Qualitative properties of bounded subsolutions of nonlinear PDEs

非线性位势理论与具有纤维正则性和充分单调性的偏微分方程的比较原理

Comparison principles for nonlinear potential theories and PDEs with fiberegularity and sufficient monotonicity

本文利用Harvey-Lawson单调性-对偶方法，在纤维正则性和充分单调性条件下，证明了变系数非线性位势理论与退化椭圆/抛物PDE的比较原理，并给出梯度依赖及最优传输等示例。

Liouville results for fully nonlinear equations modeled on Hörmander vector fields. I. The Heisenberg group

This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that …

An invariance principle for gradient flows in the space of probability measures

We seek to establish qualitative convergence results to a general class of evolution PDEs described by gradient flows in optimal transportation distances. These qualitative convergence results come from dynamical systems under the general name of LaSalle Invariance Principle. By combining some of the basic notions of gradient flow theory and dynamical systems, we are able to reproduce this invariance principle in the setting of evolution PDEs under general assumptions. We apply this abstract theory to a non-exhaustive list of examples that recover, simplify, and even extend the results in thei…

对应原理：一般势理论与非线性椭圆微分算子之间的桥梁

The Correspondence Principle: A bridge between general potential theories and nonlinear elliptic differential operators

本文提出对应原理，建立一般势理论中子方程下调和函数与非线性退化椭圆PDE粘性允许下解之间的等价关系，并探讨其对偶单调性方法在比较原理中的应用。

Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids

In this article we study the long-time behaviour of a system of nonlinear Partial Differential Equations (PDEs) modelling the motion of incompressible, isothermal and conducting modified bipolar fluids in presence of magnetic field. We mainly prove the existence of a global attractor denoted by $\A$ for the nonlinear semigroup associated to the aforementioned systems of nonlinear PDEs. We also show that this nonlinear semigroup is uniformly differentiable on $\A$. This fact enables us to go further and prove that the attractor $\A$ is of finite-dimensional and we give an explicit bounds for it…

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualita…

Qualitative properties of spatial epidemiological models

We study the qualitative properties of a spatial diffusive heterogeneous SIR model, that appears in mathematical epidemiology to describe the spread of an infectious disease in a population. The model we consider consists in a system of parabolic PDEs. In the first part of the paper, we give a criterion that ensures whether or not an epidemic propagates in a given population. We show how the features of the disease and of the population (rates of infection and of recovery, localisation and diffusivity of individuals) influence the propagation of the epidemic. In particular, we prove that there…

高对比度随机偏微分方程组：均匀化与谱理论

High-contrast random systems of PDEs: homogenisation and spectral theory

该研究针对高对比度随机系数散度型椭圆偏微分方程组建立定性均匀化与谱理论，证明异质性参数趋于零时谱一般不满足豪斯多夫收敛，并用显式例子说明其比周期系数或标量算子情形应用更广、物理效应更丰富。

全非线性偏微分方程中具有调和映射样结构的存在性及先验界

Existence and a priori bounds for fully nonlinear PDEs with a harmonic map-like structure

本文研究具有调和映射样结构的全非线性偏微分方程的存在性及先验界，提出新的存在性结果和多重性定性行为。

A variational approach to symmetry, monotonicity, and comparison for doubly-nonlinear equations

We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations. Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter go…

Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell Gauged O(3) Sigma model

This article is devoted to the study of the following semilinear equation with measure data which originates in the gravitational Maxwell gauged $O(3)$ sigma model, $$-Δu + A_0(\prod^k_{j=1}|x-p_j|^{2n_j} )^{-a} \frac{e^u}{(1+e^u)^{1+a}} = 4π\sum_{j=1}^k n_jδ_{p_j} - 4π\sum^l_{j=1}m_jδ_{q_j} \quad{\rm in}\;\; \mathbb{R}^2.\qquad(E)$$ In this equation the $\{δ_{p_j}\}_{j=1}^k$ (resp. $\{δ_{q_j}\}_{j=1}^l$ ) are Dirac masses concentrated at the points $\{p_j\}_{j=1}^k$, (resp. $\{q_j\}_{j=1}^l$), $n_j$ and $m_j$ are positive integers, and $a$ is a nonnegative real number. We set $ N=\sum^k_{j=1}…

On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds

This paper is about the influence of Geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, in particular by the study of entire graphs with prescribed mean curvature, we consider classes of coercive differential inequalities of the form $$ \mathrm{div}\left( \frac{φ(|\nabla u|)}{|\nabla u|} \nabla u\right) \ge b(x)f(u) l(|\nabla u|) \qquad \text{(respectively, $\le$ or $=$)} $$ on domains of a manifold $M$, for suitable $φ,b,f,l$, with emphasis on mean curvature type operator…