian determinantHuˆdetD2_{uinduces an-form:}

Hu dxˆdux1^ ^duxn:

Forjˆ1;. . .;n, setvjˆdux1^ ^du^ ^duxn, whereduis at thej-th factor in the

wedge product, while fori6ˆjthei-th factor isduxi. Then

@

@xj vj ˆdux1xj ^ ^du^ ^duxn

‡ ‡dux1^ ^duxj^ ^duxn ‡ ‡dux1^ ^du^ ^duxnxj : …1:3†

The first term in the right hand side (assumingj6ˆ1) will appear also in_@@_x

1v1, but with

opposite sign becausedux1xj ˆduxjx1 andduare exchanged to each other. Similarly for

the other terms in the right hand side of (1.3), except for

dux1^ ^duxj ^ ^duxnˆHu dx: Hence we have Xn jˆ1 @ @xj vj ˆn dux1^ ^duxn; that is dux1^ ^duxn ˆ 1 n Xn jˆ1 @ @xj dux1^ ^du^ ^duxn: …1:4†

Identity (1.4) yields the definition of the distributionH1u. We multiply both sides by a

test functionWand integrate by parts in the right hand side: H1u[W]ˆ 1_n Xn jˆ1 Z V W_xjdux1^ ^du^ ^duxn: …1:5†

Using Laplace theorem for determinants, this is easily seen to equal H1u[W]ˆ 1_n Xn jˆ1 Z V uxjdux1^ ^dW^ ^duxn:

As above,dWis thej-th factor.

We proceed further from (1.4). One way is to notice that for eachjˆ1;. . .;nwe have

vjˆ( 1)j 1d(u dux1^ ^duxj 1^duxj‡1^ ^duxn):

Inserting this into (1.4), multiplying by the test functionWand integrating by parts twice will lead to the definition ofH2u:

H2u[W]ˆ1_n Xn jˆ1 Z V u dux1^ ^duxj 1^dWxj ^duxj‡1^ ^duxn: …1:6†

To defineH

2u, fori6ˆj we write

vjˆ( 1)i 1d(uxidux1^ ^du^ ^duxi 1^duxi‡1^ ^duxn):

Now we sum with respect toi and insert into (1.4). In a routine manner, this gives (1.7) n(n 1)H 2u[W]ˆ ˆX i6ˆj Z V uxidux1^ ^du^ ^duxi 1^dWxj ^duxi‡1^ ^duxn:

We recall that in each termduis thej-th factor anddW_xj is thei-th factor of the wedge product.

Now we consider functions uin Sobolev spaces. If u2W2;n‡n21

loc (V), then clearly the minors of ordern 1 ofD2_{uare locally integrable with exponentn}2_=(n2 _{1), while by}

Sobolev imbedding theorem the first order derivatives of uare locally integrable with exponentn2_{. Noticing that these exponents are HoÈlder conjugate, we easily see thatH1u}

makes sense:

H1u:W2;

n2 n‡1

loc (V)!D0(V):

For H2u we recall that functions in Wloc2;n 1(V) are continuous (actually, HoÈlder continuous with exponent (n 2)=(n 1) if n>2). For H₂u we remark that if

u2W2;n‡n22

loc (V), then the minors of ordern 2 ofD2uand the first order derivatives of uare locally integrable respectively with exponentsn2_=(n2 _{4) andn}2_{=2. Therefore,}

H2u:W_loc2;n 1(V)!D0(V); H2u:W2;

n2 n‡2

loc (V)!D0(V):

We notice that in dimensionn>2 the definition ofH₂urequires thatuhas second order derivatives.

1.2. The Hessian measure.

Another concept of weak Hessian is classically introduced for defining generalized solutions of the Monge-AmpeÁre equation, see [9] and references therein; see also [18] for further developments. Now, we assume thatVRnis a convex set anduis a real-valued convex function defined onV. We denote by@uthe subdifferential (also called normal mapping) ofuand by@u(E) the image by this multifunction of the subsetEV:

@u(E)ˆ[

x2E

@u(x):

The Hessian measure ofu, which we denote byMu, is a Borel measure onVdefined by

Mu(E)ˆLn(@u(E))

…1:8†

for each Borel subsetE.Ln_{is the Lebesgue measure inR}n_{. It is well known that for}

sense that the following equality holds Z V WdMuˆ Z V WHu dx; …1:9†

for all continuous Wwith compact support.

We recall that a convex functionuis locally Lipschitz and its gradientDuhas locally bounded variation, see [6]. Henceu2We_loc2;1and in dimension 2 we can considerH2u. In

the following theorem we compare these notions.

