We prove Harnack's inequality for non-negative solutions of some degenerate elliptic operators in divergence form with the lower order term coefficients satisfying a Kato type contition.

Let G be a Lie group of polynomial volume growth, with Lie algebra g. Consider a second-order, right-invariant, subelliptic differential operator H on G, and the associated semigroup St = e-tH. We identify an ideal n' of g such that H satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of n'. The regularity is expressed as L2 estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel....

We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation ${\sum }_{i=1}^m\frac{\partial }{\partial x_i}{a}_i(x,u,\nabla u)-c_0{\left|u\right|}^{p-2}u=f(x,u,\nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $\lambda \left(\right|u\left|\right){\sum }_{i=1}^m{a}_i(x,u,\eta ){\eta }_i\ge \nu \left(x\right){\left|\eta \right|}^p,$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$.

This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...

Hölder continuity of weak solutions is studied for a nondiagonal parabolic system of singular quasilinear differential equations with matrix of coefficients satisfying special structure conditions. A technique based on estimating linear combinations of the unknowns is employed.

Let $(X,\omega )$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X,\omega )$ of the complex Monge-Ampère operator acting on $\omega$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^p$-density belong to $(X,\omega )$ and proving that $(X,\omega )$ has the...

Regularity results for elliptic systems of second order quasilinear PDEs with nonlinear growth of order $q>2$ are proved, extending results of [7] and [10]. In particular Hölder regularity of the solutions is obtained if the dimension $n$ is less than or equal to $q+2$.

If ${\left(e^{-tA}\right)}_{t>0}$ is a strongly continuous and contractive semigroup on a complex Banach space $B$, then $-{(-A)}^{\alpha }$, $0<\alpha <1$, generates a holomorphic semigroup on $B$. This was proved by K. Yosida in [7]. Using similar techniques, we present a class $H$ of Bernstein functions such that for all $f\in H$, the operator $-f(-A)$ generates a holomorphic semigroup.