Communications in Contemporary Mathematics

We investigate non-existence of non-negative dead core solutions for the problem

We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation

This paper deals with the second-order nonlinear differential equation $(a(t)|x^{\prime }{|}^{p(t)-2}{x}^{\prime }{)}^{\prime }=b(t)|x{|}^{q(t)-2}x$ involving $p(t)$-Laplacian. The existence and the uniqueness of nonoscillatory solutions of this equation in certain classes, which are related with integral conditions, are studied. Moreover, a minimal set for solutions of this equation is introduced as an extension of the concept of principal solutions for linear equations. Obtained results extend the results for equations with $p$-Laplacian.

We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as

This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings.

The aim of this note is to explain the integrability of an integrable Boltzmann billiard model, previously established by Gallavotti and Jauslin [G. Gallavotti and I. Jauslin, A theorem on Ellipses, an integrable system and a theorem of Boltzmann, preprint (2020); arXiv:2008.01955], alternatively via the viewpoint of projective dynamics. We show that the energy of a corresponding spherical problem leads to an additional first integral of the system equivalent to Gallavotti–Jauslin’s first integral. The approach also leads to a family of integrable billiard models in the plane and on the sphere defined through the planar and spherical Kepler–Coulomb problems.

A topoplane is a mild deformation of a linear hyperplane contained in a given smooth manifold that is homeomorphic to a Euclidean space. We consider solidly transsective topoplane arrangements. These collections generalize pseudohyperplane arrangements. Even though the topoplane arrangements locally look like hyperplane arrangements, the global coning procedure is absent here. The main aim of the paper is to introduce the Varchenko matrix in this context and show that the determinant has a similar factorization as in the case of hyperplane arrangements. We achieve this by suitably generalizing the strategy of Aguiar and Mahajan. We also study a system of linear equations introduced by them and describe its solution space in the context of topoplane arrangements.

Given a smooth bounded domain ${𝒟}$ in ${ℝ}^N$ with $N\ge 3$, we study the existence and the profile of positive solutions for the following elliptic Neumann problem:

We are concerned with the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita–Kato’s theorem [On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315] for the Navier–Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita–Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2$\frac{1}{2}$D flows for the Hall-MHD system (that is, 3D flows independent of the vertical variable), and establish the global existence of strong solutions, assuming only that the initial magnetic field is small. Our proofs strongly rely on the use of an extended formulation involving the so-called velocity of electron $v=u-𝜀\nabla \times B$ and as regards $2\frac{1}{2}$D flows, of the auxiliary vector-field $\mathrm{\Omega }=𝜀\omega +B$ that comes into play in the total magneto-helicity balance for the Hall-MHD system.

We explicitly express the spectral determinant of Friedrichs Dirichlet Laplacians on the 2-dimensional hyperbolic (Gaussian curvature -1) cones in terms of the cone angle and the geodesic radius of the boundary.