Narrow Results

Let ϕ(z) = (ϕ₁(z),…,ϕ_n(z)) be a holomorphic self-map of B and ψ(z) a holomorphic function on B, where B is the unit ball of ℂⁿ. Let 0 < p, s < +∞, -n - 1 < q < +∞, q+s > -1 and α ≥ 0, this paper characterizes boundedness and compactness of weighted composition operator W_ψ,ϕ induced by ϕ and ψ between the space F(p, q, s) and α-Bloch space .

This paper proves some results on negative gradient dynamical systems of Morse functions on Hilbert manifolds. It contains the compactness of flow lines, manifold structures of certain compactified moduli spaces, orientation formulas, and CW structures of the underlying manifolds.

Let $\alpha \in (0,1]$, $\beta \in [0,n)$ and $T_{\mathrm{\Omega },\beta }$ be a singular or fractional integral operator with homogeneous kernel $\mathrm{\Omega }$. In this paper, a CMO type space ${\mathrm{\text{CMO}}}_{\alpha }({ℝ}^n)$ is introduced and studied. In particular, the relationship between ${\mathrm{\text{CMO}}}_{\alpha }({ℝ}^n)$ and the Lipchitz space ${\mathrm{\text{Lip}}}_{\alpha }({ℝ}^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator ${(T_{\mathrm{\Omega },\beta })}_b^m$ on weighted Lebesgue spaces via functions in $\mathrm{\text{L}}i{p}_{\alpha }({ℝ}^n)$, and an equivalent characterization of the compactness for ${(T_{\mathrm{\Omega },\beta })}_b^m$ via functions in ${\mathrm{\text{CMO}}}_{\alpha }({ℝ}^n)$ are obtained. Some results are new even in the unweighted setting for the first-order commutators.

In this paper, a well-behaved new model of anisotropic compact star in (3+1)-dimensional spacetime has been investigated in the background of Einstein’s general theory of relativity. The model has been developed by choosing $g_{rr}$ component as Krori–Barua (KB) ansatz [Krori and Barua in J. Phys. A, Math. Gen. 8 (1975) 508]. The field equations have been solved by a proper choice of the anisotropy factor which is physically reasonable and well behaved inside the stellar interior. Interior spacetime has been matched smoothly to the exterior Schwarzschild vacuum solution and it has also been depicted graphically. Model is free from all types of singularities and is in static equilibrium under different forces acting on the system. The stability of the model has been tested with the help of various conditions available in literature. The solution is compatible with observed masses and radii of a few compact stars like Vela X-1, 4U $1608-52$, PSR J$1614-2230$, LMC X$-4$, EXO $1785-248$.

In this paper, we obtain gradient continuity estimates for viscosity solutions of ${\mathrm{\Delta }}_p^{N}u=f$ in terms of the scaling critical $L(n,1)$-norm of $f$, where ${\mathrm{\Delta }}_p^N$ is the normalized $p$-Laplacian operator. Our main result corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential ${\widetilde{𝕀}}_q^f$. Moreover, for $f\in L^m$ with $m>n$, we also obtain $C^{1,\alpha }$ estimates. This improves one of the regularity results in [A. Attouchi, M. Parviainen and E. Ruosteenoja, $C^{1,\alpha }$ regularity for the normalized $p$-Poisson problem, J. Math. Pures Appl. (9)  108(4) (2017) 553–591], where a $C^{1,\alpha }$ estimate was established depending on the $L^m$-norm of $f$ under the additional restriction that $p>2$ and $m>\mathrm{\text{max}}(2,n,\frac{p}{2})$. We also mention that differently from the approach in the above paper, which uses methods from divergence form theory and nonlinear potential theory, the method in this paper is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations 107(2) (1994) 341–350]. Moreover, for $f$ continuous, our approach also gives a somewhat different proof of the $C^{1,\alpha }$ regularity result.

