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We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and k-Hessian operators.

In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem 1 2 max q ∈ V m , p { f (q) } + 1 2 min q ∈ V m , p { f (q) } − f (p) = 0 in the Sierpiński gasket with prescribed values f (p 1) , f (p 2) and f (p 3) at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle. [ABSTRACT FROM AUTHOR]

Second order Sobolev spaces are important in applications to partial differential equations and geometric analysis, in particular to equations such as the bi-Laplacian. The main purpose of this paper is to establish some new characterizations of the second order Sobolev spaces W 2 , p ( R N ) in Euclidean spaces. We will present here several types of characterizations: by second order differences, by the Taylor remainder of first order and by the differences of the first order gradient. Such characterizations are inspired by the works of Bourgain et al. (2001) and Nguyen (2006, 2008) on characterizations of first order Sobolev spaces in the Euclidean space. [ABSTRACT FROM AUTHOR]