Estimates of solutions to linear elliptic systems and equations

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.

Exact and numerical solutions of linear and non-linear systems of fractional partial differential equations

The present study introduces a new technique of homotopy perturbation method for the solution of systems of fractional partial differential equations. The proposed scheme is based on Laplace transform and new homotopy perturbation methods. The fractional derivatives are considered in Caputo sense. To illustrate the ability and reliability of the method some examples are provided. The results ob...

On Gaussian decay estimates of solutions to some linear elliptic equations and its applications

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.

Discontinuous Solutions of Linear, Degenerate Elliptic Equations

We give examples of discontinuous solutions of linear, degenerate elliptic equations with divergence structure. These solve positively conjectures of De Giorgi.