The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

Weak solutions to the complex Hessian equation

“…[8], [15], [16], [19], [21], [22]. It is clear that if u(z) = u(x + iy) is independent of y then it is m-subharmonic if and only if it is m-convex (see [22]).…”

“…We essentially follow [8] using some ideas from [7] and the simplification from [21]. We use the notation u j = ∂u/∂z j , u j = ∂u/∂z j .…”

On a Fractional Monge–Ampère Operator

“…In that case, there is considerable work in the literature establishing the existence, uniqueness and regularity of solutions to (1.1), see [1,2,8,10] and the references therein. The main ingredients entering the theory are, roughly speaking, the following: We deduce that at a maximum of a second derivative D 2 ee u the function f must satisfy,…”

“…In fact, one may consider any concave function of the Hessian as in [2] as an infimum of affine transformations of the Laplacian, the affine transformations corresponding now to the different linearization coefficients of the function F(λ 1 , . .…”

“…In fact, the geometry of the domain is an important issue for the "inherited from the boundary" regularity theory for degenerate operators depending on the eigenvalues of the Hessian, see [2].…”