Regularity for Non-Linear PDEs

Quasilinear Elliptic Equations: regularity and existence results

speaker: Patrizia Pucci (Università degli Studi di Perugia, Dipartimento di Matematica ed Informatica)

abstract: In this talk we first present some regularity results for the solutions of the general quasilinear elliptic equation \begin{equation}\label{eg} \mbox{div}\boldsymbol A(x, u(x), Du(x))=B(x, u(x), Du(x)) \quad \mbox{in}\quad \Omega, \end{equation} where $\Omega$ is a domain of $\mathbb R^n$, not necessarily bounded, while $\boldsymbol A:\Omega\times \mathbb R\times\mathbb R^n\to \mathbb{R}^n$ and $B:\Omega\times \mathbb R\times \mathbb R^n\to \mathbb R$ are Carathéodory functions, satisfying the condition\smallskip

\noindent {\it there exist $a_1, a_4>0$ and $a_2$, $a_3\ge 0$ such that for a.a. $x\in \Omega$ and for all $(z, \bold \xi)\in \mathbb R\times\mathbb R^n$ $$\begin{array}{cc} (a) & \langle \bold A(x,z, \bold \xi), \bold \xi\rangle\ge a_1
\bold \xi
^p-a_2
z
^p-a_3; \\ \\ (b) &
\bold A(x,z, \bold \xi)
\le a_4\bold
\xi
^{p-1}. \end{array}\leqno{(A1)}$$} \noindent Hence we consider also the case when $a_3>0$ in $(a)$, which corresponds to the inhomogeneous version of problem~\eqref{eg}. The nonlinear term $B$ can be possibly singular at some point. Finally, we briefly show the existence and non-existence results for $p$--Laplace equations with multiple critical nonlinearities.

timetable:
Mon 14 Sep, 15:00 - 16:00, Aula Dini
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