CAMILLO DE LELLIS, FRANCESCO GHIRALDIN

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1 AN EXTENSION OF MÜLLER S IDENTITY Det = det CAMILLO DE LELLIS, FRANCESCO GHIRALDIN Abstract. In this paper we study the pointwise characterization of the distributional Jacobian of BnV maps. After recalling some basic notions, we will extend the well known result of Müller [9] to a more natural class of functions, using the divergence theorem to express the Jacobian as a boundary integral.. Introduction Through the rest of the paper we will assume that p and q satisfy p n, q + n p. () We first define the notion of distributional Jacobian and of BnV function. Definition.. Let Ω R m be an open set and let u L q W,p (Ω, R n ) with m n. We define j(u) as the (m n + )-current j(u), ω := ( ) n u du 2 du n ω, Ω where ω is a smooth (m n+)-form with compact support in Ω; we define the distributional Jacobian of u as the (m n)-current [Ju] := j(u). We say that a map u W,p L q belongs to BnV if its distributional Jacobian [Ju] has finite mass (and hence it can be represented by a Radon Measure). If m = n, [Ju] is a distribution and a simple calculation gives that [Ju] = div[cof( u)u], where Cof( u) is the matrix of cofactors of u. This case of Definition. was first introduced by Ball in [5], in the context of nonlinear elasticity. Subsequent works in elasticity by Šverak [22] and Müller and Spector [20] were devoted to analyze the regularity properties of such maps, such as the existence of a precise representative, the structure of the singular set and their invertibility; in particular the second paper gives an existence theory for deformations of a material body that allows for cavitation. A powerful variational theory to problems in elasticity has been developed by Giaquinta, Modica and Souček (see [2] for a detailed presentation) exploiting techniques from geometric measure theory. In some relevant situations, this latter approach and the one with the distributional Jacobian are equivalent, as shown in [7] (see also [4] for further developments in this direction). The extension of the distributional Jacobian to the case m > n is due to Jerrard and Soner in [5]. That paper initiated a program on the asymptotics of functionals of Ginzburg-Landau type, using the notion of BnV map. A quite thourough study of this problem has been pursued in [] and [2]. In this paper we prove the following theorem.

2 2 CAMILLO DE LELLIS, FRANCESCO GHIRALDIN Theorem.. Let Ω R m be an open set and let u L q W,p (Ω, R m ) be a BnV map. Let ν be the density of the absolutely continuous part of the distributional Jacobian [Ju] with respect to the Lebesgue measure: Then ν(x) = det u(x) for L m -almost every x Ω. [Ju] = νl m + [Ju] s = [Ju] a + [Ju] s. (2) The theorem can be generalized to the case m > n. Theorem.2. If u L q W,p (Ω, R n ) is a BnV map, then ν(x) = (e e m ) du n (x) for L m -a.e. x Ω (see [],.5.2 for the definition of v ω). Theorem. was originally proved by Müller in [9] assuming u W,p BnV with n2 du (x) p (3) n + Note that, by Sobolev s embedding, (3) implies that u L q for some q satisfying (). Theorem.2 was claimed by the first author in [8]. Indeed, the arguments of [8] show Theorem.2 assuming Theorem. and are outlined here in Section 3 for completeness. However, in the aforementioned paper, the first author overlooked that Müller s proof needs assumption (3) and is not valid in the full range of exponent (). Acknowledgments: The second author would like to thank Prof. Luigi Ambrosio for many useful discussions and the Universität Zürich for the generous hospitality he enjoyed during his stay. The first author is grateful to Duvan Henao for pointing out his mistake in [8]. 2. Proof of Theorem. 2.. Preliminary lemmas. Similarly to [9], Theorem. will be proved using a blow up procedure. In order to perform it we will need two preliminary lemmas. To simplify the notations, we will assume that every BnV map belongs to L q W,p, with p and q as in (). Lemma 2.. If u BnV (B R, R n ) then for L -a.e. ρ (0, R): [Ju](B ρ ) = u du 2 du n = u du 2 du n, τ dh n, (4) B ρ B ρ where τ is the simple (n )-vector orienting B ρ as the boundary of B ρ. Proof. Let for x r δ, r x ϕ δ,r (x) = for r δ x r, δ 0 elsewhere. Let f(r) := B r u du 2 du n : then f L ([0, ]) because of () and Fubini s theorem. This implies that L -a.e. r is a Lebesgue point, that is: Note also that 2δ r+δ r δ Ju, ϕ δ,r = j(u), dϕ δ,r = (5) f(s) f(r) ds 0 as δ 0. (6) u dϕ δ,r (x) du 2 du n = (7)

