# American Institute of Mathematical Sciences

July  2013, 12(4): 1687-1703. doi: 10.3934/cpaa.2013.12.1687

## On the backgrounds of the theory of m-Hessian equations

 1 Saint-Petersburg State University of Architecture and Civil Engineering, 2-nd Krasnoarmeiskaya St. 4, 190005 St. Petersburg, Russian Federation, Russian Federation

Received  July 2011 Revised  April 2012 Published  November 2012

The paper presents some pieces from algebra, theory of function and differential geometry, which have emerged in frames of the modern theory of fully nonlinear second order partial differential equations and revealed their interdependence. It also contains a survey of recent results on solvability of the Dirichlet problem for m-Hessian equations, which actually brought out this development.
Citation: Nina Ivochkina, Nadezda Filimonenkova. On the backgrounds of the theory of m-Hessian equations. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1687-1703. doi: 10.3934/cpaa.2013.12.1687
##### References:
 [1] A. D. Aleksandrov, Dirichlet problem for the equation $Det ||z_{ij}||=\varphi$, (Russian) Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr., 13 (1958), 5-24. [2] L. Caffarelli, L. Nirenberg and J. Y. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301. [3] L. Caffarelli and L. Silvestre, Smooth approximations to solutions of nonconvex fully nonlinear elliptic equations, AMS Transl., Series 2, 229 (2010), Advances in the Math. Sci., 64 (2010), Nonlinear Part. Diff. Eq. and Related Topics, 67-85. [4] H. Dong, N. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients, Algebra i Analiz, 23 (2011). [5] L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. [6] N. V. Filimonenkova, Analysis of the behavior of a solution to m-Hessian equations near the boundary of a domain, Problems in mathematical analysis, no. 45, J. Math. Sci. (N. Y.), 166 (2010), 338-356. [7] N. V. Filimonenkova, An estimate for the Hölder constant for weak solutions to m-Hessian equations in a closed domain, Vestnik St. Petersburg Univ. Math., 43 (2010), 183-190. [8] N. V. Filimonenkova, "A Quality Analysis of Weak Solutions to m-Hessian Equations," Ph.D thesis, PDMI RAS, 2010. [9] L. Garding, An inequality for hyperbolic polynomials, J. Math. Mech., 8 (1959), 957-965. [10] N. M. Ivochkina, Second order equations with d-elliptic operators, Trudy Mat. Inst. Steklov, 147 (1980), 40-56, English transl. in Proc. Steclov Inst. Math., 2 (1981). [11] N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge - Ampere type, Mat. Sb., 122 (1983), 265-275, English transl. in Math. USSR Sb., 50 (1985). [12] N. M. Ivochkina, Solution of the Dirichlet problem for some equations of Monge - Ampere type, Mat. Sb., 128 (1985), 403-415, English transl. in Math. USSR Sb., 56 (1987). [13] N. M. Ivochkina, Solution of the Dirichlet problem for the curvature equation order m, Algebra i Analiz, 2 (1990), 192-217, English transl. in Leningrad Math. J., 2 (1991). [14] N. M. Ivochkina, The Dirichlet principle in the theory of equations of Monge - Ampere type, Algebra i Analiz, 4 (1993), English transl. in St. Petersburg Math. J., 4 (1993). [15] N. M. Ivochkina, On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations, Problems in mathematical analysis, no. 50, J. Math. Sci. (N. Y.), 170 (2010), 496-509. [16] N. M. Ivochkina, N. S. Trudinger and X.-J. Wang, The Dirichlet problem for degenerate Hessian equations, Comm. Partial Differ. Equations, 29 (2004), 219-235. [17] N. M. Ivochkina and N. V. Filimonenkova, Estimate of the Hölder constant for solutions to m-Hessian equations, Problems in mathematical analysis, no. 40, J. Math. Sci. (N. Y.), 159 (2009), 67-74. [18] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., 229 (1968), 170-187. [19] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108, English transl. Math. USSR Izv., 22 (1984), 67-97. [20] D. Labutin, Potential theory for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. [21] M. Lin and N. S. Trudinger, On some inequalities for elementary symmetric functions, Bull. Austr. Math. Soc., 50 (1994), 317-326. [22] A. V. Pogorelov, "The Mincowski Multidimensional Problem," Nauka, Moscow, 1975, English transl. in New York, J.Wiley, 1978. [23] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A, 264 (1969), 413-496. [24] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rat. Mech. Anal., 111 (1990), 153-179. [25] N. S. Trudinger, Maximum principles for curvature quotient equations, J. Math. Sci. Univ. Tokyo, 1 (1994), 551-565. [26] N. S. Trudinger, Weak solutions of Hessian equations, Comm. Partial Differential Equation, 22 (1997), 1251-1261. [27] N. S. Trudinger and X.-J. Wang, Hessian measures II, Ann. of Math., 150 (1999), 579-604. [28] N. S. Trudinger and X.-J. Wang, Hessian measures I, Topol. Methods Nonlinear Anal., 10 (1997), 225-239.

