Global gradient estimates for nonlinear parabolic operators

We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of the ambient space and on the initial and boundary data. As a byproduct, one easily obtains a universal interior estimate, not depending on the parabolic data. The setting taken into account includes sourcing terms and general diffusion coefficients. The results are new, to the best of our knowledge, even in the Euclidean setting, though we treat here also the case of a complete Riemannian manifold.

The growth of gradients of solutions of some elliptic equations and Bi-Lipschicity of QCH

In this paper, we study the growth of gradients of solutions of elliptic equations, including the Dirichlet eigenfunction solutions on bounded plane convex domain. Several results related to Bi-Lipschicity of quasiconformal harmonic (qch) mappings with respect to quasi-hyperbolic and euclidean metrics, are proved. In connection with the subject, we announce a few results concerning the so called interior estimate, including Proposition 1.1. In addition, a short review of the subject is given.