3 AN EXTENSION OF MÜLLER S IDENTITY Det = det 3 = δ r r δ dt u du 2 du n = B t δ r r δ ( ) u du 2 du n dl (t), (8) B t hence at every Lebesgue point Ju, ϕ δ,r u du 2 du n ; B r (9) on the other hand by Lebesgue dominated convergence theorem [Ju], ϕ δ,r [Ju](B r ), that proves the proposition. Definition 2.. Let u BnV (Ω, R n ) and let x 0 B R Ω. We define u (y) := u(x 0 + y) u(x 0 ). (0) Lemma 2.2. Let u be as above and set δ a (x) := a(x x 0 ). Then [Ju ] = δ n #[Ju]. () Proof. Let φ C c (B ) be a test function. = ( ) n = ( ) n Ω u (x 0 + y) u (x 0 ) B u (x) u (0) n+ [J(u )], φ = j(u ), dφ = det( u 2 (x 0 + y),..., u n (x 0 + y), φ(y))dy = ( ( )) x x0 det u(x), φ dx = [ ( )] x x0 j(u), d φ = n = ( ) x x0 [Ju], φ. n In particular, taking the supremum over {φ C c (B ) : φ } we have [Ju ] = n δ #[Ju], Ju = n δ # Ju. Note that the Radon-Nikodym decomposition commutes with the push forward: [Ju ] a = n δ #[Ju]a, [Ju ] s = n δ #[Ju]s. This property, together with the previous observation, allows to conclude that r > 0 [Ju ] s (B r (x 0 )) = [Ju]s (B r (x 0 )) n. (2)

4 4 CAMILLO DE LELLIS, FRANCESCO GHIRALDIN 2.2. Proof of.. To simplify the notation we denote by (u h ) the sequence (u h ), h =. h We wish to apply formula (4) to the blow-up sequence (u h ) around a good point x 0 [Ju h ](B ρ (x 0 )) = u hdu 2 h du n h (3) B ρ(x 0 ) and let h tend to to obtain ν(x 0 ) B ρ = (L x) L 2 L n = B ρ(x 0 ) B ρ(x 0 ) where L := u(0) and η is the exterior unit normal to B ρ. (L x) cof(l) k η k = det(l) B ρ, (4) Step : by standard properties of Sobolev functions (see [3], [], [0]), L n -almost every point x 0 Ω satisfies the following properties: (a) { } lim [Ju] s (B r 0 r n r (x 0 )) + ν(x) ν(x 0 ) dx B r(x 0 ) (b) u is approximately continuous at x 0 and in particular lim u(x) u(x r 0 r n 0 ) p dx = 0. B r(x 0 ) = 0 ; From now on we fix x 0 satisfying (a) and (b) and, without loss of generality, we assume x 0 = 0. Observe first of all that condition (a) and equation (2) imply [Ju h ](B r (0)) = h n [Ju](B r (0)) = o() + hn ν(y)dy ν(0) B h r r > 0. (5) B r h (0) Step 2: We observe that, being (u h ) a sequence, there is a set of radii ρ (0, ) of full measure such that (4) holds for every h. Moreover by (b), using Fubini s and Fatou s Theorems, for almost every ρ there exists a subsequence (not relabeled and possibly depending on ρ) such that u h L := u(0) in L p ( B ρ ). We fix now a radius ρ such that all the properties above hold and we do not relabel the relevant subsequence. Hence du 2 h du n h L 2 L n (6) in L p n ( Bρ ), since it is sufficient to rewrite the difference as du 2 h du n h L 2 L n = i L 2 (du i k L i ) du n k. (7) In the borderline case p = (n ), the convergence is improved to the first Hardy space H ( B ρ ) because of the Coifman-Lions-Meyer-Semmes estimate (see [6]): dv 2 dv n, τ H ( B ρ) C dv 2 L n ( B ρ)... dv n L n ( B ρ). (8) Suppose first of all that p > n. Then by the Poincaré s inequality and the Sobolev embedding theorem, the sequence (u h ) is equicontinuous, with the estimate u h L x C h C α ( B ρ) C u h L L p ( B ρ) 0.

