1. INTRODUCTION. u yy. u yx

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1 Rend. Mat. Acc. Lincei s. 9, v. 16: ) Analisi funzionale. Ð On weak Hessian determinants. Nota di LUIGI D'ONOFRIO, FLAIA GIANNETTI eluigi GRECO, 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, see e.g. [3-5, 17, 10, 1]. We begin with the case of functions of two variables. Let R be an open set and u be a function defined on with second order derivatives. We denote by H u the Hessian determinant H u ˆ det D u ˆ uxx u xy : If u is sufficiently smooth, the following identities can be easily checked: H u ˆ u x u yy ) x u x u xy ) y ˆ u y u xx ) y u y u xy ) x u yx u yy ˆ 1 uu xx) yy 1 uu yy) xx uu xy ) xy ˆ u x u y ) xy 1 u x ) yy 1 u y ) xx: On the other hand, each of the above expressions can be used to define a Schwartz distribution on, provided u belongs to an appropriate Sobolev space. The first one we consider is the distribution of order zero defined by the rule H 0 u[w] ˆ H u x; y) W x; y) dxdy for all test functions W D ). Of course, the natural assumption for defining H 0 u is u W ; ), as it implies that the Hessian determinant H u is ally integrable on. *) Nella seduta del 16 giugno 005.

2 160 L. D'ONOFRIO ET AL. Other distributions will result integrating formally by parts. Precisely, we define H 1 u[w] ˆ u x u xy W y u x u yy W x ) dxdy ˆ u y u xy W x u y u xx W y ) dxdy H u[w] ˆ 1 uu xx W yy uu yy W xx uu xy W xy ) dxdy H u[w] ˆ 1 u x u y W xy u x W yy u y W xx) dxdy for all W D ). In general, H 1 u is a distribution of order one. It is well defined for u W ;4 3. Indeed, by Sobolev imbedding u x ; u y L 4, thus u xu xy and u x u yy are ally integrable and the first integral in the definition of H 1 u converges. The same argument shows that the second integral also converges for u W ;4 3. We remark that the two integrals give the same distribution. Therefore, we can also write H 1 u[w] ˆ 1 Wx u x u yy u y u xy ) W y u xx u y u xy u x ) dxdy: REMARK 1.1. The definition of H 1 u is related to the well-known concept of weak or distributional) Jacobian introduced by Ball []. Actually, H 1 u is precisely the weak Jacobian of the gradient map Du. To justify the definitions of H u and H u we again use Sobolev imbeddings: W ; W;4 3 1:1 W;1 W e ;1 C0 W 1; where we denoted by W e ;1 the space of W1;1 -functions whose second order distributional derivatives are Radon measures on. Therefore, clearly H u makes sense for u W ;1, or even for u W e ;1 with obvious interpretation of u xx, u yy and u xy Radon measures). Surprisingly, H u makes sense for u W1; and does not require second order derivatives. Thus, we are imposing weaker and weaker conditions on u. 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, if u W e ;1, we can define both H u and H u, and we have 1: H u[w] ˆ H u[w] for all W D ).

3 ON WEAK HESSIAN DETERMINANTS Weak Hessian in R n. To deal with functions of several variables, we will use the formalism of differential forms which is quite effective. Let R n be an open subset and let u C 1 ). The Hessian determinant H u ˆ det D u induces a n-form: H udxˆ du x1 ^ ^du xn : For j ˆ 1;...; n, set v j ˆ du x1 ^^du ^^du xn, where du is at the j-th factor in the wedge product, while for i 6ˆ j the i-th factor is du xi. v j ˆ du x1 j ^^du ^^du xn j 1:3 du x1 ^^du xj ^^du xn du x1 ^^du ^^du xn x j : The first term in the right hand side assuming j 6ˆ 1) will appear also v 1, but 1 opposite sign because du x1 x j ˆ du xj x 1 and du are exchanged to each other. Similarly for the other terms in the right hand side of 1.3), except for du x1 ^^du xj ^^du xn ˆ H udx: Hence we have X v j ˆ ndu x1 ^ ^du xn jˆ1 j that is du x1 ^ ^du xn ˆ 1 X 1:4 du x1 ^^du ^^du xn : jˆ1 j Identity 1.4) yields the definition of the distribution H 1 u. We multiply both sides by a test function W and integrate by parts in the right hand side: H 1 u[w] ˆ 1 X n 1:5 W n xj du x1 ^^du ^^du xn : jˆ1 Using Laplace theorem for determinants, this is easily seen to equal H 1 u[w] ˆ 1 X n u xj du x1 ^^dw ^^du xn : n jˆ1 As above, dw is the j-th factor. We proceed further from 1.4). One way is to notice that for each j ˆ 1;...; n we have v j ˆ 1) j 1 d udu x1 ^^du xj 1 ^ du xj 1 ^^du xn ) : Inserting this into 1.4), multiplying by the test function W and integrating by parts twice will lead to the definition of H u: H u[w] ˆ 1 X n 1:6 udu x1 ^^du xj 1 ^ dw n xj ^ du xj 1 ^^du xn : jˆ1

