Analysis and Mathematical Physics最新文献

In this paper, we are concerned with the existence of infinitely many nontrivial solutions to the following semilinear degenerate elliptic equation

where (V: {mathbb {R}}^Nrightarrow {mathbb {R}}) is a potential function and allowed to be vanishing at infinitely, (f: {mathbb {R}}^Ntimes {mathbb {R}}rightarrow {mathbb {R}}) is a given function and (Delta _lambda ) is the strongly degenerate elliptic operator. Under suitable assumptions on the potential V and the nonlinearity f,  some results on the multiplicity of solutions are proved. The proofs are based on variational methods, in particular, on the well-known mountain pass lemma of Ambrosetti–Rabinowitz. Due to the vanishing potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results (Alves and Souto in J Differ Equ 254:1977–1991, 2013; Hamdani in Asia-Eur J Math 13:2050131, https://doi.org/10.1142/S1793557120501314, 2020; Luyen in Commun Math Anal 22:61–75, 2019; Luyen and Tri in J Math Anal Appl 461:1271–1286, 2018; Tang in J Math Anal Appl 401:407–415, 2013; Toon and Ubilla in Discrete Contin Dyn Syst 40:5831–5843, 2020).

We prove sharp upper bounds for the first and second non-trivial eigenvalues of the Neumann Laplacian in two classes of domains: parallelograms and domains of constant width. This gives in particular a new proof of an isoperimetric inequality for parallelograms recently obtained by A. Henrot, A. Lemenant and I. Lucardesi.

In this paper we start the study of Schur analysis for Cauchy–Fueter regular quaternionic-valued functions, i.e. null solutions of the Cauchy–Fueter operator in ({mathbb {R}}^4). The novelty of the approach developed in this paper is that we consider axially regular functions, i.e. functions spanned by the so-called Clifford-Appell polynomials. This type of functions arises naturally from two well-known extension results in hypercomplex analysis: the Fueter mapping theorem and the generalized Cauchy–Kovalevskaya (GCK) extension. These results allow one to obtain axially regular functions starting from analytic functions of one real or complex variable. Precisely, in the Fueter theorem two operators play a role. The first one is the so-called slice operator, which extends holomorphic functions of one complex variable to slice hyperholomorphic functions of a quaternionic variable. The second operator is the Laplace operator in four real variables, that maps slice hyperholomorphic functions to axially regular functions. On the other hand, the generalized CK-extension gives a characterization of axially regular functions in terms of their restriction to the real line. In this paper we use these two extensions to define two notions of rational function in the regular setting. For our purposes, the notion coming from the generalized CK-extension is the most suitable. Our results allow to consider the Hardy space, Schur multipliers and their relation with realizations in the framework of Clifford-Appell polynomials. We also introduce two notions of regular Blaschke factors, through the Fueter theorem and the generalized CK-extension.

We give a probabilistic representation of the solution to a semilinear elliptic Dirichlet problem with general (discontinuous) boundary data. The boundary behaviour of the solution is in the sense of the controlled convergence initiated by A. Cornea. Uniqueness results for the solution are also provided.

We present a new completely elementary model that describes the creation, annihilation, and motion of non-interacting electrons and positrons along a line. It is a modification of the model known under the names Feynman checkers or one-dimensional quantum walk. It can be viewed as a six-vertex model with certain complex weights of the vertices. The discrete model is consistent with the continuum quantum field theory, namely, reproduces the known expected charge density as the lattice step tends to zero. It is exactly solvable in terms of hypergeometric functions. We introduce interaction resembling Fermi’s theory and establish perturbation expansion.

In this paper, we prove bilinear sparse domination bounds for a wide class of Fourier integral operators of general rank, as well as oscillatory integral operators associated to Hörmander symbol classes (S^m_{rho ,delta }) for all (0le rho le 1) and (0le delta < 1), a notable example is the Schrödinger operator. As a consequence, one obtains weak (1, 1) estimates, vector-valued estimates, and a wide range of weighted norm inequalities for these classes of operators.

In this paper we give a complete description of surjective linear isometries between Banach spaces of absolutely continuous functions on arbitrary (not necessarily compact) subsets of the real line with respect to the sum-norm. We also use this description to study approximate local isometries and approximate 2-local isometries on these spaces. In particular, we present generalizations of all known results concerning such isometries, and obtain the reflexivity and 2-reflexivity of the isometry group of absolutely continuous function spaces in a noncompact framework.

We employ a new function class called B-function to create a new version of fractional Hermite–Hadamard and trapezoid type inequalities on the right-hand side that involves h-convex and (psi )-Hilfer operators. We also provide new midpoint-type inequalities using h-convex functions.

Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space G/H with reductive decomposition (mathfrak {{g}} = mathfrak {{h}} oplus mathfrak {{m}}), we consider rollings of (mathfrak {{m}}) over G/H without slip and without twist, where G/H is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space Q which is tangent to a certain distribution. By considering an H-principal fiber bundle (overline{pi }:overline{Q}rightarrow Q) over the configuration space equipped with a suitable principal connection, rollings of (mathfrak {{m}}) over G/H can be expressed in terms of horizontally lifted curves on (overline{Q}). The total space of (overline{pi }:overline{Q}rightarrow Q) is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of (mathfrak {{m}}) over G/H as solutions of an explicit, time-variant ordinary differential equation (ODE) on (overline{Q}), the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in G/H is the projection of a one-parameter subgroup in G. Lie groups and Stiefel manifolds are discussed as examples.

Wilhelm Weber’s electrodynamics is an action-at-a-distance theory which has the property that equal charges inside a critical radius become attractive. Weber’s electrodynamics inside the critical radius can be interpreted as a classical Hamiltonian system whose kinetic energy is, however, expressed with respect to a Lorentzian metric. In this article we study the Schrödinger equation associated with this Hamiltonian system, and relate it to Weyl’s theory of singular Sturm–Liouville problems.