相关论文: Bochner formula and Bernstein type estimates on lo…
Faber-Krahn functions provide lower bounds on the first Dirichlet eigenvalue of the Laplacian and are useful because they imply heat kernel upper bounds. In this paper, we are interested in Faber-Krahn functions and heat kernel estimates…
A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincar\'e's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this…
In this paper we study numerical approximations of the evolution problem for the nonlocal $p$-Laplacian operator with homogeneous Neumann boundary conditions on inhomogeneous random convergent graph sequences. More precisely, for networks…
We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully…
Elliptic homogenization is used to determine coarse-grained properties of materials with features on small scales for heat transfer and elasticity. When microstructural features of a material have rapid, periodic fluctuations, the solution…
This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate…
We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our…
We consider a boundary value problem in unbounded 2D doubly periodic composite with circular inclusions having arbitrary constant conductivities. By introducing complex potentials, the boundary value problem for the Laplace equation is…
In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and…
Recently, Ali et al. posed several open problems concerning extremal graphs with respect to the ABS index. These problems involve characterizing graphs that attain the maximum ABS index within specific graph classes, including: connected…
In this paper, we look at a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov…
We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and…
Consider a finite connected graph denoted as $G=(V, E)$. This study explores a generalized Chern-Simons Higgs model, characterized by the equation: $$ \Delta u = \lambda e^u (e^u - 1)^{2p+1} + f,$$ where $\Delta$ denotes the graph…
Formal Laplace operators are analyzed for a large class of resistance networks with vertex weights. The graphs are completed with respect to the minimal resistance path metric. Compactness and a novel connectivity hypothesis for the…
We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both…
This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical…
This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which…
We construct parabolic analogues of (global) eigenvarieties, of patched eigenvarieties and of (local) trianguline varieties, that we call respectively Bernstein eigenvarieties, patched Bernstein eigenvarieties, and Bernstein paraboline…
We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We use the formula to derive lower…
We study the numerical approximation of a class of degenerate parabolic stochastic partial differential equations on non-compact metric graphs, which naturally arise in the asymptotic analysis of Hamiltonian flows under small noise…