On two symmetries in the theory of $m$-Hessian operators
Keywords
Partial differential fully nonlinear operators, m-Hessian operators, skew symmetry, symmetric functions, Hessian integrals, isoperimetric variational problem, Poincar\'e-type inequalitiesAbstract
The modern theory of fully nonlinear operators had been inspired by the skew symmetry of minors in cooperation with the symmetry of symmetric functions. We present some consequences of this interaction for $m$-Hessian operators. One of them is setting of the isoperimetric variational problem for Hessian integrals. The $m$-admissible minimizer is found that allows a new simple proof of the well-known Poincaré-type inequalities for Hessian integrals. Also a new set of inequalities, generated by a special finite set of functions, is presented.References
J.M. Ball, Convexity conditions and existence theorems in nonlinear elastisity, Arch. Ration. Mech. Anal. 63 (1977), no. 4, 339–403.
E. Bedford and B.A. Taylor, Variational properties of the complex Monge–Ampère equation I: Dirichlet principle, Duke Math. J. 45 (1978), 375–405.
L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations III. Functions of the Hessian, Acta Math. 155 (1985), 261–301.
L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations V. The Dirichlet problem for Weingarten hypersurfaces, Comm. Pure Appl. Math. 41 (1988), 47–70.
K.S. Chou (K. Tso), On a real Monge–Ampère functional, Invent. Math. 101 (1990), 425–448.
K.S. Chou and X.-J. Wang, A variational theory of the Hessian equations, Comm. Pure Appl. Math. 54 (2001), 1029–1064.
D.-P. Covei, The Keller–Osserman problem for the k-Hessian operator, 2015, https: //arxiv.org/abs/1508.04653.
L.C. Evans, Classical solutions of fully nonlinear convex second order elliptic equations, Comm. Pure Appl. Math. 25 (1982), 333–363.
N.M. Ivochkina, On the possibility of integral formulae in Rn , Zap. Nauchn. Sem. LOMI 52 (1975), 35–51.
N.M. Ivochkina, Second order equations with d-elliptic operators, Trudy Mat. Inst. Steklov 147 (1980), 45–56 (in Russian); English transl. Proc. Steklov Inst. Math. 2 (1981).
N.M. Ivochkina, A description of the stability cones generated by differential operators of Monge–Ampère type, Mat. Sb. 122 (1983), 265–275 (in Russian); English transl. Math. USSR Sb. 50 (1985).
N.M. Ivochkina, Variational problems connected to Monge–Ampère type operators, Zap. Nauchn. Semin. LOMI 167 (1988), 186–189.
N.M. Ivochkina, From Gårding cones to p-convex hypersurfaces, J. Math. Sci. 201 (2014), 634–644.
N.M. Ivochkina, On some properties of the positive m-Hessian operators in C 2 (Ω), J. Fixed Point Th. Appl. 14 (2014), no. 1, 79–90.
N.M. Ivochkina and N.V. Filimonenkova, On the backgrounds of the theory of mHessian equations, Comm. Pure Appl. Anal. 12 (2013), no. 4, 1687–1703.
N.M. Ivochkina and N.V. Filimonenkova, On algebraic and geometric conditions in the theory of Hessian equations, J. Fixed Point Theory Appl. 16 (2015), no. 1, 11–25.
N.M. Ivochkina, S.I. Prokof’eva, and G.V. Yakunina, The Gårding cones in the modern theory of fully nonlinear second order differential equations, J. of Math. Sci. 184 (2012), no. 3, 295–315.
L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957–965.
N.V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk. SSSR Ser. Mat. 47 (1983), 75–108 (in Russian); English transl. Math. USSR Izv. 22 (1984), 67–97.
V.V. Lychagin, Contact geometry and second-order nonlinear differential equations, Russian Math. Surveys 34 (1979), no. 1, 149–180.
A.V. Pogorelov, The Minkowski multidimensional problem, “Nauka”, Moscow, 1975 (in Russian); English transl. New York, J. Wiley (1978).
R.C. Reilly On the Hessian of a function and the curvatures of its graph, Michigan Math. J. 20 (1973/1974), 373–383.
H. Rund, Integral formulae associated with Euler–Lagrange operators of multiple integral problems in the calculus of variations, Aequationes Math. 11 (1974), no. 2/3, 212–229.
N.S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal. 111 (1990), 153–170.
N.S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151–164.
N.S. Trudinger and X.-J. Wang, Hessian measures I, Topol. Methods Nonlinear Anal. 10 (1997), 225–239.
N.S. Trudinger and X.-J. Wang, A Poincaré type inequality for Hessian integrals, Calc. Var. Partial Differenytial Equations 6 (1998), 315–328.
I.E. Verbitsky, The Hessian Sobolev inequality and its extensions, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 6165–6179.
X.-J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J. 43 (1994), 25–54.
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