THEOREM 1.2. If u is in the domain of two of the above weak formulations of the

Hessian, these formulation coincide.

As an illustration, in dimension nˆ2 if u is convex, then MuˆH2uˆH2u.

Theorem 1.2 will be an easy consequence of the continuity properties we discuss in the next section.

2. CONTINUITY PROPERTIES

One of our concern is to elucidate on the continuity properties of those weak formulations of the Hessian. First, we note trivially thatH1,H2andH2are continuous

in their corresponding domains with respect to the strong convergences. More precisely:

Ifuh !uinW2; n2 n‡1

loc , thenH1uh!H1uin the sense of distributions. This means

thatH1uh[W]!H1u[W], for allW2D(V);

Ifuh!uinWloc2;n 1, thenH2uh!H2uin the sense of distributions.

Fornˆ2, ifuh!uinWloc1;2, thenH2uh !H2uin the sense of distributions. Forn>2, ifuh!uinW2;

n2

n‡2

loc , thenH2uh!H2uin the sense of distributions.

It is also easy to prove convergence in D0 _{of the weak Hessians of u}

h coupling a suitable weak convergence of the minors of the Hessian matrix D2_u

h with a strong convergence of the functionuhor its first order derivativesDuh. As an example, we have

If Duh!Du strongly inL n2

2

locand (forn>2) the minors of order n 2 ofD2uh converge to corresponding minors ofD2_{uweakly inL}n2 4n2

loc , thenH2uh !H2uin the

sense of distributions.

Next, we examine continuity properties with respect to weak convergences. It is enough to consider the weakest formulation;i.e.H

2. We first consider the casenˆ2. In

[10] it is shown by an example that H

2 is not continuous with respect to weak

convergence inW_loc1;2. Here we present another simple example of this type. EXAMPLE2.1. The functions

clearly converge uniformly to zero andDuhweaklyinL1. On the other hand,H2uhare not converging to zero in the sense of distributions. To see this we consider a test function

Wsupported in the unit diskDR2_{, and compute}

H₂uh[W]ˆ Z D

(4xyWxy 2x2Wyy 2y2Wxx)cos2h(x2‡y2)dx dy: Integrating in polar co-ordinates and then by parts, we find that

lim h H 2uh[W]ˆ Z D (2xyW_xy x2_W yy y2Wxx)dx dyˆ2 Z D Wdx dy:

The above example and the following arguments indicate clearly that in order to ensure convergence ofH

2uh toH2uone really needs strong convergence inWloc1;2. THEOREM 2.2. Let uh 2Wloc2;1, hˆ1;2;. . ., and u2Wloc2;1 be given. Then the dis-tributionH2uhˆH2uhconverge toH2uˆH2u under each of the assumptions below:

(a)uh*u weakly in Wloc2;1;

(b)the sequence fu_hg is bounded in W2;1

loc and converges to u uniformly on compact subsets.

The proof follows easily when we realize that both assumptions (a) and (b) yield

uh!ustrongly inWloc1;2. Actually, we have

LEMMA2.3. (a))(b)) u_h!u strongly in W_loc1;2. We remark that both imbeddings ofW_loc2;1C0_andW2;1

loc Wloc1;2are not compact, in the sense that bounded sequences inW_loc2;1may not have subsequences converging locally uniformly, or strongly inW1;2

loc.

EXAMPLE 2.4. Given a function inu2W2;1(R2) with u60 and lim

z!1u(z)ˆ0, we

define uh(z)ˆu(hz), 8h2N. Then uh(z)!0, 8z2R2 f0g, Duh!0 in L1,

kD2_u

hk1ˆ kD2uk1for allh2NandD2uh* Cdweakly in the sense of measures, with

Ca constant matrix. Moreover fuhg has no subsequence converging locally uniformly and, askDu_hk₂ˆ kDuk₂ >0 for allh, no subsequence strongly converging inW1;2

loc. On the other hand, the assumption of weak convergence inW_loc2;1is essentially stronger than mere boundedness. By Ascoli-ArzelaÁ theorem, under assumption (a) we can prove uniform convergenceuh!uon compact subsets, hence condition (b) holds.

PROOF OFLEMMA2.3. There is no loss of generality in assuminguˆ0, otherwise we consider the functionsuh u.

Let us prove (a))(b). For simplicity, we assumeVˆ]0;1[2. ForW2C1 0 (V), we

definevh ˆWuh,hˆ1;2;. . ., and we have

Dv_hˆu_hDW‡WDu_h;

D2_v

hˆuhD2W‡DWDuh‡DuhDW‡WD2uh; hencevh*0 weakly inW2;1(V).

