On a class of equations of monge-ampere type

Abstract

We prove that the Dirichlet problem is solvable in a generalized sense ]or a class of nonlinear elliptic equations and related equations of Monge-Ampdre type. Bibliography: 1~ titles. v u 1 [ Jm-l( )= n-m+i jv(u~);'~-l[u]dx' m=2,...,n, f~ where/~ = 1, and the operators #m-l," m = 1, .. . , n, are defined by recursion as the variational derivatives for the functionals J,~,_2(u); u U ~ u 1= It turns out [9, 11], that #V~[u] --~:m[A, u]with A(p) = vp-~ 1, where vpp is the Hessian matrix for v(p). In particular for the function v = X/1 + p2 of the matrix A = X/1 + p2(~hj. + pipj) we Obtain the curvature operator of order m with a vector k defining the principal curvatures of the graph of the function u(x). If v = p2/2, then A = E, the vector k is the vector of eigenvalues of the matrix u~, and we are dealing with operators of Monge-Amp&re type. It might be more correct to call them operators of Laplacian type.

An interior estimate of the gradient of a solution to the Dirichlet problem for equations of curvature type

“…The equations with such matrices were introduced in [7], where it was proved that the Dirichlet problem is solvable for the curvature-type equations of order m, i.e.. Y = Sm\(:). The present article can be regarded as a continuation of [7], which makes it possible to pass to the class of functions 5 t-introduced in [4][5][6].…”

“…(2) at interior points of the domain fl, we apply the method proposed in [7] for the function 5 c = Sra\ (ran). The method is based on the classical maximum principle for elliptic equations and is connected with the differentiation of equations.…”

An optimal property ofl-minimal surfaces in viscosity interpretation

“…Obviously, the right-hand inequality in (13) In [19] the solvability of the problem (6), (4) was proved in the viscosity sense for f > O, v = (l +p2)~/2 /s, s > 1. Formulate a theorem that is a consequence of these results.…”

Estimate of the gradient of a solution of the Dirichlet problem for Monge-Ampère type equations in a domain with nonsmooth boundary

“…to [9], devoted to the problem (3), (4) in the case (s --2)uiuj aij = Oj --a~ -__(.~; : ~)u2, s > 1.…”

“…In [9] the existence of a solution to the problem (3)-(5) is established: In [9] the existence of a solution to the problem (3)-(5) is established:…”