Solution of the Dirichlet Problem for Some Equations of Monge-Ampère Type

An Alexsandrov type theorem fork-convex functions

“…Then u is differentia We almost everywhere C" and diU 6 BV1 OC (R"), for all PROOF: Observe that for k > n/2, we can take n < q < nk/(n -A; ) and by the gradient estimate (2.2), we conclude that A:-convex functions are differentiable £ " almost available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700038260 [6] where $ E fdx we denote the mean value (C n (E))~l J E f dx. [u] is represented by the k-Hessian operator F k [u].…”

“…The study of fc-Hessian operators was initiated by Caffarelli, Nirenberg and Spruck [2] and Ivochkina [6] and further developed by Trudinger and Wang [10,12,13,14,15].…”

Oblique boundary value problems for augmented Hessian equations I

“…To show F5, (independently of F4), we may simply estimate, using F2 and (1.10), for r 0 ∈ and F(r ) < b, 9) for some constant δ 0 , depending on F and b, if F(r 0 ) is sufficiently large. To show F4, we first note from (1.10) that F(tr) is nondecreasing in t for t > 0 so if the point r lies on the ray from the vertex 0 through r 0 ∈ , we have F(tr) ≥ F(r 0 ) for all t ≥ |r 0 |/|r |.…”

The k-Yamabe problem

“…The regularity of k-admissible solutions was established by Caffarelli, Nirenberg and Spruck [10]. See also Ivochkina [33] for some special cases, and Trudinger [65] for Hessian quotient equations. According to Caffarelli, Nirenberg and Spruck [10], a function is k-admissible if the eigenvalues of the Hessian matrix D 2 u lie in the convex cone…”