4 16 L. D'ONOFRIO ET AL. To define H u, for i 6ˆ j we write v j ˆ 1) i 1 d u xi du x1 ^^du ^^du xi 1 ^ du xi 1 ^^du xn ) : Now we sum with respect to i and insert into 1.4). In a routine manner, this gives 1.7) n n 1) H u[w] ˆ ˆ X u xi du x1 ^^du ^^du xi 1 ^ dw xj ^ du xi 1 ^^du xn : i6ˆj We recall that in each term du is the j-th factor and dw xj is the i-th factor of the wedge product. n ; n 1 Now we consider functions u in Sobolev spaces. If u W ), then clearly the minors of order n 1ofD u are ally integrable with exponent n = n 1), while by Sobolev imbedding theorem the first order derivatives of u are ally integrable with exponent n. Noticing that these exponents are HoÈlder conjugate, we easily see that H 1 u makes sense: n ; n 1 H 1 u: W )! D 0 ) : For H u we recall that functions in W ;n 1 ) are continuous actually, HoÈ lder continuous with exponent n )= n 1) if n > ). For H u we remark that if n ; n u W ), then the minors of order n ofd u and the first order derivatives of u are ally integrable respectively with exponents n = n 4) and n =. Therefore, H u: W ;n 1 )! D 0 ) ; H u: W; n n )! D 0 ) : We notice that in dimension n > the definition of H u requires that u has second order derivatives. 1.. 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 that R n is a convex set and u is a real-valued convex function defined on. We denote the subdifferential also called normal mapping) of u and E) the image by this multifunction of the subset E E) ˆ x) : xe The Hessian measure of u, which we denote by Mu, is a Borel measure on defined by 1:8 Mu E) ˆ L E)) for each Borel subset E. L n isthelebesguemeasureinr n.itiswellknownthatfor u C ) the Hessian measure coincides with the Hessian determinant H u in the

5 ON WEAK HESSIAN DETERMINANTS 163 sense that the following equality holds 1:9 W dmu ˆ W H udx; for all continuous W with compact support. We recall that a convex function u is ally Lipschitz and its gradient Du has ally bounded variation, see [6]. Hence u W e ;1 and in dimension we can consider H u.in the following theorem we compare these notions. THEOREM 1.. If u is in the domain of two of the above weak formulations of the Hessian, these formulation coincide. As an illustration, in dimension n ˆ ifu is convex, then Mu ˆ H u ˆ H u. Theorem 1. will be an easy consequence of the continuity properties we discuss in the next section.. CONTINUITY PROPERTIES One of our concern is to elucidate on the continuity properties of those weak formulations of the Hessian. First, we note trivially that H 1, H and H are continuous in their corresponding domains with respect to the strong convergences. More precisely: ; n n 1 If u h! u in W, then H 1 u h! H 1 u in the sense of distributions. This means that H 1 u h [W]! H 1 u[w], for all W D ); If u h! u in W ;n 1, then H u h! H u in the sense of distributions. For n ˆ, if u h! u in W 1;, then H u h! H u in the sense of distributions. n ; n For n >, if u h! u in W, then H u h! H u in the sense of distributions. It is also easy to prove convergence in D 0 of the weak Hessians of u h coupling a suitable weak convergence of the minors of the Hessian matrix D u h with a strong convergence of the function u h or its first order derivatives Du h. As an example, we have If Du h! Du strongly in L n and for n > ) the minors of order n ofd u h converge to corresponding minors of D u weakly in L n n 4, then H u h! H u in 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. We first consider the case n ˆ. In [10] it is shown by an example that H is not continuous with respect to weak convergence in W 1;. Here we present another simple example of this type. EXAMPLE.1. The functions u h ˆ u h x; y) ˆ 1 h sin h x y )

