Related papers: Square function and heat flow estimates on domains

We give matching upper and lower bounds for the Dirichlet heat kernel of a Schr\"odinger operator $\Delta+W$ in the domain above the graph of a bounded Lipschitz function, in the case when $W$ decays away from the boundary faster than…

Using the theory of $L^p$-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value…

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations (*) $Pu+\partial_tu=f$ on $\Omega\times I $, where $P$ is a nonlocal operator, and $\Omega \subset R^n$,…

We establish a connection between anomalous heat conduction and anomalous diffusion in one dimensional systems. It is shown that if the mean square of the displacement of the particle is $<\Delta x^2> =2Dt^{\alpha} (0<\alpha\le 2)$, then…

We outline here a simple mathematical introduction to the notions of multipoles for a general extensive property $\Pi$ from the point of view of continuum mechanics. Classically, $\Pi$ is the electric charge, but the theory is not limited…

In this paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators. By the…

It is known that the couple formed by the two dimensional Brownian motion and its L\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is…

For an operator generating a group on $L^p$ spaces transference results give bounds on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction…

We prove that for a general diffusion process, certain assumptions on its behavior \emph{only within a fixed open subset} of the state space imply the existence and sub-Gaussian type off-diagonal upper bounds of the \emph{global} heat…

The Langevin theory of the polarization of a dilute collection of dipoles by an external field is often included in introductory solid state physics and physical chemistry curricula. The average polarization is calculated assuming the…

We characterize all the solutions of the heat equation that have their (spatial) equipotential surfaces which do not vary with the time. Such solutions are either isoparametric or split in space-time. The result gives a final answer to a…

An experimental and numerical smoothed particle hydrodynamics (SPH) analysis was performed for the convective flow arising from a horizontal, thin cylindrical heat source enclosed in a glycerin-filled, slender enclosure at low Rayleigh…

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply…

In this paper we give the q-analogue of the higher-order Bessel operators studied by M. I. Klyuchantsev [12] and A. Fitouhi, N. H. Mahmoud and S. A. Ould Ahmed Mahmoud [3]. Our objective is twofold. First, using the q-Jackson integral and…

The heat equation with inverse square potential on both half-lines of $\mathbb{R}$ is discussed in the presence of \emph{bridging} boundary conditions at the origin. The problem is the lowest energy (zero-momentum) mode of the transmission…

The aim of this article is to establish two-sided Gaussian bounds for the heat kernels on the unit ball and simplex in $\mathbb{R}^n$, and in particular on the interval, generated by classical differential operators whose eigenfunctions are…

We consider heat operators on a convex domain $\Omega$, with a critically singular potential that diverges as the inverse square of the distance to the boundary of $\Omega$. We establish a general boundary controllability result for such…

We investigate geometric analysis on metric measure spaces equipped with asymmetric distance functions. Extending concepts from the symmetric case, we introduce upper gradients and corresponding $L^q$-energy functionals as well as…

Heating plates describe the transfer of heat from actuators to a target object. In other words, they separate the heat sources and heated object and can be further used to apply a specific heat distribution on this object. Therefore, an…

In this work we investigate the inverse problem of recovering one point source in the heat equation from sparse boundary measurement, i.e., the flux data at several points on the boundary. We prove the unique recovery of the location and…