5 AN EXTENSION OF MÜLLER S IDENTITY Det = det 5 Here C h is the average of u h on B ρ. Since B ρ du 2 h... dun h = 0, we conclude ( ) [Ju h ](B ρ ) = u hdu 2 h du n h = u h Ch du 2 h du n h (9) B ρ B ρ (L x) L 2 L n = det(l) B ρ. (20) B ρ Finally if p = n we use the John-Nirenberg embedding and Poincaré s inequality [u h C h L x] BMO + u h C h L x L C u h L L n ( B ρ) 0. Recall that, by Fefferman s Theorem, BMO is the dual space of H and thus fg C ([f] BMO + f L ) g H (2) whenever f g is integrable (see [2], chapter IV; take into account that the original Theorem of Fefferman, proved in R n, must be suitably modified to our situation where the domain is a compact manifold, see [3]). We thus infer that ( ) u h Ch du 2 h du n h (L x) L 2 L n = det(l) B ρ. (22) B ρ B ρ 3. Proof of Theorem.2 Given a normal current T N k (R m ) and a Lipschitz map π : R m R l with k l, we can define a weakly*-measurable map x T, π, x N k l (R m ), uniquely characterized by the property R m T, π, x ψ(x)dx = T (ψ π)dπ ψ C c (R m ), (this is the so-called slicing of the current, see for instance [4], Thm 5.6). In [8], the first author proved a slicing theorem for Jacobians, namely: Theorem 3.. Let i x : R k {x} R k the natural injection of R k into R m, and let π : R m k R k R m k a projection, with k n; we denote by u x the trace u(x, ) = u i x. Then [Ju], π, x = ( ) (m k)n i x #[Ju x ]. (23) Moreover this slicing property holds separately for the absolutely continuous part and the singular part of [Ju]. This theorem allows to pass from Theorem. to Theorem.2 Proof of Theorem.2. Set π(x) = (x,..., x m n ), and y = (x m n+,..., x n ), then, by Theorem. and Theorem 3., [Ju] a, fdπ = [Ju] a dπ, f = [Ju] a, π, x (f)dl m n (x) = R ( m n ) = ( ) (m n)n det( y u(x, y))f(x, y)dl n (y) dl m n (x) = R m n R n = det( y u(x, y))f(x, y)dy dπ = f e e m du du n, dπ dl m. R m R m

6 6 CAMILLO DE LELLIS, FRANCESCO GHIRALDIN It s then sufficient to write a generic form as ω = I f Idx I. It is easy to show that for every A GL(n, R) it holds [J(u A)] = deg(a) (A # )[Ju] where deg(a) is the sign of the determinant of A. If then I is a multiindex of length m n, and π I (x) = (x i,..., x i m n ), we let A be a permutation matrix satisfying π = π I A. Then [Ju] a, f I dπ I = deg(a) [J(u A)] a dπ, f I A = = deg(a) f I A e e m d(u A) d(u n A), d(π I A) dl m = R m = deg(a) A (f I du du n dπ I ) = f I du du n dπ I. R m R m This concludes the proof. References [] G. Alberti, S. Baldo, and G. Orlandi. Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS), 5(3):275 3, [2] G. Alberti, S. Baldo, and G. Orlandi. Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J., 54(5):4 472, [3] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, [4] L. Ambrosio and B. Kirchheim. Currents in metric spaces. Acta Math., 85(): 80, [5] J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63(4): , 976/77. [6] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes. Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9), 72(3): , 993. [7] S. Conti and C. De Lellis. Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2(3):52 549, [8] C. De Lellis. Some fine properties of currents and applications to distributional Jacobians. Proc. Roy. Soc. Edinburgh Sect. A, 32(4):85 842, [9] C. De Lellis. Some remarks on the distributional Jacobian. Nonlinear Anal., 53(7-8):0 4, [0] L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 992. [] H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 53. Springer-Verlag New York Inc., New York, 969. [2] M. Giaquinta, G. Modica, and J. Souček. Cartesian currents in the calculus of variations. I, II, volume 37, 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 998. [3] D. Goldberg. A local version of real Hardy spaces. Duke Math. J., 46():27 42, 979. [4] D. Henao. Variational Modelling of Cavitation and Fracture in Nonlinear Elasticity. PhD thesis, University of Oxford, [5] R. L. Jerrard and H. M. Soner. Functions of bounded higher variation. Indiana Univ. Math. J., 5(3): , [6] C. Mora-Corral and D. Henao. Fracture surfaces and the regularity of inverses for BV deformations. Preprint OxPDE-09/2. Available at [7] C. Mora-Corral and D. Henao. Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Rational Mech. Anal. To appear. [8] C. Mora-Corral and D. Henao. Lusin s condition and the distributional determinant for deformations with finite energy. To Appear.

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