6 164 L. D'ONOFRIO ET AL. clearly converge uniformly to zero and Du h weakly in L 1. On the other hand, H u h are not converging to zero in the sense of distributions. To see this we consider a test function W supported in the unit disk D R, and compute H u h[w] ˆ 4xyW xy x W yy y W xx ) cos h x y ) dxdy: D Integrating in polar co-ordinates and then by parts, we find that lim H u h[w] ˆ xyw xy x W yy y W xx ) dxdy ˆ W dxdy: h D D The above example and the following arguments indicate clearly that in order to ensure convergence of H u h to H u one really needs strong convergence in W1; THEOREM.. Let u h W ;1,hˆ 1; ;..., and u W;1 be given. Then the distribution H u h ˆ H u h converge to H u ˆ H u under each of the assumptions below: a) u h * u weakly in W ;1 ; b) the sequence fu h g is bounded in W ;1 subsets.. and converges to u uniformly on compact The proof follows easily when we realize that both assumptions a) and b) yield u h! u strongly in W 1;. Actually, we have LEMMA.3. a) ) b) ) u h! u strongly in W 1;. We remark that both imbeddings of W ;1 C0 and W ;1 W1; the sense that bounded sequences in W ;1 uniformly, or strongly in W 1;. are not compact, in may not have subsequences converging ally EXAMPLE.4. Given a function in u W ;1 R ) with u 6 0 and lim u z) ˆ 0, we z!1 define u h z) ˆ u hz), 8h N. Then u h z)! 0, 8z R f0g, Du h! 0 in L 1, kd u h k 1 ˆkD uk 1 for all h N and D u h * Cd weakly in the sense of measures, with C a constant matrix. Moreover fu h g has no subsequence converging ally uniformly and, as kdu h k ˆkDuk > 0 for all h, no subsequence strongly converging in W 1;. On the other hand, the assumption of weak convergence in W ;1 is essentially stronger than mere boundedness. By Ascoli-ArzelaÁ theorem, under assumption a) we can prove uniform convergence u h! u on compact subsets, hence condition b) holds. PROOF OF LEMMA.3. There is no loss of generality in assuming u ˆ 0, otherwise we consider the functions u h u.

7 ON WEAK HESSIAN DETERMINANTS 165 Let us prove a) ) b). For simplicity, we assume ˆ ]0; 1[. For W C0 1 ), we define v h ˆ Wu h, h ˆ 1; ;..., and we have Dv h ˆ u h DW WDu h ; D v h ˆ u h D W DW Du h Du h DW WD u h ; hence v h * 0 weakly in W ;1 ). The point is that weak convergence of fu h g gives equi-integrability of fdu h g. It is well known that W ;1 ) is continuously imbedded into C ), see [1, p. 100]. If v is a continuous function of class W ;1 ) vanishing then for all x 1 ; y 1 ) and x ; y )in, we have x 1 y 1 x 1 y jv x 1 ; y 1 ) v x ; y )j4 ds v xy s; t) dt ds v xy s; t) dt : y 0 In particular, kvk 1 4kv xy k 1 : These inequalities yield equiboundedness and equicontinuity of any bounded family of functions in W ;1 0 ) with equi-integrable second order derivatives. We are in a position to apply Ascoli-ArzelaÁ theorem and get v h! 0 uniformly in. This implies uniform convergence u h! 0 on compact subsets of. IfK is compact, it suffices to choose W 1onK. Now we assume that condition b) holds and prove strong convergence in W 1; ). If v W ;1 0 ), integrating by parts we find jdvj ˆ v Dv4kvk 1 kd vk 1 : Therefore, defining v h ˆ Wu h as above, we see that v h! 0 strongly in W 1; ). To deduce from this strong convergence of u h on an arbitrary compact K, as before we only need to take W 1onK. Notice that b) implies strong convergence of fu h g in W 1; can be seen also as a consequence of inequality 1.16) of [13]. p Theorem. can be slightly generalized. x 0 THEOREM.5. Let fu h g be a sequence bounded in W e ;1 and converging to u uniformly on compact subsets. Then u W e ;1 and H u h ˆ H u h converge to H u ˆ H u in the sense of distributions. Now we assume n >. We have the following continuity property. THEOREM.6. If u h * u weakly in W ;p n for some p > n, then H u h! H u in the sense of distributions.

