Nonlinear Elliptic and Parabolic Equations of the Second Order

Behavior near the boundary of positive solutions of second order parabolic equations. II

“…For cylinders Q = C r , r > 0, the constant N = N (n, ν, δ r ). This theorem was proved in [KS], see also [K,Chap. 4], for the divergence case it was proved in [M1], [M2], see also [FSt].…”

Martingale representation theorem for theG-expectation

“…We claim that the celebrated regularity result of Krylov [11] (Theorem 1, section 6.3, page 292) applies and that v ǫ i (t, x 1 , . .…”

“…Moreover, it is constructed so that all its derivatives with respect to γ are bounded on all of the space. Hence this nonlinearity G ǫ can be directly shown to belong to the class of functions considered in the Definition 5.5.1 of [11]. Moreover, in the notation of Theorem 1 of Section 6.3 in [11] (page 292), the domain Q = (0, 1) × R d .…”

“…Hence this nonlinearity G ǫ can be directly shown to belong to the class of functions considered in the Definition 5.5.1 of [11]. Moreover, in the notation of Theorem 1 of Section 6.3 in [11] (page 292), the domain Q = (0, 1) × R d . Therefore, this theorem applies to yield existence and interior regularity.…”

“…Now suppose that the solution u(t, x) of (2.2) is smooth. Indeed, we can approximate the equation (2.2) so that the approximating equation admits smooth solutions as proved by [11]. This is done in the Appendix.…”