The point is that weak convergence offuhggives equi-integrability offDuhg. It is well known that W2;1_{(V) is continuously imbedded into C(V), see [1, p. 100]. If v is a}

continuous function of classW2;1_{(V) vanishing on@V, then for all (x}

1;y1) and (x2;y2) in V, we have jv(x1;y1) v(x2;y2)j4 Zx1 0 ds Zy1 y2 vxy(s;t)dt ‡ Zx1 x2 ds Zy2 0 vxy(s;t)dt : In particular, kvk₁4kvxyk1:

These inequalities yield equiboundedness and equicontinuity of any bounded family of functions inW₀2;1(V) with equi-integrable second order derivatives. We are in a position to apply Ascoli-ArzelaÁ theorem and get vh!0 uniformly in V. This implies uniform convergenceuh!0 on compact subsets ofV. IfKVis compact, it suffices to choose

W1 onK.

Now we assume that condition (b) holds and prove strong convergence inW_loc1;2(V). If

v2W₀2;1(V), integrating by parts we find Z V jDvj2ˆ Z V vDv4kvk₁kD2_vk 1:

Therefore, defining vhˆWuh as above, we see that vh!0 strongly in W1;2(V). To deduce from this strong convergence ofuhon an arbitrary compactKV, as before we only need to takeW1 onK.

Notice that (b) implies strong convergence of fuhg inWloc1;2 can be seen also as a

consequence of inequality (1.16) of [13]. p

Theorem 2.2 can be slightly generalized.

THEOREM2.5. Letfu_hgbe a sequence bounded inWe_loc2;1and converging to u uniformly

on compact subsets. Then u2We_loc2;1andH2uh ˆH2uhconverge toH2uˆH2u in the sense of distributions.

Now we assumen>2. We have the following continuity property.

THEOREM2.6. If u_h *u weakly in W_loc2;pfor some p>_nn_‡22 , thenH₂u_h!H₂u in the

PROOF. As p> n

2

n‡2 the imbedding Wloc2;pW1; n2

2

loc is compact. Then Duh!Du strongly in Ln22

loc. On the other hand, the minors of order n 2 of D2uh converge to corresponding minors of D2_{u in the sense of distributions, see [11, Theorem 8.2.1,}

p. 173], and are bounded inLnp2

loc, where we notice that_n p₂> n 2

n2 ₄>1, hence they

are weakly converging inLnp2

loc and therefore we can pass to the limit in the expression

definingH

2u[W]. p

REMARK2.7. In [5] it is shown that the conditionp> n

2

n‡2in Theorem 2.6 is sharp, in the sense that the conclusion cannot be drawn in general (in dimensionn>2) assuming

uh*u weakly in W2; n2 n‡2

loc :

…2:1†

With this condition, we still have weak convergence inLn2 4n2

loc of the minors of ordern 2

ofD2_u

h, but we need to reinforce (2.1) to guarantee the strong convergence inL n2

2 locof first

derivatives. For 1<n

2<r<n, we have the Sobolev imbeddings

W_loc2;rW_loc1;r C0;2 nr

loc ;

neither of which is compact. Clearly, ifuh!ustrongly inW1;r

loc, then it converges also strongly in the HoÈlder space. On the other hand, by the interpolation inequality (5) of [16], for a bounded sequencefu_hginW2;r

locthe convergence in the HoÈlder space implies strong convergence of first derivatives, that is, the two convergences are equivalent. For

rˆ_nn_‡2₂, we have the following generalization of Theorem 2.6, in the spirit of Theorem 2.5:

Let condition(2.1)hold together with uh!u strongly in C0;1

2

n

loc . ThenH2uhconverge

toH

2u in the sense of distributions.

We conclude this section by recalling some continuity properties of the Hessian measure, see [9].

THEOREM2.8. Let u_h, hˆ1;2;. . ., and u be convex functions and let the sequence u_h

converge to u uniformly on compact subsets. Then Muh converges to Mu weaklyin the

sense of measure, that is,

Z V WdMuh ! Z V WdMu for all continuous functionsWwith compact support.

It is clear that in the two-dimensional context Theorem 2.8 follows from Theorem 2.5.

3. THEL1-PART OF THEHESSIAN

One concern of the study of the weak formulations of Jacobian and Hessian determinant is to relate them with the point-wise definitions. In [14] it is proved that the point-wise Jacobian coincides with the distributional one, provided the latter is a locally integrable function. More generally, assuming that the distributional Jacobian is a Radon measure, then the point-wise Jacobian is the density of its absolutely continuous part with respect to the Lebesgue measure. In a different direction, in [8] the identity between distributional and point-wise Jacobian is established without assuming a priori that the former is a distribution of order 0, but under suitable integrability conditions for the differential matrix.