8 166 L. D'ONOFRIO ET AL. PROOF. Asp > n the imbedding W;p is compact. Then Du n h! Du strongly in L n. On the other hand, the minors of order n ofd u h converge to corresponding minors of D u in the sense of distributions, see [11, Theorem 8..1, p. 173], and are bounded in L p p n, where we notice that n > n n > 1, hence they 4 are weakly converging in L p n and therefore we can pass to the limit in the expression defining H u[w]. p W1;n REMARK.7. In [5] it is shown that the condition p > in Theorem.6 is sharp, in n the sense that the conclusion cannot be drawn in general in dimension n > ) assuming :1 u h * u weakly in W n n ; n : With this condition, we still have weak convergence in L n n 4 of the minors of order n of D u h, but we need to reinforce.1) to guarantee the strong convergence in L n of first derivatives. For 1 < n < r < n, we have the Sobolev imbeddings W ;r W1;r C 0; n r ; neither of which is compact. Clearly, if u h! u strongly in W 1;r, then it converges also strongly in the HoÈ lder space. On the other hand, by the interpolation inequality 5) of [16], for a bounded sequence fu h g in W ;r the convergence in the HoÈ lder space implies strong convergence of first derivatives, that is, the two convergences are equivalent. For r ˆ n, we have the following generalization of Theorem.6, in the spirit of n Theorem.5: Let condition.1) hold together with u h! u strongly in C 0;1 n. Then H u h converge to H u in the sense of distributions. We conclude this section by recalling some continuity properties of the Hessian measure, see [9]. THEOREM.8. Let u h,hˆ 1; ;..., and u be convex functions and let the sequence u h converge to u uniformly on compact subsets. Then Mu h converges to Mu weakly in the sense of measure, that is, W dmu h! W dmu for all continuous functions W with compact support. It is clear that in the two-dimensional context Theorem.8 follows from Theorem.5.

9 ON WEAK HESSIAN DETERMINANTS THE L 1 -PART OF THE HESSIAN 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 ally 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 in R n can be a set of any prescribed Hausdorff dimension less than n. The result of [14] immediately gives that the point-wise Hessian is the density of the n ; n 1 absolutely continuous part of H 1 u for u W, assuming that H 1 u is a measure. This is generalized in [10] showing that the point-wise Hessian is the regular part of the distribution H u. A further extension is given in [1] for H u. Here we remark that an analogous result holds for the Hessian measure Mu. Given a Radon measure m on, we denote by m a the density of its absolutely continuous part with respect to Lebesgue measure. We recall that m a x) ˆ lim m r h x) ; for a:e: x ; h where fr h g is a sequence of mollifiers. If u is a convex function, its second order derivative D u is a matrix-valued) measure. THEOREM 3.1. For every convex function u the density of the absolutely continuous part of the Hessian measure Mu is det D u) a, that is 3:1 Mu) a ˆ det D u) a : In particular, det D u) a is ally integrable. Denoting Mu by det D u, we formally rewrite 3.1) as det D u) a ˆ det D u) 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.. Let f h L 1 ), h ˆ 1; ;...; be functions verifying f h50, 8h, f h x)! f x) a.e. and f h * m weakly in the sense of measures. Then we have f x)4m a x) a.e. in.

