Ricerc@Sapienza

We prove boundedness and continuity for solutions to the Dirichlet problem for the equation
[
-Div(a(x,
abla u))=h(x,u)+mu,,quadhbox{in }OmegasubsetR^N,,
]
where the left--hand side is a Leray--Lions operator from $W_0^{1,p}$ into $W^{-1,p'}(Omega)$, with $1

We investigate the size of the regular set for small perturbations of some classes of strong large solutions to the
Navier–Stokes equation. We consider perturbations of the data that are small in suitable weighted L2 spaces but can be
arbitrarily large in any translation invariant Banach space. We give similar results in the small data setting.

In this article we prove that the singular set of Dirichlet-minimizing Q-valued functions is countably .m2/-rectifiable and we give upper bounds for the .m2/-dimensional Minkowski content of the set of singular points with multiplicity Q.

We construct Lipschitz Q-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of 2-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3-dimensional area minimizing cones.

We study weak solutions of the problem
$$
\begin{dcases*}
\ - \Delta u = \frac{\lambda}{|x|^2} u + u^p & \ \ \ in \ $\Omega \backslash\{0\}$\\
\ u \geq 0 & \ \ \ in \ $\Omega \backslash\{0\}$\\
\ u|_{\partial \Omega} =0 &
\end{dcases*}
$$
where $\Omega \subseteq \real^N$ is a smooth bounded domain containing the origin, $N \geq 3$, $1

This book is the reprint of the 1998 edition of Elliptic Partial Differential Equations of Second Order by David Gilbarg and Neil S. Trudinger that originally appeared in 1977.