Abstract
We present a few connections between the notion of Div-free/Div-BV symmetric tensors, with geometrical topics such as convex bodies or minimal surfaces. In passing, we establish some related results about the Cofactor map and the geometrical mean of positive definite matrices.
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References
W. K. Allard. On the first variation of a varifold. Annals of Math., 95 (1972), pp 417–491. On the first variation of a varifold: Boundary behaviour. Annals of Math., 101 (1975), pp 418–446.
W. J. Firey. Blaschke sums of convex bodies and mixed bodies. Proc. Colloq. Convexity (Copenhagen 1965), Københavns Univ. Mat. Inst., 1967, pp 94–101.
L. Gårding. An inequality for hyperbolic polynomials. J. Math. Mech., 8 (1959), pp 957–965.
E. Gagliardo. Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat., 7 (1958), pp 102–137.
R. Kupferman, A. Schachar. A geometric perspective on the Piola identity in Riemannian settings. J. Geom. Mech.11 (2019), 59–76.
J. H. Michael, L. M. Simon. Sobolev and mean value inequalities on generalized submanifolds of \({\mathbb R}^n\). Comm. Pure & Appl. Math., 26 (1973), pp 361–379.
E. Lutwak. Volume of mixed bodies. Trans. of the Amer. Math. Soc., 294 (1986), pp 487–500.
A. V. Pogorelov. The Minkowski multidimensional problem. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.; Halsted Press (John Wiley & Sons), New York–Toronto–London (1978).
D. Serre. Divergence-free positive symmetric tensors and fluid dynamics. Annales de l’Institut Henri Poincaré (analyse non linéaire), 35 (2018), pp 1209–1234.
D. Serre. Compensated integrability. Applications to the Vlasov–Poisson equation and other models of mathematical physics. J. Math. Pures & Appl., 127 (2019), pp 67–88.
D. Serre. Hard spheres dynamics: Weak vs strong collisions. Arch. Rat. Mech. Anal., 240 (2021), pp 243–264.
D. Serre. Symmetric Divergence-free tensors in the Calculus of Variations. Comptes Rendus, Mathématique, 360 (2022), pp. 653–663.
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I am indebted to the anonymous referee, who let me know a relevant piece of literature.
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Serre, D. (2023). Divergence-Free Tensors and Cofactors in Geometry and Fluid Dynamics. In: Morel, JM., Teissier, B. (eds) Mathematics Going Forward . Lecture Notes in Mathematics, vol 2313. Springer, Cham. https://doi.org/10.1007/978-3-031-12244-6_32
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DOI: https://doi.org/10.1007/978-3-031-12244-6_32
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