Compactness Principle Research Articles

Existence of solutions for critical (p,q)-Laplacian equations in ℝN

In this paper, we are mainly interested in existence properties for a class of nonlinear PDEs driven by the ([Formula: see text])-Laplace operator where the reaction combines a power-type nonlinearity at critical level with a subcritical term. In addition, nonnegative nontrivial weights and a positive parameter [Formula: see text] are included in the nonlinearity. An important role in the analysis developed is played by the two potentials. Precisely, under suitable conditions on the exponents of the nonlinearity, first a detailed proof of the tight convergence of a sequence of measures is given, then the existence of a nontrivial weak solution is obtained provided that the parameter [Formula: see text] is far from [Formula: see text]. Our proofs use concentration compactness principles by Lions and Mountain Pass Theorem by Ambrosetti and Rabinowitz.

Compactness Principles for Topological Vector Spaces

Abstract We show that the many of the canonical quantifications of interrelated concepts, centered around compactness in the setting of metric spaces, can be easily generalized to the setting of topological linear spaces. Among other things, we obtain a generalization of the Hausdorff Total Boundedness Principle, of the Grothendieck Compactness Principle, as well as of the Convex Compactness Principle for topological vector spaces.

Positive ground state of coupled planar systems of nonlinear Schrödinger equations with critical exponential growth

In this paper, we prove the existence of a positive ground state solution to the following coupled system involving nonlinear Schrödinger equations: { − Δ u + V 1 ( x ) u = f 1 ( x , u ) + λ ( x ) v , x ∈ R 2 , − Δ v + V 2 ( x ) v = f 2 ( x , v ) + λ ( x ) u , x ∈ R 2 , where λ , V 1 , V 2 ∈ C ( R 2 , ( 0 , + ∞ ) ) and V 1 ( x ) have critical exponential growth in the sense of Trudinger–Moser inequality. The potentials V 1 ( x ) and V 2 ( x ) satisfy a condition involving the coupling term λ ( x ) , namely 0 < λ ( x ) ≤ λ 0 V 1 ( x ) V 2 ( x ) . We use non-Nehari manifold, Lions's concentration compactness and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap regularity lifting argument and L q -estimates we get regularity and asymptotic behavior. Our results improve and extend the previous results.

On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces

<abstract><p>In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.</p></abstract>

A Ball-Marsden-Slemrod type result for semi-linear equations with analytic semigroups

We extend a well-known non-controllability result of Ball, Marsden, and Slemrod on infinite-dimensional bilinear systems (with bounded control term) to control-affine semi-linear systems whose linear part generates an analytic semigroup and whose control term is possibly unbounded as well. Here control inputs are assumed to lie in some suitable Lp-space, p > 1. The result allows an application to PDEs whose control terms include not only the state but also lower oder derivatives compared to the uncontrolled leading linear part. The proof relies on an abstract compactness principle for parameter dependent fixed point maps.

POSITIVE SOLUTIONS FOR A NONLOCAL PROBLEM WITH CRITICAL SOBOLEV EXPONENT IN HIGHER DIMENSIONS

This paper is devoted to a nonlocal problem involving critical Sobolev exponent and negative nonlocal term. By virtue of a cut-off technique and the concentration compactness principle, we prove the existence and asymptotic behavior of positive solutions for the considered problem. In particular, our results generalize the existence results of positive solutions to higher dimensions <i>N</i> ≥ 5.

A simple design of uniform LED illumination using catadioptric collimator and freeform lenslet array

We introduce a compact lenslet array principle that takes advantage of freeform optics to deploy a light distributor, beneficial for highly efficient, inexpensive, low energy consumption light-emitting diode (LED) lighting system. We outline here a simple strategy for designing the freeform lens that makes use of an array of the identical plano-convex lenslet. The light is redistributed from such lenslet, hinging on the principle of optical path length conservation, and then delivered to the receiver plane. The superimposing of such illumination area from every lenslet occurs on the receiver plane, in which the non-uniform illumination area located in the boundary should have the same dimension as the size of the freeform lenslet array. Such an area, insofar, is negligible due to their small size, which is the crux of our design, representing a large departure from the former implementations. Based on simulations that assess light performance, the proposed design exhibited the compatibility for multiple radiation geometries and off-axis lighting without concern for the initial radiation pattern of the source. As simulated, the LED light source integrated with such proposed freeform lenslet array revealed high luminous efficiency and uniformity within the illumination area of interest were above 70% and 85%, respectively. Such novel design was then experimentally demonstrated to possess a uniformity of 75% at hand, which was close to the simulation results. Also, proposed indoor lighting was implemented in comparison with the commercial LED downlight and LED panel, whereby the energy consumption, number of luminaires and illumination performance were assessed to show the advantage of our simplified model.

