Analysis of PDEs (math.AP)

Abstract: In these notes we want, in addition to presenting some new results, to both clean up and refine some reflections on a couple of articles published on paper a few years ago, 2019-20. These papers concerned integral sufficient conditions on $u\,,$ $\n u\,,$ and mixed, to guarantee the equality of the energy, EE in the sequel, for solutions of the Navier-Stokes equations under the classical non-slip boundary condition. Concerning the $\n u$ case a crucial role was enjoyed by a previous well known Berselli and Chiodaroli's pioneering 2019 work on the subject. The above three papers are the main sources of these notes. References will be mostly concentrated on their direct relation to the above papers at the time of pubblication. More recent results will be not stated throughout the article. However, in the last section, the reader will be suitably sent to the more recent bibliography. Below, we also turn back to the innovative interpretation of some main parameters which allowed to overcome their apparent incongruence. Non-Newtonian fluids were also considered in our 2019 paper, maybe for the first time in the above Berselli-Chiodaroli's particular $\n u\,$ context. However we will stick mostly to the Newtonian case since in the end we come to the conclusion that there are not particular additional obstacles to extend the present results from Newtonian to non-Newtonian fluids. Hence we avoid to go further in this direction.

Abstract: We provide a precise description of the set of normalized ground state solutions (NGSS) for the class of elliptic equations: $$ -\Delta u - \lambda u + V (| x |) u - f (| x |, u) = 0,\quad\text{in}\quad \mathbb{R}^n,\ n\geq 1. $$ In particular, we show that under suitable assumptions on $V$ and $f$, the NGSS is unique for all the masses except for at most a finite number. Moreover, we prove that when unique, the NGSS $u_c$ is a smooth function of the mass $c.$ Our method is as follow: using the NGSS for a given mass $c$, we construct an exhaustive list of potential candidates to the minimization problem for masses close to $c$, and we develop a strategy how to pick the right one. In particular, if there is a unique NGSS for a given mass $c_0,$ then this uniqueness property is inherited for all the masses $c$ close to $c_0.$ Our method is general and applies to other equations provided that some key properties hold true.

Abstract: We study $s$-dependence for minimizing $W^{s,n/s}$-harmonic maps $u\colon \mathbb{S}^n \to \mathbb{S}^\ell$ in homotopy classes. Sacks--Uhlenbeck theory shows that, for each $s$, minimizers exist in a generating subset of $\pi_{n}(\S^\ell)$. We show that this generating subset can be chosen locally constant in $s$. We also show that as $s$ varies the minimal $W^{s,n/s}$-energy in each homotopy class changes continuously. In particular, we provide progress to a question raised by Mironescu and Brezis--Mironescu.

Abstract: The aim of these notes is to give a complete self-contained account of the current state of the art in the regularity for planar minimizers and critical points of the Mumford-Shah functional.