Analysis and Mathematical Physics最新文献

We provide conditions under which we prove for measure-valued transport equations with non-linear reaction term in the space of finite signed Radon measures, that positivity is preserved, as well as absolute continuity with respect to Lebesgue measure, if the initial condition has that property. Moreover, if the initial condition has (L^p) regular density, then the solution has the same property.

In this paper, we investigate the multiplicity of solutions for the following nonlinear fractional Schrödinger-Poisson system of Kirchhoff type:

where (s, tin (0,1)), (Omega subset mathbb {R}^N) is a smooth bounded domain containing 0 with Lipschitz boundary, (left( -Delta right) ^{gamma }) ((gamma =s,t)) is the fractional Laplace operator, (lambda ) is a positive parameter, (0le alpha<2s<N), (2<r<2theta<4<q<2_{alpha }^{*}) and (f(x)in L^{frac{2_alpha ^*}{2_alpha ^*-r}}(Omega )) is positive almost everywhere in ({Omega }). By using variational methods, we get over some tricky difficulties stemming from degenerate feature of Kirchhoff term. As a result, by employing the Nehari manifold method, under some certain conditions, we prove that the above system has at least two distinct positive solutions for (lambda ) small.

Recently,Ikoma [8] considered optimal constants and extremisers for the 2-dimensional Dirac equation using the spherical harmonics decomposition. Though its argument is valid in any dimensions (d ge 2), the case (d ge 3) remains open since it leads us to too complicated calculation: determining all eigenvalues and eigenvectors of infinite dimensional matrices. In this paper, we give optimal constants and extremisers of smoothing estimates for the 3-dimensional Dirac equation. In order to prove this, we construct a certain orthonormal basis of spherical harmonics. With respect to this basis, infinite dimensional matrices actually become block diagonal and so that eigenvalues and eigenvectors can be easily found. As applications, we obtain the equivalence of the smoothing estimate for the Schrödinger equation and the Dirac equation, and improve a result by Ben-Artzi and Umeda [3].

The principal result in this note is a strengthened version of Kadison’s transitivity theorem for unital JB(^*)-algebras, showing that for each minimal tripotent e in the bidual, ({mathfrak {A}}^{**}), of a unital JB(^*)-algebra ({mathfrak {A}}), there exists a self-adjoint element h in ({mathfrak {A}}) satisfying (ele exp (ih)), that is, e is bounded by a unitary in the principal connected component of the unitary elements in ({mathfrak {A}}). This new result opens the way to attack new geometric results, for example, a Russo–Dye type theorem for maximal norm closed proper faces of the closed unit ball of ({mathfrak {A}}) asserting that each such face F of ({mathfrak {A}}) coincides with the norm closed convex hull of the unitaries of ({mathfrak {A}}) which lie in F. Another geometric property derived from our results proves that every surjective isometry from the unit sphere of a unital JB(^*)-algebra ({mathfrak {A}}) onto the unit sphere of any other Banach space is affine on every maximal proper face. As a final application we show that every unital JB(^*)-algebra ({mathfrak {A}}) satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of ({mathfrak {A}}) onto the unit sphere of any other Banach space Y admits an extension to a surjective real linear isometry from ({mathfrak {A}}) onto Y. This extends a contribution by M. Mori and N. Ozawa who have proved the same result for unital C(^*)-algebras.

Let (mathcal {F}) be a family of meromorphic functions in a domain D, and (mathcal {F}_k) be a family of kth derivative functions of all (fin mathcal {F}). In this paper, we study normality relationships between (mathcal {F}) and (mathcal {F}_k), and obtain some normality criteria. Some applications of our results are given.

We study the ground-state properties of one-dimensional fluids of classical (i.e., non-quantum) particles interacting pairwisely via a potential, at the fixed particle density (rho ). Restricting ourselves to periodic configurations of particles, two possibilities are considered: an equidistant chain of particles with the uniform spacing (A=1/rho ) and its simplest non-Bravais modulation, namely a bipartite lattice composed of two equidistant chains, shifted with respect to one another. Assuming the long range of the interaction potential, the equidistant chain dominates if A is small enough, (0<A<A_c). At a critical value of (A=A_c), the system undergoes a continuous second-order phase transition from the equidistant chain to a bipartite lattice. The energy and the order parameter are singular functions of the deviation from the critical point (A-A_c) with universal (i.e., independent of the model’s parameters) mean-field values of critical exponents. The tricritical point at which the curve of continuous second-order transitions meets with the one of discontinuous first-order transitions is determined. The general theory is applied to the Lennard-Jones model with the (n, m) Mie potential for which the phase diagram is constructed. The inclusion of a hard-core around each particle reveals a non-universal critical phenomenon with an m-dependent critical exponent.

In this paper necessary and sufficient conditions on a measure (mu ) guaranteeing the boundedness of the multilinear fractional integral operator (T_{gamma , mu }^{(m)}) (defined with respect to a measure (mu )) from the product of Lorentz spaces (prod _{k=1}^m L^{r_k, s_k}_{mu }) to the Lorentz space (L^{p,q}_{mu }(X)) are established. The results are new even for linear fractional integrals (T_{gamma , mu }) (i.e., for (m=1)). From the general results we have a criterion for the validity of Sobolev–type inequality in Lorentz spaces defined for non-doubling measures. Finally, we investigate the same problem for Morrey-Lorentz spaces. To prove the main result we use the boundedness of the multilinear modifies maximal operator (widetilde{mathcal {M}}).

We study homogenization problem for non-autonomous parabolic equations of the form (partial _t u=L(t)u) with an integral convolution type operator L(t) that has a non-symmetric jump kernel which is periodic in spatial variables and in time. It is assumed that the space-time scaling of the environment is not diffusive. We show that asymptotically the spatial and temporal evolutions of the solutions are getting decoupled, and the homogenization result holds in a moving frame.

We study the spectral properties of a Schrödinger operator, in presence of a confining potential given by the distance squared from a fixed compact potential well. We prove continuity estimates on both the eigenvalues and the eigenstates, lower bounds on the ground state energy, regularity and integrability properties of eigenstates. We also get explicit decay estimates at infinity, by means of elementary nonlinear methods.

In this work, we will study the existence, uniqueness and exponential stability of almost periodic solutions to the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We clarify the existence and uniqueness of such solutions of the linear equation by utilizing the dispersive and smoothing estimates of the heat semigroup. Thereafter, we use the fixed point arguments to investigate for the case of the semi-linear equation by utilizing the results of the linear case. Finally, we invoke the Gronwall’s inequality to point out the exponential stability.