Analysis and Applications

In this paper, we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.

Using deep neural networks to solve partial differential equations (PDEs) has attracted a lot of attention recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on the deep Ritz method (DRM) for second-order elliptic equations with Dirichlet, Neumann and Robin boundary conditions, respectively. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep neural networks with smooth activation functions including logistic and hyperbolic tangent functions. Our results show how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of the number of training samples.

In this paper, we consider a nonlinear Schrödinger system with fractional derivative and quadratic interaction. Using constrained variational methods, we prove that this problem has a prescribed solution. Our contribution is that we can deal with a case in the supercritical range.

This paper is concerned with the well-posedness and optimal control problem of a reaction–diffusion system for an epidemic susceptible–exposed–infected–recovered–susceptible mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates $u_e$ and $u_i$ of contagion resulting from the contact with both asymptomatic and symptomatic persons, respectively, we optimize the number of exposed and infected individuals at a final time $T$ of the controlled evolution of the system. More precisely, we search for the optimal $u_i$ and $u_e$ such that the number of infected plus exposed does not exceed at the final time a threshold value $\mathrm{\Lambda }$, fixed a priori. We prove here the existence of optimal controls in a proper functional framework and we derive the first-order necessary optimality conditions in terms of the adjoint variables.

In this paper, we focus on the Zakharov–Kuznetsov (ZK) equation in the $n$-dimensional setting with $n\ge 2$ and investigate its smoothness properties. We extend the well-known regularity propagation phenomenon observed in the 2D and 3D cases, where the regularity of the initial data on certain half-spaces propagates with infinite speed, to the case where the regularity of the initial data is measured on a fractional scale. To achieve this, we introduce new localization formulas that enable us to describe the regularity of the solution on a specific class of subsets in Euclidean space. This work provides insights into the regularity behavior of solutions of the ZK equation in higher dimensions and with more general initial data.

The paper is concerned with the change of probability measures $\mathbf{\mu }$ along non-random probability measure-valued trajectories ${\nu }_t$, $t\in [-1,1]$. Typically solutions to non-linear partial differential equations (PDEs), modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map $\nu \equiv {\nu }_0\mapsto {\nu }_t$ does not exit the state space, for $t\in [-1,0]$ or for $t\in [0,1]$, the Radon–Nikodym derivative $d\mathbf{\mu }\circ {\nu }_t/d\mathbf{\mu }$ is determined. It is also investigated how Fréchet differentiability of the solution map of the PDE can contribute to the existence of this Radon–Nikodym derivative. The first application is a certain Boltzmann type equation. Here, the Fréchet derivative of the solution map is calculated explicitly and quasi-invariance is established. The second application is a PDE related to the asymptotic behavior of a Fleming–Viot type particle system. Here, it is demonstrated how quasi-invariance can be used in order to derive a corresponding integration by parts formula.