The symmetric $p$-Laplace operator enters various models in mathematical physics,
such as incompressible materials with power-type hardening and non-Newtonian fluids. In this and related models, the usual gradient $\nabla u$ of the displacement field $u$ is replaced by the symmetric gradient $\mathcal{E}u=\frac 12 (\nabla u +(\nabla u)^T)$, which results in a higher coupling of the system.

We present regularity results for the solutions of such systems, including global and maximal regularity.

This talk is based on joint work with Andrea Cianchi, Lars Diening, and Fa Peng.

Nonlocal operators like the fractional Laplacian play a prominent role in physical models like the Boltzmann equation. Associated nonlocal Dirichlet problems and the regularity of their solutions have been extensively studied in the past decades.

We establish optimal boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of fractional order.
Namely, we consider Lévy operators that are symmetric and their Fourier symbols are comparable to $|\xi|^{2s}$ in $\mathbb{R}^d$.
This was only known when the kernel of the operator (or Lévy measure) is either homogeneous or comparable to that of the fractional Laplacian, with different proofs in each case.
Our new proofs extend both at the same time, and work in a very general class of domains, under a $C^1$-Dini-type condition.

This talk is based on a joint article with Xavier Ros-Oton.

We consider the one-dimensional nonlinear Dirac equation with cubic scalar self interaction, commonly known as the Gross--Neveu model. This equation admits solitary wave solutions of the form $\phi_{m,\omega}(x)e^{-i\omega t}$, where $m>0$ is the mass and $0<|\omega|<m$. In this talk, we discuss the spectral stability of non-relativistic solitary waves, namely the regime $\omega\in(m-\epsilon,m)$ for some sufficiently small $\epsilon>0$. We prove that, for $\epsilon$ sufficiently small, the essential spectrum of the linearized operator is given by $i\bigl((-\infty,-m+\omega)\cup(m-\omega,\infty)\bigr)$, while the point spectrum consists only of the symmetry-induced eigenvalues: a zero eigenvalue associated with translations and eigenvalues at $\pm 2\omega i$ arising from the underlying $SU(1,1)$ symmetry. We further establish limiting absorption principles at the thresholds of the essential spectrum, $\pm i(m-\omega)$, as well as at the embedded thresholds $\pm i(m+\omega)$. Our approach is based on analyzing the Schur complement of the linearized Dirac operator and relating it to a relatively compact perturbation of the linearized one-dimensional cubic nonlinear Schr\"odinger operator about its solitary wave. This is joint work with Nabile Boussa{\"\i}d, Andrew Comech, and Jonas Lührmann.

We consider on arbitrary Riemannian manifolds the Leibenson equation \begin{equation*} \partial _{t}u=\Delta _{p}u^{q}. \end{equation*} This equation is also known as doubly nonlinear evolution equation. It comes from hydrodynamics where it describes filtration of a turbulent compressible liquid in porous medium. We show that, under optimal restrictions on $p$ and $q$, non-negative bounded solutions to this equation have finite propagation speed. The talk is based on joint work with Alexander Grigor'yan.