Estimates for the Multiplicative Square Function of Solutions to Nondivergence Elliptic Equations

Hölder Estimates for Solutions of Degenerate Nondivergence Elliptic and Parabolic Equations

We deal with a class of nondivergence type elliptic and parabolic equations degenerating at the coordinate hyperplanes. Assuming that the degeneration is coordinatewise and varies regularly, we prove the Hölder continuity of solutions. Also, the approximative solutions are considered. §

Boundary Estimates for Positive Solutions to Second Order Elliptic Equations

Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the Hopf-Oleinik type estimates and the boundary Lipschitz estimates for solutions. These conditions are sharp even for harmonic functions.

Pointwise Estimates of Solutions to Semilinear Elliptic Equations and Inequalities

We obtain sharp pointwise estimates for positive solutions to the equation −Lu+V u = f , where L is an elliptic operator in divergence form, q ∈ R\{0}, f ≥ 0 and V is a function that may change sign, in a domain Ω in R, or in a weighted Riemannian manifold.

A Priori Estimates of Positive Solutions for Sublinear Elliptic Equations

In this paper, a priori estimates of positive solutions for sublinear elliptic equations are given in terms of thicknesses of domains. To this end, a supersolution is constructed by a composite function of a solution to an ordinary differential equation and a distance function. The results work efficiently in the case where the domain is an exterior or an interior of a convex set.

New Jacobi Elliptic Function Solutions for the Zakharov Equations