The inner structure of compact stars is checked from theoretical as well as observational points of view. In this paper, we determine the possible radii of six compact stars: two binary millisecond pulsars, namely PSR J1614-2230 and PSR J1903+327, studied by [P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts and W. T. Hessels, Nature467, 1081 (2010)] and four X-ray binaries, namely Cen X-3, SMC X-1, Vela X-1 and Her X-1 studied by [M. L. Rawls et al., Astrophys. J.730, 25 (2011)]. Interestingly, we see that density of the star does not vanishes at the boundary though it is maximum at the center which implies that these compact stars may be treated as strange stars rather than neutron stars. We propose a stiff equation of state (EoS) relating to pressure with matter density. We also obtain compactness (u) and surface redshift (Z${}_s$) for the above-mentioned stars and compare it with the recent observational data.

This paper is devoted to study static spherically symmetric model in the presence of charged perfect fluid. This is the generalization of neutral perfect fluid (when there is no charge) through the solution of Einstein Maxwell equations. For this purpose, we consider a suitable form of gravitational potential $g_{tt}$ and the electric field $E(r)$, already used in the literature. The value of mass-radius ratio or compactness $u=\frac{M}{R}$, which depends upon the chosen model exceeds the value $\frac{4}{9}$ corresponding to neutral stars. The most important feature of the current study is to use the Bardeen model geometry instead of usual Reissner–Nordström model for the matching conditions. In this case the energy density and pressure remain positive, bounded and monotonically decreasing whereas electric field is monotonically increasing. Also the causality condition, i.e. the magnitude of speed of sound must be less than the speed of light, is satisfied. Moreover, the behavior of all the physical parameters at the center and on surface of star of mass $M\sim M_{\odot }$ and for Her X-1 are tabulated. All the results by graphical analysis and tabular information suggest that Bardeen model provides physically realistic stellar structures.

We have demonstrated here that protein compactness, which we define as the ratio of the accessible surface area of a protein to that of the ideal sphere of the same volume, is one of the factors determining the mechanism of protein folding. Proteins with multi-state kinetics, on average, are more compact (compactness is 1.49 ± 0.02 for proteins within the size range of 101–151 amino acid residues) than proteins with two-state kinetics (compactness is 1.59 ± 0.03 for proteins within the same size range of 101–151 amino acid residues). We have shown that compactness for homologous proteins can explain both the difference in folding rates and the difference in folding mechanisms.

In this paper, the authors study the compactness for higher order commutators of oscillatory singular integral operators with rough kernels satisfying L^q-Dini conditions.

In this paper, we show the version of Pego's theorem on locally compact abelian groups. This theorem, [R. L. Pego, Compactness in L² and the Fourier transform, Proc. Amer. Math. Soc.95 (1985) 252–254], gives a characterization of precompact sets of L² in terms of the Fourier transform.

In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of $L^2$ in terms of the Fourier transform.

Many classical results about compactness in functional analysis can be derived from suitable inequalities involving distances to spaces of continuous or Baire one functions: this approach gives an extra insight to the classical results as well as triggers a number of open questions in different exciting research branches. We exhibit here, for instance, quantitative versions of Grothendieck's characterization of weak compactness in spaces C(K) and also of the Eberlein–Šmulyan and Krein–Šmulyan theorems. The above results specialized in Banach spaces lead to several equivalent measures of non-weak compactness. In a different direction we envisage a method to measure the distance from a function f ∈ ℝ^X to B₁(X) — space of Baire one functions on X — which allows us to obtain, when X is Polish, a quantitative version of the well-known Rosenthal's result stating that in B₁(X) the pointwise relatively countably compact sets are pointwise compact. Other results and applications are commented too.

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 4, and h be a Holdër continuous function on M. We prove multiplicity of changing sign solutions for equations like Δ_g u + hu = |u|^{2* - 2} u, where Δ_g is the Laplace–Beltrami operator and 2* = 2n/(n - 2) is critical from the Sobolev viewpoint.

These notes are concerned with the $L^2$-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed Cauchy–Riemann (CR)-submanifolds of ${ℂ}^n$ of hypersurface type. This class of submanifolds generalizes that of boundaries of pseudoconvex domains. We first discuss briefly the CR-geometry of general CR-submanifolds and then specialize to this class. Next, we review the basic $L^2$-theory of the tangential CR operator and the associated complex Green operator(s) on these submanifolds. After these preparations, we discuss recent results on compactness and regularity in Sobolev spaces of the complex Green operator(s).

