We establish new quantitative Hessian integrability estimates for viscosity supersolutions of fully nonlinear elliptic operators. As a corollary, we show that the optimal power ε=ε(λ,Λ,n) in celebrated W2,ε–regularity estimate satisfies(1+23(1−λΛ))n−1ln⁡n4⋅(λΛ)n−1≤ε≤nλ(n−1)Λ+λ, where n≥3 is dimension and 0<λ<Λ are ellipticity constants. In particular, (Λλ)n−1ε(λ,Λ,n) blows-up, as n→∞; previous ...

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show existence a universal number 0<ε<1, depending only on ellipticity constants and dimension, such that if u is viscosity solution F(D2u)=f(x)∈Lp, then, u∈W1+ε,p, with appropriate estimates. Our strategy suggests sort feature nonli...

Let $u$ denote a solution to rotationally invariant Hessian equation $F(D^2u)=0$ on bounded simply connected domain $\Omega\subset R^2$, with constant Dirichlet and Neumann data $\partial \Omega$. In this paper we prove that if is real analytic not identically zero, then radial $\Omega$ disk. The fully nonlinear operator $F\not\equiv 0$ of general type, in particular, assumed be elliptic. We al...

This paper develops a novel probabilistic framework for stochastic nonlinear and uncertain control problems. The proposed exploits the Kullback–Leibler divergence to measure between distribution of closed-loop behavior dynamical system predefined ideal distribution. To facilitate derivation analytic solution randomized controllers systems, transformation methods are applied such that dynamics c...