ABSTRACT. We derive an a prioriC 2,α estimate for solutions of the fully non-linear elliptic equationF(D 2 u) = 0, provided the level setΣ={M|F(M)=0} satisfies: (a)Σ∩{M|TrM= t} is strictly convex for all constants t; (b) the angle between the identity matrixI and the normalFij toΣis strictly positive on the non-convex part of Σ. Moreover, we do not need any convexity assumption onF in the course of the proof for the two dimensional case, as the classical result indicates. 1.

In this paper we study Hölder regularity for the first and second derivatives of continuous viscosity solutions of fully nonlinear equations of the form (1.1) F(D 2 u) = 0. It is well known that viscosity solutions of (1.1) are C 1,α for some 0 < α < 1, and

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1

We consider the symmetric non-local Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1|y − x | −d−α ≤ J(x, y) ≤ κ2|y − x | −d−β for 0 < α < β < 2, |x − y | < 1. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F). We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.

: Polynomial bounds for the coefficient of fi-mixing are established for stochastic dynamical systems under weak recurrency assumptions. The method is based on direct evaluations for certain functionals of hitting-times of the system under consideration. Key-words: mixing coefficients, stochastic differential equations, stability of Markov chains (R'esum'e : tsvp) 19 Bolshoy Karetnii (Ermolovoy str.), Institute of Information Transmission Problems, Moscow, Russia This paper was supported by INTAS grants # 93-0894, and # 93-1585 CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Centre National de la Recherche Scientiøque Institut National de Recherche en Informatique (URA 227) Université de Rennes 1 Insa de Rennes et en Automatique unité de recherche de Rennes Sur les bornes polynomiales des coefficients de m'elange pour des 'equations diff'erentielles stochastiques R'esum'e : Sous la condition de recurrence faible on etablit des bornes polynomiales pour le coefficient de fi-melange...

A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in non-divergence form in a bounded cylinder Q =\Omega \Theta (0; T ) which vanish on @xQ = @\Omega \Theta (0; T ) ; where\Omega is a bounded Lipschitz domain in R n . This inequality is applied to the proof of the Holder continuity of the quotient of two positive solutions vanishing on a portion of @xQ: 1.

this article, we will study polynomial growth solutions to a uniformly elliptic operator of non-divergence form defined on R

Abstract. Hölder estimates for spatial second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions. 1.

. We study the obstacle problem for fully nonlinear second-order uniformly elliptic operators. We can show the existence of a continuous viscosity solution in the general setting,and we can show C 1;1 -regularity of the viscosity solution when the operator is convex or concave. C 1;ff -regularity of the free boundary is established when the operator is convex in any dimension or concave in two dimensions. In this paper,we consider the least viscosity super solution satisfying the following equation F (D 2 u) 0 in\Omega u = 0 on @\Omega u OE in\Omega (0.1) where ffl F is uniformly elliptic i.e. jjP jj F (A + P ) \Gamma F (P ) jjP jj when P 0 ffl OE 2 C 2;ff (\Omega\Gamma ; OE ! 0 on @\Omega OE(x o ) ? 0 for some x o 2\Omega ffl F (D 2 OE) and DF (D 2 OE) do not vanish simultaneously. When F is a linear operator,many authors studied the existence and the regularity of the weak solution,and the regularity of the free boundary. Hans Lewy and Guido Stampacchia con...

We consider second-order linear elliptic operators of nondivergence type which are intrinsically defined on Riemannian manifolds. Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabré’s result and, as a consequence, we give another proof to the Harnack inequality of Yau for positive harmonic functions on Riemannian manifolds with nonnegative Ricci curvature using the nondivergence structure of the Laplace operator. 1. Introduction and