10 168 L. D'ONOFRIO ET AL. PROOF. For any continuous function W50 with compact support, by Fatou lemma we find W fdx4 lim W f h dx ˆ W dm : h This inequality with arbitrary W implies that f L 1 ). Moreover, considering a sequence fr h g of mollifiers, for a.e. x we have f x) ˆ lim f r h x)4 lim m r h x) ˆ m a x) : h h p More generally, if m h, h ˆ 1; ;..., are nonnegative Radon measures and m h * m, then lim inf m h ) a 4m a : h PROOF OF THEOREM 3.1. By convolution, we can find a sequence of smooth convex functions u h such that u h! u uniformly on compact subsets of, D u h! D u) a a.e. in and hence also det D u h! det D u) a a.e. in. Notice that for all h we have Mu h ˆ det D u h. Recalling the continuity property of Theorem.8, by Lemma 3. we get det D u) a 4 Mu) a ; a:e: in : To prove the opposite inequality, we use the change of variable formula, see [7, Theorem 1, p. 7]. Accordingly, there exists a Borel set R u such that u is twice differentiable in R u, L n n R u ) ˆ 0, and 3: det D u) a dx ˆ N u y; A) dy ; A R n for every measurable subset A of. Here N u y; A) denotes the Banach's indicatrix N u y; A) ˆ #f x A \ R u : Du x) ˆ y g : From 3.) we deduce det D u) a dx5l n Du A \ R u ) ˆ Mu A \ R u ) ; A which clearly implies det D u) a 5 Mu) a : p REMARK 3.3. Taking into account the continuity property of Hessian measures stated in Theorem.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.

11 ON WEAK HESSIAN DETERMINANTS 169 REFERENCES [1] R.A. ADAMS, Sobolev spaces. Pure and Applied Mathematics, vol. 65, Academic Press, [] J.M. BALL, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63, n. 4, , [3] J.M. BALL - J.C. CURIE - P.J. OLER, Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal., 41, 1981, [4] R. COIFMAN - P.-L. LIONS - Y. MEYER - S. S EMMES, Compensated compactness and Hardy spaces. J. Math. Pures Appl., 7, 1993, [5] B. DACOROGNA - F. MURAT, On the optimality of certain Sobolev exponents for the weak continuity of determinants. J. Funct. Anal., 105, n. 1, 199, 4-6. [6] L.C. EANS - R.F. GARIEPY, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL 199. [7] M. GIAQUINTA -G.MODICA -J.SOUC Ï EK, Area and the area formula. Rend. Sem. Mat. Fis. Milano, 6, 199, [8] L. GRECO, A remark on the equality det Df ˆ Det Df. Differential Integral Equations, 6, n. 5, 1993, [9] C.E. GUTIE  RRE, The Monge-AmpeÁre Equation. BirkhaÈuser, Boston 001. [10] T. IWANIEC, On the concept of the weak Jacobian and Hessian. Report. Univ. JyvaÈskylaÈ, 83, 001, [11] T. IWANIEC - G. MARTIN, Geometric Function Theory and Nonlinear Analysis. Oxford University Press, Oxford 001. [1] J. MALY Â, From Jacobians to Hessian: distributional form and relaxation. In: Proceedings of the Trends in the Calculus of ariations Parma, September 15-18, 004). Riv. Mat. Univ. Parma, in press; nw0z/publications/05-cna-003/003abs/05-cna-003.pdf. [13] C. MIRANDA, Su alcuni teoremi di inclusione. Annales Polonici Math., 16, 1965, [14] S. MUÈ LLER, Det ˆ det, a remark on the distributional determinant. C.R. Acad. Sci. Paris, SeÂr. I Math., 311, 1990, [15] S. MUÈ LLER, On the singular support of the distributional determinant. Ann. Inst. H. Poincare Anal. Non LineÂaire, 10, 1993, [16] L. NIRENBERG, An extended interpolation inequality. Ann. Sc. Norm. Sup. Pisa Sci. fis. mat., 0, 1966, [17] P.J. OLER, Hyper-Jacobians, determinantal ideals and weak solutions to variational problems. Proc. Roy. Soc. Edinburgh Sect. A, 95, 1983, n. 3-4, [18] N.G. TRUDINGER, Weak solutions of Hessian equations. Comm. Partial Differential Equations,, 1997, n. 7-8, Pervenuta il 15 febbraio 005, in forma definitiva il 31 maggio 005. L. D'Onofrio: Dipartimento di Statistica e Matematica per la Ricerca Economica UniversitaÁ degli Studi di Napoli «Parthenope» ia Medina, NAPOLI [email protected] F. Giannetti, L. Greco: Dipartimento di Matematica e Applicazioni «R. Caccioppoli» UniversitaÁ degli Studi di Napoli «Federico II» ia Cintia NAPOLI [email protected] [email protected]

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