Some remarks on sign changing solutions of a quasilinear elliptic equation in two variables

We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain vanishing properties of sign changing solutions to such a Dirichlet problem. Our method is applicable in the plane.Comment: In v2, we discuss and fix a gap that was found in our method. However, our results are weaker after this modification. Can be considered part II of arXiv:1205.0785, with which it overlap

Phragm\'en-Lindel\"of theorem for infinity harmonic functions

We investigate a version of the Phragm\'en-Lindel\"of theorem for solutions of the equation $\Delta_\infty u=0$ in unbounded convex domains. The method of proof is to consider this infinity harmonic equation as the limit of the $p$-harmonic equation when $p$ tends to $\infty$

Arithmetic three-spheres theorems for quasilinear Riccati type inequalities

We consider arithmetic three-spheres inequalities to solutions of certain second order quasilinear elliptic differential equations and inequalities with a Riccati-type drift term.Comment: to appear in Journal d'Analyse Math\'ematiqu

Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincar\'e inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss-Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss-Green formula we introduce a suitable notion of the interior normal trace of a regular ball

Vanishing properties of sign changing solutions to p-Laplace type equations in the plane

We study the nonlinear eigenvalue problem for the p-Laplacian, and more general problem constituting the Fucik spectrum. We are interested in some vanishing properties of sign changing solutions to these problems. Our method is applicable in the plane.Comment: In v2, we discuss and fix a gap that was found in our method. However, our results are weaker after this modificatio

Aspects of area formulas by way of Luzin, Rad\'o, and Reichelderfer on metric measure spaces

We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincar\'e inequality and are equipped with a doubling measure. The properties we have selected to study are those that are related to area formulas