Nina Uraltseva

Two-Phase Obstacle Problem

AbstractOn the basis of the monotonicity formula due to Alt, Caffarelli, and Friedman, the boundedness of the second-order derivatives D2u of solutions to the equation

Convex-monotone hulls and an estimate of the maximum of the solution of a parabolic equation

\Delta u = \lambda _ + \chi _{\{ u >0\} } - \lambda _ - \chi _{\{ u < 0\} } {\text{ }}in{\text{ }}D

Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

is proved, where D is a domain in Rn, Δ is the Laplace operator, χΩ is the characteristic function of the set Ω ⊂ Rn, λ+ and λ- are nonnegative constants such that λ+ + λ- >0. Bibliography: 4 titles.

Limit smoothness of the solutions of variational inequalities under convex constraints on the boundary of the domain

In the upper half of the unit ball B+ = {\x\ 0}, let u and Omega (a domain in R-+(n) = {X is an element of R-n : x(1) > 0}) solve the following overdetermined problem: Deltau = chi(Omega) in B+, u ...

On the global solutions of the parabolic obstacle problem

N. V. Krylovs estimate for the maximum of the solution of a linear parabolic equation is extended to a larger class of operators.In this connection one investigates some properties of convex and convex-monotone hulls.

Evolution of Nonparametric Surfaces with Speed Depending on Curvature, III. Some Remarks on Mean Curvature and Anisotropic flows

One investigates the smoothness of the solutions of variational inequalities, connected with second-order linear diagonal elliptic systems under convex constraints on the solution at the boundary of the domain. One establishes the Holder continuity of the first derivatives of the solutions up to the boundary of the domain.

A nonlinear problem with an oblique derivative for parabolic equations

AbstractLet u and Ω solve the problem