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##### References:
 [1] A. D. Aleksandrov, Dirichlet problem for the equation $Det ||z_{ij}||=\varphi$, (Russian) Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr., 13 (1958), 5-24. [2] L. Caffarelli, L. Nirenberg and J. Y. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301. [3] L. Caffarelli and L. Silvestre, Smooth approximations to solutions of nonconvex fully nonlinear elliptic equations, AMS Transl., Series 2, 229 (2010), Advances in the Math. Sci., 64 (2010), Nonlinear Part. Diff. Eq. and Related Topics, 67-85. [4] H. Dong, N. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients, Algebra i Analiz, 23 (2011). [5] L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. [6] N. V. Filimonenkova, Analysis of the behavior of a solution to m-Hessian equations near the boundary of a domain, Problems in mathematical analysis, no. 45, J. Math. Sci. (N. Y.), 166 (2010), 338-356. [7] N. V. Filimonenkova, An estimate for the Hölder constant for weak solutions to m-Hessian equations in a closed domain, Vestnik St. Petersburg Univ. Math., 43 (2010), 183-190. [8] N. V. Filimonenkova, "A Quality Analysis of Weak Solutions to m-Hessian Equations," Ph.D thesis, PDMI RAS, 2010. [9] L. Garding, An inequality for hyperbolic polynomials, J. Math. Mech., 8 (1959), 957-965. [10] N. M. Ivochkina, Second order equations with d-elliptic operators, Trudy Mat. Inst. Steklov, 147 (1980), 40-56, English transl. in Proc. Steclov Inst. Math., 2 (1981). [11] N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge - Ampere type, Mat. Sb., 122 (1983), 265-275, English transl. in Math. USSR Sb., 50 (1985). [12] N. M. Ivochkina, Solution of the Dirichlet problem for some equations of Monge - Ampere type, Mat. Sb., 128 (1985), 403-415, English transl. in Math. USSR Sb., 56 (1987). [13] N. M. Ivochkina, Solution of the Dirichlet problem for the curvature equation order m, Algebra i Analiz, 2 (1990), 192-217, English transl. in Leningrad Math. J., 2 (1991). [14] N. M. Ivochkina, The Dirichlet principle in the theory of equations of Monge - Ampere type, Algebra i Analiz, 4 (1993), English transl. in St. Petersburg Math. J., 4 (1993). [15] N. M. Ivochkina, On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations, Problems in mathematical analysis, no. 50, J. Math. Sci. (N. Y.), 170 (2010), 496-509. [16] N. M. Ivochkina, N. S. Trudinger and X.-J. Wang, The Dirichlet problem for degenerate Hessian equations, Comm. Partial Differ. Equations, 29 (2004), 219-235. [17] N. M. Ivochkina and N. V. Filimonenkova, Estimate of the Hölder constant for solutions to m-Hessian equations, Problems in mathematical analysis, no. 40, J. Math. Sci. (N. Y.), 159 (2009), 67-74. [18] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., 229 (1968), 170-187. [19] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108, English transl. Math. USSR Izv., 22 (1984), 67-97. [20] D. Labutin, Potential theory for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. [21] M. Lin and N. S. Trudinger, On some inequalities for elementary symmetric functions, Bull. Austr. Math. Soc., 50 (1994), 317-326. [22] A. V. Pogorelov, "The Mincowski Multidimensional Problem," Nauka, Moscow, 1975, English transl. in New York, J.Wiley, 1978. [23] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A, 264 (1969), 413-496. [24] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rat. Mech. Anal., 111 (1990), 153-179. [25] N. S. Trudinger, Maximum principles for curvature quotient equations, J. Math. Sci. Univ. Tokyo, 1 (1994), 551-565. [26] N. S. Trudinger, Weak solutions of Hessian equations, Comm. Partial Differential Equation, 22 (1997), 1251-1261. [27] N. S. Trudinger and X.-J. Wang, Hessian measures II, Ann. of Math., 150 (1999), 579-604. [28] N. S. Trudinger and X.-J. Wang, Hessian measures I, Topol. Methods Nonlinear Anal., 10 (1997), 225-239.
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