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Unformatted text preview: WEAK NOTIONS OF JACOBIAN DETERMINANT AND RELAXATION GUIDO DE PHILIPPIS Abstract. In this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely the distributional Jacobian and the relaxed total variation , which in general could be different. We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals. 1. Introduction The aim of this paper is to study weak notions of Jacobian determinant , det Du , for maps u : Ω ⊂ R n → R n in th Sobolev class W 1 ,p for some p . If u is a diffeomorphism, the change of variable formula and Lebesgue differentiation Theorem give a clear geometric meaning to det Du ( x ), it is the “infinitesimal” change of volume due to the deformation u . If u is not a bijective map the area formulas (in the unoriented version (1.1) or in the oriented one (1.2)) Z Ω  det Du ( x )  dx = Z R n N ( u, Ω ,y ) dy (1.1) Z Ω det Du ( x ) dx = Z R n deg( u, Ω ,y ) dy ( 1 ) (1.2) relate the integral of the Jacobian determinant to how many times the image of u covers the target space. If u is merely a Sobolev map it is still possible to consider the area formula and the “pointwise” Jacobian det Du (see [28]) , however if u is not sufficiently regular (more precisely if u ∈ W 1 ,p and p < n ) this gives only a partial information about the behaviour of u . If p ≥ n , H¨ older inequality implies det Du ∈ L p n ; moreover the map W 1 ,p → L p n u 7→ det Du is continuous if we endow both spaces with the strong topology. What is more surprising is that if p > n this map is still (sequentially) continuous also if we endow the spaces with the weak topology (see Theorem 2.2). If p = n we don’t have continuity if we consider the L 1 weak topology for Jacobians, however we have it if we consider the weak* topology (see for example [12, chapter 8]), however if det Du k ≥ 0 we still have continuity with respect to the L 1 weak topology (see [10, 38]). 1 N ( u, Ω ,y ) = # { x ∈ Ω: u ( x ) = y } is the Banach indicatrix function, while deg( u, Ω ,y ) is the Brouwer degree of u (see subsection 2.2). 1 2 GUIDO DE PHILIPPIS This implies the semicontinuity with respect to the weak convergence of the functional: u 7→ TV ( u, Ω) := Z Ω  det Du  (1.3) and, more in general, of polyconvex functionals, i.e. the ones that can be represented as the integral of a convex function of the minors of the gradient: F ( u, Ω) = Z Ω g ( Du, M 1 ( Du ) ,..., det Du ) . If p < n , in general det Du is not a summable function; moreover also if det Du ∈ L 1 we lose continuity and semicontinuity properties. In particular it is possible to see that: • the function u ( x ) = x  x  is in W 1 ,p ( B ) for any p < n , det Du = 0 almost ev erywhere but for any sequence of smooth functions strongly converging to u in W 1 ,p ( B ) ∩ L ∞ with p > n 1 we have: det Du k * *ω n δ ( 2 ) in the sense of distributions....
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 Spring '10
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