In this paper, we study the dynamical behavior of nonautonomous, almost-periodic discrete FitzHugh–Nagumo system defined on infinite lattices. We prove that the nonautonomous infinite-dimensional system has a uniform attractor which attracts all solutions uniformly with respect to the translations of external terms. We also establish the upper semicontinuity of uniform attractors when the infinite-dimensional system is approached by a family of finite-dimensional systems. This paper is based on a uniform tail method, which shows that, for large time, the tails of solutions are uniformly small with respect to bounded initial data as well as the translations of external terms. The uniform tail estimates play a crucial role for proving the uniform asymptotic compactness of the system and the upper semicontinuity of attractors.

We study the Dirichlet boundary value problem on a bounded domain Ω ⊂ ℝ^N. For 2 ≤ N ≤ 7, we characterize compactness for solutions sequence in terms of spectral informations. As a by-product, we give an uniqueness result for λ close to 0 and λ* in the class of all solutions with finite Morse index, λ* being the extremal value associated to the nonlinear eigenvalue problem.

In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in [B. Kirchheim, S. Müller and V. Šverák, Studying Nonlinear PDE by Geometry in Matrix Space (Springer, 2003), Sec. 7], and many of its properties have already been shown in [A. Lorent and G. Peng, Null Lagrangian measures in subspaces, compensated compactness and conservation laws, Arch. Ration. Mech. Anal. 234(2) (2019) 857–910; A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156]. In particular, in [A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156], it is shown that the differential inclusion does not contain any $T_4$ configurations. Here, we continue that study by showing that the differential inclusion does not contain $T_5$ configurations.

In this paper, we examine properties of the division topology on the set of positive integers introduced by Rizza in 1993. The division topology on $\mathbb{N}$ with the division order is an example of $T_0$-Alexandroff topology. We mainly concentrate on closures of arithmetic progressions and connected and compact sets. Moreover, we show that in the division topology on $\mathbb{N}$, the continuity is equivalent to the Darboux property.

In this survey, we present several results on the regularizing effect, rigidity and approximation of 2D unit-length divergence-free vector fields. We develop the concept of entropy (coming from scalar conservation laws) in order to analyze singularities of such vector fields. In particular, based on entropies, we characterize lower semicontinuous line-energies in 2D and we study by Γ-convergence method the associated regularizing models (like the 2D Aviles–Giga and the 3D Bloch wall models). We also present some applications to the analysis of pattern formation in micromagnetics. In particular, we describe domain walls in the thin ferromagnetic films (e.g. symmetric Néel walls, asymmetric Néel walls, asymmetric Bloch walls) together with interior and boundary vortices.

We propose a toy model for self-organized road traffic taking into account the state of orderliness in drivers’ behavior. The model is reminiscent of the wide family of generalized second-order models (GSOM) of road traffic. It can also be seen as a phase-transition model. The orderliness marker is evolved along vehicles’ trajectories and it influences the fundamental diagram of the traffic flow. The coupling we have in mind is non-local, leading to a kind of “weak decoupling” of the resulting $2\times 2$ system; this makes the mathematical analysis similar to the analysis of the classical Keyfitz–Kranzer system. Taking advantage of the theory of weak and renormalized solutions of one-dimensional transport equations [Panov, 2008], which we further develop on this occasion in the Appendix, we prove the existence of admissible solutions defined via a mixture of the Kruzhkov and the Panov approaches; note that this approach to admissibility does not rely upon the classical hyperbolic structure for $2\times 2$ systems. First, approximate solutions are obtained via a splitting strategy; compactification effects proper to the notion of solution we rely upon are carefully exploited, under general assumptions on the data. Second, we also address fully discrete approximation of the system, constructing a BV-stable Finite Volume numerical scheme and proving its convergence under the no-vacuum assumption and for data of bounded variation. As a byproduct of our approach, an original treatment of local GSOM-like models in the BV setting is briefly discussed, in relation to discontinuous-flux LWR models.