Nonlinear perturbations of a periodic fractional Laplacian with supercritical growth

Our main goal is to explore the existence of positive solutions for a class of nonlinear fractional Schrödinger equation involving supercritical growth given by $$ (- \Delta)^{\alpha} u + V(x)u=p(u),\quad x\in \mathbb{R^N},\ N \geq 1. $$ We analyze two types of problems, with $V$ being periodic and asymptotically periodic; for this we use a variational method, a truncation argument and a concentration compactness principle.

Do population density, socio-economic ranking and Gini Index of cities influence infection rates from coronavirus? Israel as a case study.

A prominent characteristic of the COVID-19 pandemic is the marked geographic variation in COVID-19 prevalence. The objective of the current study is to assess the influence of population density and socio-economic measures (socio-economic ranking and the Gini Index) across cities on coronavirus infection rates. Israel provides an interesting case study based on the highly non-uniform distribution of urban populations, the existence of one of the most densely populated cities in the world and diversified populations. Moreover, COVID19 challenges the consensus regarding compact planning design. Consequently, it is important to analyze the relationship between COVID19 spread and population density. The outcomes of our study show that ceteris paribus projected probabilities to be infected from coronavirus rise with population density from 1.6 to 2.72% up to a maximum of 5.17–5.238% for a population density of 20,282–20,542 persons per square kilometer (sq. km.). Above this benchmark, the anticipated infection rate drops up to 4.06–4.50%. Projected infection rates of 4.06–4.50% are equal in cities, towns and regional councils (Local Authorities) with the maximal population density of 26,510 and 11,979–13,343 persons per sq. km. A possible interpretation is that while denser cities facilitate human interactions, they also enable and promote improved health infrastructure. This, in turn, contributes to medical literacy, namely, elevated awareness to the benefits associated with compliance with hygienic practices (washing hands), social distancing rules and wearing masks. Findings may support compact planning design principles, namely, development of dense, mixed use, walkable and transit accessible community design in compact and polycentric regions. Indeed, city planners should weigh the costs and benefits of many risk factors, including the COVID19 pandemic.

Existence and concentration of positive solutions for a critical p&amp;q equation

Abstract We show existence and concentration results for a class of p&amp;q critical problems given by − d i v a ϵ p | ∇ u | p ϵ p | ∇ u | p − 2 ∇ u + V ( z ) b | u | p | u | p − 2 u = f ( u ) + | u | q ⋆ − 2 u in R N , $$-div\left(a\left(\epsilon^{p}|\nabla u|^{p}\right) \epsilon^{p}|\nabla u|^{p-2} \nabla u\right)+V(z) b\left(|u|^{p}\right)|u|^{p-2} u=f(u)+|u|^{q^{\star}-2} u\, \text{in} \,\mathbb{R}^{N},$$ where u ∈ W 1,p (ℝ N ) ∩ W 1,q (ℝ N ), ϵ &gt; 0 is a small parameter, 1 &lt; p ≤ q &lt; N, N ≥ 2 and q * = Nq/(N − q). The potential V is positive and f is a superlinear function of C 1 class. We use Mountain Pass Theorem and the penalization arguments introduced by Del Pino &amp; Felmer’s associated to Lions’ Concentration and Compactness Principle in order to overcome the lack of compactness.

Existence for fractional [formula omitted] systems with critical and Hardy terms in [formula omitted

This paper deals with the existence of entire nontrivial solutions for fractional (p,q) systems with critical Hardy terms in RN. Existence of solutions for system (S) is derived via the mountain pass lemma, whose application is pretty involved and requires some variants of the concentration–compactness principle of Lions and of the concentration–compactness principle at infinity of Chabrowski for vectorial fractional spaces, here proved. Indeed, the “triple loss of compactness” in (S), caused by the simultaneous presence of Hardy and critical terms in the whole RN, forces to study the exact behaviour of the PS sequences, in the spirit of Lions. We also present a simplified version (S˜) of the main system (S), which is anyway interesting, since the existence proof for (S˜) is simpler and more direct. Finally, both existence theorems improve or complement previous results for fractional (p,q) scalar and vectorial problems.

Existence and Symmetry of Solutions for a Class of Fractional Schrödinger–Poisson Systems

In this paper, we investigate a class of Schrödinger–Poisson systems with critical growth. By the principle of concentration compactness and variational methods, we prove that the system has radially symmetric solutions, which improve the related results on this topic.

Remarks on blow-up criteria for the derivative nonlinear Schrödinger equation under the optimal threshold setting

We study the Cauchy problem of the mass critical nonlinear Schrödinger equation with derivative with the 4 π mass. One has the global well-posedness in H 1 whenever “the mass is strictly less than 4 π ” or whenever “the mass is equal to 4 π and the momentum is strictly less than zero”. In this paper, by the concentration compact principle as originally done by Kenig-Merle, we obtain the limiting profile of blow up solutions with the critical 4 π mass.

