m-Hessian Equations Research Articles

Solutions to weighted complex m-Hessian equations on domains in [formula omitted

In this paper, we first study the comparison principle for the operator Hχ,m. This result is used to solve certain weighted complex m-Hessian equations.

Geometry of m-Hessian Equations

In the process of developing the modern theory of fully nonlinear, second-order partial differential equations, new geometric characteristics of surfaces naturally appeared. The implementation of these characteristics in terms of the classical differential geometry leads to significant technical difficulties. This paper provides a review of the necessary methodological reform and demonstrates a new differential geometric techniques by an example of constructing boundary barriers for m-Hessian equations.

Weak Solutions to the Complex m-Hessian Equation on Open Subsets of $${{\mathbb {C}}}^{n}$$

In this paper, we prove the existence of weak solutions to the complex m-Hessian equations in the class \({\mathcal {D}}_{m}(\Omega )\) on an open subset \(\Omega \) of \({\mathbb {C}}^n\). In the end of the paper we give an example shows that in the unit ball \({\mathbb {B}}^{2}(0,1)\subset {\mathbb {C}}^{2}\) the complex Monge-Ampere equation \((dd^{c} .)^{2}=\mu \) is solvable but the complex Hessian equation \(H_{1}(.)=\mu \) has not any weak solutions where \(\mu \) is a nonnegative Radon measure on \({\mathbb {B}}^{2}(0,1)\).

On New Structures in the Theory of Fully Nonlinear Equations

We describe the current state of the theory of equations with m-Hessian stationary and evolution operators. It is quite important that new algebraic and geometric notions appear in this theory. In the present work, a list of those notions is provided. Among them, the notion of m-positivity of matrices is quite important; we provide a proof of an analog of Sylvester’s criterion for such matrices. From this criterion, we easily obtain necessary and sufficient conditions for existence of classical solutions of the first initial boundary-value problem for m-Hessian evolution equations. The asymptotic behavior of m-Hessian evolutions in a semibounded cylinder is considered as well.

Attractors of m-Hessian Evolutions

We study the asymptotic behavior of C 2-evolutions u = u(x, t) under a given action of the m-Hessian evolution operators and boundary conditions. We obtain sufficient (close to necessary) conditions for the convergence of solutions to the first initial-boundary value problem for the m-Hessian evolution equations to stationary functions as t → ∞. Bibliography: 18 titles.

On some properties of the positive m-Hessian operators in C 2(Ω)

We consider the p-Hessian operator Tp[u] := trpuxx on the set \({\{u \in C^2(\bar{\Omega}), T_p[u] \geq \nu > 0, 1 \leq p \geq 0\}}\) and extend the common approach to the uniqueness and existence theorems. For instance, we prove the uniqueness of solutions to the Dirichlet problem for m-Hessian equations in \({C^2(\bar{\Omega})}\) for constant boundary data. We also give nonexistence theorems and, in addition, we establish the extremal properties of the functionals Ip[u; f] in \({C^2(\bar{\Omega})}\) and we derive some applications.

On the backgrounds of the theory of m-Hessian equations

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.

On the classical solvability of the Dirichlet problem for nondegenerate m-Hessian equations

We study the classical solvability of m-Hessian equations. We give a complete proof of the existence of a classical solution under minimal assumptions on the data of the Dirichlet problem and systematize methods for studying fully nonlinear Hessian equations. Bibliography: 18 titles.

On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations

We construct an a priori estimate of the seminorm $$ {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} $$ for solutions to the problem $$ {F_m}\left[ u \right] = f;\quad \left. u \right|{_{\partial \Omega }} = \Phi $$ in terms of $$ \mathop {{\inf }}\limits_\Omega f > 0 $$ , $$ \mathop {{\sup }}\limits_{\partial \Omega } \left| {{f_x}} \right| $$ , $$ {\left\| {f_{xx}^{-} } \right\|_{{L^n}\left( \Omega \right)}} $$ , diam Ω, $$ {\left\| {\partial \Omega, \Phi } \right\|_{{C^3}}} $$ , $$ {\left\| u \right\|_{{C^2}\left( \Omega \right)}} $$ . For this purpose, we analyze the methods due to L. C. Evans, N. V. Krylov, M. V. Safonov, and N. Trudinger. Bibliography: 10 titles.

An estimate for the Hölder constant for weak solutions to m-Hessian equations in a closed domain

The smoothness properties of weak solutions to the Dirichlet problem for m-Hessian equations are studied. Namely, fully nonlinear second-order equations of the form $$ tr_m u_{xx} = f^m $$ in a domain Ω ⊂ Rn are considered; here, trmuxx is the sum of all principal minors of the matrix uxx for 1 ⩽ m ⩽ n. Such equations were considered in the 1980s by N.M. Ivochkina, L. Nirenberg, L. Caffarelly, J. Spruck, and N.V. Krylov for f ∈ C4(\( \bar \Omega \)), f > 0. In 1997, N. Trudinger introduced the notion of an approximative solution to an m-Hessian equation and outlined the proof of its local Holder regularity for f belonging to one of the Lebesgue spaces. The behavior of an approximative solution to the Dirichlet problem near the boundary provided that f ∈ Ln(Ω), f ⩾ 0, and f ∈ Lp for p ⩾ n near the boundary is analyzed. It is shown that the solution approaches the boundary value at rate dα, where d is the distance to the boundary and 0 < α < 1. The dependence of α on p is also described. Moreover, the simplest proof of the Holder regularity of a weak solution in a closed domain is given. Methods developed by O.A. Ladyzhenskaya and N.N. Ural’tseva and Trudinger’s approach are used.

Analysis of the behavior of a weak solution to m-Hessian equations near the boundary of a domain

We consider the Dirichlet problem for the m-Hessian equations F m [u] = f in a domain Ω and analyze the behavior of approximate solutions at the boundary of Ω. We show that the growth rate for weak solutions towards to the boundary locally depends on the summability exponent of f or on the fact whether f belongs to a certain Morrey type space near the boundary. The result obtained can be used for estimating the Holder constant for weak solutions in the closed domain. Bibliography: 11 titles.

Estimate of the Hölder constant for solutions to m-Hessian equations

An a priori estimate for the Holder constant with exponent 0 < α < 1 is obtained for smooth solutions to m-Hessian equations in a closed domain with (m − 1)-convex boundary of class C 2. The Holder constant depends on the L p -norm, p > [n(n + 1)]/2, of the right-hand side of the equation, and the estimate remains valid for approximate solutions. Bibliography: 4 titles.

Phragmén–Lindelöf type theorem for m-hessian equations

We derive an interior estimate for the gradient of a solution u to the m-Hessian equation with a certain right-hand side. The estimate depends on the oscillation of u and properties of the right-hand side of the equation. The proof is based on a modification of some ideas of Trudinger (1997). As a consequence of the main result, a theorem of Fragmen–Lindelof type is obtained for solutions to the m-Hessian equations in the entire space $$ \mathbb{R}^n $$ . Bibliography: 4 titles.

Lemma about increase of approximate solutions to the Dirichlet problem for m-Hessian equations

We prove an assertion about the increase of a solution, weak in the sense of Trudinger, to the Dirichlet problem for m-Hessian equations with the righthand side in L q , q > n(n + 1)/(2m). We estimate the ratio between the increment of the solution along the normal and the distance to the boundary of a domain. This assertion is also proved for some class of degenerate linear elliptic equations of second order. Bibliography: 7 titles.

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem.