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.