Related papers: Li-Yau and Harnack estimates for nonlocal diffusio…
By using coupling arguments, Harnack type inequalities are established for a class of stochastic (functional) differential equations with multiplicative noises and non-Lipschitzian coefficients. To construct the required couplings, two…
In the paper, we derive Li-Yau gradient estimates and Souplet Zhang type estimates of the following equation \begin{equation*} \begin{split} u_t= \Delta_\xi p+\lambda u+A(u) , \end{split} \end{equation*} on complete noncompact metric…
This paper is devoted to studying the weak Harnack inequalities for nonlocal double phase functionals by using expansion of positivity, whose prototype is $$ \iint_{\mathbb{R}^n\times\mathbb{R}^n}…
These notes are written up after my lectures at the University of Pittsburgh in March 2014 and at Tsinghua University in May 2014. My objective is the $\infty$-Laplace Equation, a marvellous kin to the ordinary Laplace Equation. The…
We establish quantitative estimates for solutions $u(t,x)$ to the fractional nonlinear diffusion equation, $\partial_t u +(-\Delta)^s (u^m)=0$ in the whole range of exponents $m>0$, $0<s<1$. The equation is posed in the whole space…
This is a lecture notes for a mini-course in Department of Mathematics, Ghent University, 14 Mar.-25 Mar. 2023.
We consider fractional variants of divergence form problems with highly oscillatory local coefficients. We characterise the convergence of these coefficients by means of classical $H$-convergence covering the local behaviour of the…
By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary). This inequality as well as the…
We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the…
The propagation of primary discontinuities in initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Based on the (quasi-) Weierstra{\ss} form for regular matrix pencil, a complete characterization of…
We study master equations of the form $$(\partial_t+L)^su=f\quad\hbox{in}~\mathbb{R}\times\Omega$$ where $L$ is a divergence form elliptic operator and $\Omega\subseteq\mathbb{R}^n$. These are nonlocal equations of order $2s$ in space and…
Multidimensional hypoelliptic diffusions arise naturally in different fields, for example to model neuronal activity. Estimation in those models is complex because of the degenerate structure of the diffusion coefficient. In this paper we…
This paper is concerned with the spatial propagation of nonlocal dispersal equations with bistable or multistable nonlinearity in exterior domains. We obtain the existence and uniqueness of an entire solution which behaves like a planar…
This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time…
In this paper, we establish a new global Hessian matrix estimate for heat-type equations on Riemannian manifolds using a Bismut-type Hessian formula. Our results feature fully explicit coefficients as well as delay / growth rate functions.…
We establish Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear anisotropic elliptic equations exhibiting non-standard growth conditions. A primary example of such operators is the degenerate anisotropic…
For the logarithmically singular parabolic equation \[ u_t-\Delta\ln u=0\qquad\text{weakly in}\ \ E\times(0,T], \] we establish a Harnack type estimate in the $L^1_{loc}$ topology, and we show that the solutions are locally analytic in the…
This is a cursory review of diffractive studies ( mostly elastic scattering)at the LHC with physical interpretation of the experimental data and comments to related problems of theory.
Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper we…
By constructing a coupling with unbounded time-dependent drift, dimension-free Harnack inequalities are established for a large class of stochastic differential equations with multiplicative noise. These inequalities are applied to the…