Higher Regularity of the p-Poisson Equation in the Plane
A talk in the Oberseminar Analysis series by
Anna Khripunova-Balci
| Abstract: | We consider non-linear, degenerate $p$-Poisson equation
\begin{align*}
-\mathrm{div}(A(\nabla u))=- \mathrm{div} (|\nabla
u|^{{p-2}}\nabla u)= -{\rm div} F.
\end{align*}
In recent years it has been discovered that such equations allow for optimal regularity. The non-linear mapping
$F \mapsto A(\nabla u)$ satisfies surprisingly the linear, optimal estimate
$\|A(\nabla u)\|_X \le c\, \|F\|_X$ for several choices of
spaces $X$. In particular, this estimate holds for Lebesgue
spaces $L^q$ (with $q \geq p'$), spaces of bounded mean oscillations
and Hölder spaces $C^{0,\alpha}$ (for some $\alpha>0$).
In this talk we show that we can extend this theory to Sobolev and
Besov spaces of (almost) one derivative. Our result are restricted to
the case of the plane, since we use complex analysis in our
proof. Moreover, we are restricted to the super-linear
case $p \geq 2$, since the result fails if $p < 2$.
The talk is based on joint work with Lars Diening and Marcus Weimar. |