Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2207.04051

Nonlocal Harnack inequalities in the Heisenberg group

Giampiero Palatucci·Mirco Piccinini
View original source ↗Read source document ↗Export metadata ↓

Abstract

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $\mathbb{H}^n$, whose prototype is the Dirichlet problem for the $p$-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is $p=2$, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent $s$ goes to $1$.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Giampiero Palatucci, Mirco Piccinini. 2023-01-11. Nonlocal Harnack inequalities in the Heisenberg group. https://arxiv.org/abs/2207.04051

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Two-layers neural networks for Schr{ö}dinger eigenvalue problems

The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of high-dimensional Schr{ö}dinger eigenvalue problems with smooth interaction potentials and Neumann boundary condition on the unit cube in any dimension. More precisely, any eigenfunction associated to the lowest eigenvalue of the Schr{ö}dinger operator is a unit L 2 norm minimizer of the associated energy. Using Barron's representation of the solution with a probability measure defined on the set of parameter values and following the approach initially suggested by Bach and Chizat [1], the energy is minimized thanks to a constrained gradient curve dynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. We prove the existence of solutions to this constrained gradient curve. Furthermore, we prove that, if it converges, the represented function is then an eigenfunction of the considered Schr{ö}dinger operator. At least up to our knowledge, this is the first work where this type of analysis is carried out to deal with the minimization of non-convex functionals.

math.AP↗
arXiv2026

Validity of Prandtl Expansion for Steady Compressible Navier-Stokes-Fourier Flows

Assume no-slip boundary conditions for the velocity field and either insulated or Dirichlet boundary conditions for the temperature field in a steady compressible fluid. In the inviscid limit $\v \rightarrow 0$, we develop a mathematical framework for the uniform-in-$\v$ remainder estimate for the linear steady compressible Navier-Stokes-Fourier equations around a Prandtl layer profile with both velocity and thermal layers, which leads to the validity of the Prandtl layer expansion.

math.AP↗
arXiv2026

Long time behaviour of Mean Field Games with fractional diffusion

In this paper we study the long time behaviour of mean field games systems with fractional diffusion, modeling the case that the individual dynamics of the players is driven by independent jump processes and controlled through the drift term, while being confined by an external field in order to guarantee ergodicity. In the case of globally Lipschitz, locally uniformly convex Hamiltonian, and weakly coupled costs satisfying the Lasry-Lions monotonicity condition, we prove that there is a unique solution $(u_T,m_T)$ to the mean field game problem in $(0,T)$ and we show that, if $T$ is sufficiently large, $(u_T,m_T)$ satisfies the so-called turnpike property, namely it is exponentially close to the (unique) stationary ergodic state for any proportionally long intermediate time.

math.AP↗