Analysis and Mathematical Physics最新文献

This work addresses the resolvent convergence of generalized MIT bag operators to Dirac operators with zigzag type boundary conditions. We prove that the convergence holds in strong but not in norm resolvent sense. Moreover, we show that the only obstruction for having norm resolvent convergence is the existence of an eigenvalue of infinite multiplicity for the limiting operator. More precisely, we prove the convergence of the resolvents in operator norm once projected into the orthogonal of the corresponding eigenspace.

We are interested in four-dimensional Dirac–Klein–Gordon equations, a fundamental model in particle physics. The main goal of this paper is to establish global existence of solutions to the coupled system and to explore their long-time behavior. The results are valid uniformly for mass parameters varying in the interval [0, 1].

We exhaustively classify the Lie reductions of the real dispersionless Nizhnik equation to partial differential equations in two independent variables and to ordinary differential equations. Lie and point symmetries of reduced equations are comprehensively studied, including the analysis of which of them correspond to hidden symmetries of the original equation. If necessary, associated Lie reductions of a nonlinear Lax representation of the dispersionless Nizhnik equation are carried out as well. As a result, we construct wide families of new invariant solutions of this equation in explicit form in terms of elementary, Lambert and hypergeometric functions as well as in parametric or implicit form. We show that Lie reductions to algebraic equations lead to no new solutions of this equation in addition to the constructed ones. Multiplicative separation of variables is used for illustrative construction of non-invariant solutions.

We obtain a Lie theoretic intrinsic characterization of the connected and simply connected solvable Lie groups whose regular representation is a factor representation. When this is the case, the corresponding von Neumann algebras are isomorphic to the hyperfinite (textrm{II}_infty ) factor, and every Casimir function is constant. We thus obtain a family of geometric models for the standard representation of that factor. Finally, we show that the regular representation of any connected and simply connected solvable Lie group with open coadjoint orbits is always of type (textrm{I}), though the group needs not be of type (textrm{I}), and include some relevant examples.

This paper is concerned with a class of periodic Schrödinger lattice systems with spectrum 0 and saturable nonlinearities. The existence of ground state solitons of the systems under weak assumptions is obtained. The main novelties are as follows. (1) Some new sufficient conditions for the existence of ground state solitons under the “spectral endpoint” assumption are constructed. (2) Our “non-monotonic” conditions make the proofs of the boundedness of the (PS) sequences to be easier. (3) Our result extends and improves the related results in the literature. Besides, some examples are given to illuminate our result.

Let (1<qle p le rle infty ) and (tau in (0,infty ]). Besov–Bourgain–Morrey spaces ({mathcal {M}}dot{B}^{p,tau }_{q,r}({mathbb {R}}^n)) in the special case where (tau =r), extending what was introduced by J. Bourgain, have proved useful in the study related to the Strichartz estimate and the non-linear Schrödinger equation. In this article, by cleverly mixing the norm structures of grand Lebesgue spaces and Besov–Bourgain–Morrey spaces and adding an extra exponent (theta in [0,infty )), the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces ({mathcal {M}}dot{B}^{p,tau }_{q),r,theta }({mathbb {R}}^n)). The authors explore their various real-variable properties including pre-dual spaces and the Gagliardo–Peetre and the ± interpolation theorems. Via establishing some equivalent quasi-norms of ({mathcal {M}}dot{B}^{p,tau }_{q),r,theta }({mathbb {R}}^n)) related to Muckenhoupt (A_1({mathbb {R}}^n))-weights, the authors then obtain an extrapolation theorem of ({mathcal {M}}dot{B}^{p,tau }_{q),r,theta }({mathbb {R}}^n)). Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of ({mathbb {R}}^n), the authors establish the sharp boundedness on ({mathcal {M}}dot{B}^{p,tau }_{q),r,theta }({mathbb {R}}^n)) of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator.

This paper explores strongly parabolic-elliptic systems within Orlicz–Sobolev spaces. It introduces the concept of capacity solutions and emphasizes the establishment of existence and regularity of solutions through rigorous proofs. Specifically, it addresses the existence of capacity solutions for a strongly nonlinear coupled system without reliance on the (Delta _2)-condition for the N-function. This system, akin to a modified thermistor problem, concerns the determination of variables representing the temperature within a conductor and the associated electrical potential.

In the setting of Carnot groups, we propose an approach of taming singularities to get coercive inequalities. To this end, we develop a technique to introduce natural singularities in the energy function U in order to force one of the coercivity conditions. In particular, we explore explicit constructions of probability measures on Carnot groups which secure Poincaré and even Logarithmic Sobolev inequalities. As applications, we get analogues of the Dyson–Ornstein–Uhlenbeck model on the Heisenberg group and obtain results on the discreteness of the spectrum of related Markov generators.

This paper mainly deals with the Sturm–Liouville operator

acting in (L_{w}^{2}left( Gamma right) ,) where (Gamma ) is a metric graph. We establish a relationship between the bottom of the spectrum and the positive solutions of quantum graphs, which is a generalization of the classical Allegretto–Piepenbrink theorem. Moreover, we prove the Persson-type theorem, which characterizes the infimum of the essential spectrum.