Narrow Results

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.

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 .

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.

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).

We discuss compactness of the $\overline{\partial }$-Neumann operator in the setting of weighted $L^2$-spaces on ${ℂ}^n.$ In addition we describe an approach to obtain the compactness estimates for the $\overline{\partial }$-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich–Kondrachov lemma.

In this paper, we obtain compactness theorems for Yang–Mills–Higgs fields on vector bundle $E$ over compact Riemannian manifold $M$, $\mathrm{\text{dim}}M>4$, with general Higgs-like potential $W:ℝ\to [0,\infty )$.

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.

In this paper, we study the inner structure of some neutron stars from theoretical as well as observational points of view. We calculate the probable radii, compactness (u) and surface redshift (Z${}_s$) of five neutron stars (X-ray binaries) namely 4U 1538-52, LMC X-4, 4U 1820-30, 4U 1608-52, EXO 1745-248. Here, we propose a stiff equation of state (EoS) of matter distribution which relates pressure with matter density. Finally, we check the stability of such kind of theoretical structure.

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.

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 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.

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of “global Jacobian” detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a $\mathrm{\Gamma }$-convergence expansion at the second order. The second-order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity $1$. Compactness results are also shown for the magnetization and the corresponding global Jacobian.

In this paper (strong) continuity of domain-free information algebras and labeled information algebras are introduced. Relationships between domain-free information algebras and labeled information algebras are mainly explored. It is shown that continuity and compactness can be preserved under certain canonical correspondences between domain-free information algebras and labeled information algebras. Some equivalent characterizations and examples for continuous information algebras are also given.

It is believed that the family of Riemannian manifolds with negative curvatures is much richer than that with positive curvatures. In fact there are many results on the obstruction of furnishing a manifold with a Riemannian metric whose curvature is positive. In particular any manifold admitting a Riemannian metric whose Ricci curvature is bounded below by a positive constant must be compact. Here we investigate such obstructions in terms of certain functional inequalities which can be considered as generalized Poincaré or log-Sobolev inequalities. A result of Saloff-Coste is extended.

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for fourth order critical equations like P_gu = u^2♯-1, where is a Paneitz–Branson operator with constant coefficients b and c, u is required to be positive, and is critical from the Sobolev viewpoint. We prove that such equations are compact on locally conformally flat manifolds, unless b lies in some closed interval associated to the spectrum of the smooth symmetric (2,0)-tensor field involved in the definition of the geometric Paneitz–Branson operator.

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 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.

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is $n\ge 7$ and the trace-free part of the second fundamental form is nonzero everywhere on the boundary.

For a conformally compact Poincaré–Einstein manifold $(X,g_{+})$, we consider two types of compactifications for it. One is $ḡ={\rho }^2{g}_{+}$, where $\rho$ is a fixed smooth defining function; the other is the adapted (including Fefferman–Graham) compactification ${ḡ}_s={\rho }_s^2{g}_{+}$ with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove that for a set of conformally compact Poincaré–Einstein manifolds $\{(X,g_{+}^{(i)})\}$ with conformal infinity of positive Yamabe type, $\{{ḡ}^{(i)}\}$ is compact in $C^{k,\alpha }(\overline{X})$ topology if and only if $\{{ḡ}_s^{(i)}\}$ is compact in some $C^{l,\beta }(\overline{X})$ topology, provided that ${ḡ}^{(i)}{|}_{TM}={ḡ}_s^{(i)}{|}_{TM}={ĝ}^{(i)}$ and ${ĝ}^{(i)}$ has positive scalar curvature for each $i$. See Theorem 1.1 and Corollary 1.1 for the exact relation of $(k,\alpha )$ and $(l,\beta )$.

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.