We mention also that in [15] examples are given showing that the support of the singular part of the distributional Jacobian inRn_{can be a set of any prescribed Hausdorff} dimension less thann.

The result of [14] immediately gives that the point-wise Hessian is the density of the absolutely continuous part ofH1uforu2W2;

n2 n‡1

loc , assuming thatH1uis a measure. This is generalized in [10] showing that the point-wise Hessian is the regular part of the distributionH2u. A further extension is given in [12] forH2u. Here we remark that an

analogous result holds for the Hessian measureMu. Given a Radon measuremonV, we denote by m_a the density of its absolutely continuous part with respect to Lebesgue measure. We recall that

ma(x)ˆlim_h mrh(x); for a:e: x2V;

where fr_hg is a sequence of mollifiers. If u is a convex function, its second order derivativeD2_{uis a (matrix-valued) measure.}

THEOREM3.1. For every convex function u the density of the absolutely continuous part

of the Hessian measure Mu isdet(D2_u)

a, that is (Mu)aˆdet(D2u)a:

…3:1†

In particular,det(D2_{u)ais locally integrable.}

DenotingMubydetD2_{u, we formally rewrite (3.1) as}

(detD2_u)

aˆdet(D2u)a:

Therefore, we can state the result of Theorem 3.1 by saying that the two operations of computing the Hessian determinant and of passing to the density of the absolutely continuous part commute.

To prove Theorem 3.1, we begin with a lemma.

LEMMA 3.2. Let f_h2L1_loc(V), hˆ1;2;. . .; be functions verifying f_h50, 8h,

fh(x)!f(x)a.e. and fh* mweakly in the sense of measures. Then we have f(x)4ma(x)

PROOF. For any continuous functionW50 with compact support, by Fatou lemma we find Z V Wf dx4lim h Z V Wfhdxˆ Z V Wdm:

This inequality with arbitrary W implies that f 2L1

loc(V). Moreover, considering a

sequencefr_hgof mollifiers, for a.e.x2Vwe have

f(x)ˆlim

h f rh(x)4limh mrh(x)ˆma(x):

p

More generally, ifm_h,hˆ1;2;. . ., are nonnegative Radon measures andm_h* m, then lim inf

h (mh)a4ma:

PROOF OFTHEOREM3.1. By convolution, we can find a sequence of smooth convex functionsuhsuch thatuh!uuniformly on compact subsets ofV,D2uh!(D2u)aa.e. in

V and hence also detD2_u

h !det(D2u)a a.e. in V. Notice that for all h we have

Mu_hˆdetD2_u

h. Recalling the continuity property of Theorem 2.8, by Lemma 3.2 we get

det(D2_u)

a4(Mu)a; a:e:inV:

To prove the opposite inequality, we use the change of variable formula, see [7, Theorem 1, p. 72]. Accordingly, there exists a Borel set RuV such that uis twice differentiable inRu,Ln(V_ZnRu)ˆ0, and A det(D2_u)adxˆ Z Rn Nu(y;A)dy; …3:2†

for every measurable subsetAofV. HereNu(y;A) denotes the Banach's indicatrix

Nu(y;A)ˆ#f x2A\Ru : Du(x)ˆyg: From (3.2) we deduce Z A det(D2_u) adx5Ln Du(A\Ru)ˆMu(A\Ru); which clearly implies

det(D2_u)a5(Mu)a:

p

REMARK3.3. Taking into account the continuity property of Hessian measures stated in Theorem 2.8, it is interesting to notice that weak convergence of measures does not imply point-wise convergence of the densities of absolutely continuous parts. As an example, we recall that the Lebesgue integral of a continuous function is the limit of integral sums, which can be considered as integrals with respect to purely atomic measures.

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Pervenuta il 15 febbraio 2005, in forma definitiva il 31 maggio 2005.

L. D'Onofrio: Dipartimento di Statistica e Matematica per la Ricerca Economica UniversitaÁ degli Studi di Napoli «Parthenope» Via Medina, 40 - 80100 NAPOLI luigi.donofrio@uniparthenope.it F. Giannetti, L. Greco: Dipartimento di Matematica e Applicazioni «R. Caccioppoli» UniversitaÁ degli Studi di Napoli «Federico II» Via Cintia - 80126 NAPOLI giannett@unina.it luigreco@unina.it