# 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 & 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.   Google Scholar [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.   Google Scholar [3] L. Caffarelli and L. Silvestre, Smooth approximations to solutions of nonconvex fully nonlinear elliptic equations,, AMS Transl., 229 (2010), 67.   Google Scholar [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).   Google Scholar [5] L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations,, Comm. Pure Appl. Math., 35 (1982), 333.   Google Scholar [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, 166 (2010), 338.   Google Scholar [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.   Google Scholar [8] N. V. Filimonenkova, "A Quality Analysis of Weak Solutions to m-Hessian Equations,", Ph.D thesis, (2010).   Google Scholar [9] L. Garding, An inequality for hyperbolic polynomials,, J. Math. Mech., 8 (1959), 957.   Google Scholar [10] N. M. Ivochkina, Second order equations with d-elliptic operators,, Trudy Mat. Inst. Steklov, 147 (1980), 40.   Google Scholar [11] N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge - Ampere type,, Mat. Sb., 122 (1983), 265.   Google Scholar [12] N. M. Ivochkina, Solution of the Dirichlet problem for some equations of Monge - Ampere type,, Mat. Sb., 128 (1985), 403.   Google Scholar [13] N. M. Ivochkina, Solution of the Dirichlet problem for the curvature equation order m,, Algebra i Analiz, 2 (1990), 192.   Google Scholar [14] N. M. Ivochkina, The Dirichlet principle in the theory of equations of Monge - Ampere type,, Algebra i Analiz, 4 (1993).   Google Scholar [15] N. M. Ivochkina, On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations,, Problems in mathematical analysis, 170 (2010), 496.   Google Scholar [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.   Google Scholar [17] N. M. Ivochkina and N. V. Filimonenkova, Estimate of the Hölder constant for solutions to m-Hessian equations,, Problems in mathematical analysis, 159 (2009), 67.   Google Scholar [18] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions,, J. Reine Angew. Math., 229 (1968), 170.   Google Scholar [19] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75.   Google Scholar [20] D. Labutin, Potential theory for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.   Google Scholar [21] M. Lin and N. S. Trudinger, On some inequalities for elementary symmetric functions,, Bull. Austr. Math. Soc., 50 (1994), 317.   Google Scholar [22] A. V. Pogorelov, "The Mincowski Multidimensional Problem,", Nauka, (1975).   Google Scholar [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.   Google Scholar [24] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, Arch. Rat. Mech. Anal., 111 (1990), 153.   Google Scholar [25] N. S. Trudinger, Maximum principles for curvature quotient equations,, J. Math. Sci. Univ. Tokyo, 1 (1994), 551.   Google Scholar [26] N. S. Trudinger, Weak solutions of Hessian equations,, Comm. Partial Differential Equation, 22 (1997), 1251.   Google Scholar [27] N. S. Trudinger and X.-J. Wang, Hessian measures II,, Ann. of Math., 150 (1999), 579.   Google Scholar [28] N. S. Trudinger and X.-J. Wang, Hessian measures I,, Topol. Methods Nonlinear Anal., 10 (1997), 225.   Google Scholar

show all references

##### 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.   Google Scholar [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.   Google Scholar [3] L. Caffarelli and L. Silvestre, Smooth approximations to solutions of nonconvex fully nonlinear elliptic equations,, AMS Transl., 229 (2010), 67.   Google Scholar [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).   Google Scholar [5] L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations,, Comm. Pure Appl. Math., 35 (1982), 333.   Google Scholar [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, 166 (2010), 338.   Google Scholar [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.   Google Scholar [8] N. V. Filimonenkova, "A Quality Analysis of Weak Solutions to m-Hessian Equations,", Ph.D thesis, (2010).   Google Scholar [9] L. Garding, An inequality for hyperbolic polynomials,, J. Math. Mech., 8 (1959), 957.   Google Scholar [10] N. M. Ivochkina, Second order equations with d-elliptic operators,, Trudy Mat. Inst. Steklov, 147 (1980), 40.   Google Scholar [11] N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge - Ampere type,, Mat. Sb., 122 (1983), 265.   Google Scholar [12] N. M. Ivochkina, Solution of the Dirichlet problem for some equations of Monge - Ampere type,, Mat. Sb., 128 (1985), 403.   Google Scholar [13] N. M. Ivochkina, Solution of the Dirichlet problem for the curvature equation order m,, Algebra i Analiz, 2 (1990), 192.   Google Scholar [14] N. M. Ivochkina, The Dirichlet principle in the theory of equations of Monge - Ampere type,, Algebra i Analiz, 4 (1993).   Google Scholar [15] N. M. Ivochkina, On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations,, Problems in mathematical analysis, 170 (2010), 496.   Google Scholar [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.   Google Scholar [17] N. M. Ivochkina and N. V. Filimonenkova, Estimate of the Hölder constant for solutions to m-Hessian equations,, Problems in mathematical analysis, 159 (2009), 67.   Google Scholar [18] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions,, J. Reine Angew. Math., 229 (1968), 170.   Google Scholar [19] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75.   Google Scholar [20] D. Labutin, Potential theory for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.   Google Scholar [21] M. Lin and N. S. Trudinger, On some inequalities for elementary symmetric functions,, Bull. Austr. Math. Soc., 50 (1994), 317.   Google Scholar [22] A. V. Pogorelov, "The Mincowski Multidimensional Problem,", Nauka, (1975).   Google Scholar [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.   Google Scholar [24] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, Arch. Rat. Mech. Anal., 111 (1990), 153.   Google Scholar [25] N. S. Trudinger, Maximum principles for curvature quotient equations,, J. Math. Sci. Univ. Tokyo, 1 (1994), 551.   Google Scholar [26] N. S. Trudinger, Weak solutions of Hessian equations,, Comm. Partial Differential Equation, 22 (1997), 1251.   Google Scholar [27] N. S. Trudinger and X.-J. Wang, Hessian measures II,, Ann. of Math., 150 (1999), 579.   Google Scholar [28] N. S. Trudinger and X.-J. Wang, Hessian measures I,, Topol. Methods Nonlinear Anal., 10 (1997), 225.   Google Scholar
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