On asymptotic expansions and approximation schemes for the $p$-Laplacian
A talk in the Nonlocal Equations: Analysis and Numerics series by
Félix del Teso from Madrid
| Abstract: | The aim of this talk is to introduce the topic of asymptotic expansions and
approximation schemes for the following $p$-Laplacian type operators:
\begin{align*}
\Delta_p u := \nabla(|\nabla u|^{p-2} \nabla u) \quad \text{ and } \Delta_p^N u := \frac{1}{|\nabla u|^{p-2} } \Delta_p u.
\end{align*}
We will present the results in collaboration with J. J. Manfredi and M. Parviainen ([3]). Here, we show a unified framework to prove convergence of
approximation schemes for boundary value problems regarding $\Delta_p^N$, the so-called
normalized $p$-Laplacian, which has to be treated in the context of viscosity
solutions.
While for $\Delta_p^N$, asymptotic expansion and finite difference discretizations were
very well known, this was not the case for $\Delta_p$. In the second part of the talk,
we will present such results. This is a work in collaboration with E. Lindgren
([2,3]). Here, we introduce new asymptotic expansions and finite difference
discretizations and show convergence of approximation schemes for associated
problems.
[1] del Teso, Félix; Lindgren, Erik; A mean value formula for the variational
$p$-Laplacian. NoDEA Nonlinear Differential Equations Appl., 28 (2021),
no. 3, Paper No. 27, 33 pp.
[2] del Teso, Félix; Lindgren, Erik; A finite difference method for the variational $p$-Laplacian. Journal of Scientific Computing, 90 (2022), Article No. 67, 31pp.
[3] del Teso, Félix; Manfredi, Juan J.; Parviainen, Mikko; Convergence of dynamic programming principles for the $p$-Laplacian. Advances in Calculus of Variations, Ahead of print. (2019). Within the CRC this talk is associated to the project(s): A7 |