Socio-Spatial Changes in Danwei Neighbourhoods: A Case Study of the AMS Danwei Compound in Hefei, China

This paper analysed socio-spatial changes in old urban neighbourhoods (Danwei compounds) in Chinese cities as a result of two major national level reforms: the Reform and Opening Up and the Urban Housing reform in 1978 and 1997, respectively. Existing research indicates fundamental changes have taken place in the political, economic and social aspects of Danwei compounds. However, there is a paucity of research on micro-level changes. To understand how these reforms have affected the social-spatial schema of Danwei Compounds, the study utilised mapping, key person interviews and field observation in AMS compound, Hefei city of Anhui province. This paper compared the AMS Danwei Compound before and after the reforms in terms of public spaces, building features and compound management. The study found that the AMS Danwei Compound has experienced a significant reduction in public space, an increase in building density and a reconfiguration of compound management actors. The study suggests the need for local planning authorities and government to pay attention to planning and design of the old city core by emphasising improvement in public spaces, attention to compact design principles for urban neighbourhood planning, and establishment of local community management body.

Existence of solutions for a class of critical Kirchhoff type problems involving Caffarelli-Kohn-Nirenberg inequalities

In this paper, we study the existence of a nontrival weak solution for a class of Kirchhoff type problems with singular potentials and critical exponents. The proofs are essentially based on an appropriated truncated argument, Caffarelli-Kohn-Nirenberg inequalities, combined with a variant of the concentration compactness principle. We also get a priori estimates of the obtained solution

A ξ-weak Grothendieck compactness principle

AbstractFor 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

A Remark on the Concentration Compactness Principle in Critical Dimension

AbstractWe prove some refinements of the concentration compactness principle for Sobolev space W1, n on a smooth compact Riemannian manifold of dimension n. As an application, we extend Aubin's theorem for functions on with zero first‐order moments of the area element to the higher‐order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher‐order Sobolev spaces. © 2020 Wiley Periodicals LLC.

More insights into the Trudinger–Moser inequality with monomial weight

In this paper we present a detailed study of critical embeddings of weighted Sobolev spaces into weighted Orlicz spaces of exponential type for weights of monomial type. More precisely, we give an alternative proof of a recent result by N. Lam [NoDEA 24(4), 2017] showing the optimality of the constant in the Trudinger–Moser inequality. We prove a Poincare inequality for this class of weights. We show that the critical embedding is optimal within the class of Orlicz target spaces. Moreover, we prove that it is not compact, and derive a corresponding version of P.-L. Lions’ principle of concentrated compactness.

Nonlinear dispersive equations: classical and new frameworks

The purpose of this manuscript associated to the Golden Jubilee of the IME-USP is to present selected material from the author’s scientific contribution dealing with nonlinear phenomena of dispersive type. That study has been a source for modern research in the dynamic of traveling wave solutions of different type and it has been disseminated through publication of scientific articles and/or books. Here, we provide the reader with information about current research in stability theory for nonlinear dispersive equations and possible developments. Also, I hope it inspires future developments in this important and fascinating subject. In this manuscript we consider the following topics: stability theory of solitary waves and the applicability of the concentration–compactness principle, the existence and orbital (in)stability of periodic traveling wave solutions for nonlinear dispersive models, nonlinear Schrodinger and Korteweg–de Vries models on star-shaped metric graphs. The use of tools of the theory of spaces of Hilbert, the spectral theory for unbounded self-adjoint operators, Sturm–Liouville’s theory, variational methods, analytic perturbation theory of operators, and the extension theory of symmetric operators are pieces fundamental in our study. The methods presented in this manuscript have prospect for the study of the dynamic of solutions for nonlinear evolution equations around of different traveling waves profiles which may appear in non-standard environments such as star-shaped metric graphs.

Proposal for environmental guidelines about a religious foundation university since the global compact principles

The University whose foundational principles are based on religiosity contemplates the preservation of the environment from its theoretical principles, in this sense it was pertinent to evaluate the Institutional principles postulated in relation to the contemporary trends established by means of the encyclical “Laudato si”. Similarly, this article focused on analyzing how these converge with University Social Responsibility. As a result of the above, the preservation and care of the environment was found as a point of agreement, understanding this as the reflection of the manifestation of God. On the other hand, an environmental pre-diagnosis of the University was carried out, evaluating the consumption from three edges, energy consumption, water consumption and the generation of waste according to the main interest groups, finding seasonality with respect to the increase in use of them, that is, consumption is concentrated in the case of water in the intermediate months, while light consumption increases at the end of the academic semesters. Finally, through the environmental matrix of the Global Reporting Initiative (GRI), the environmental aspects of the university were analyzed and measurement indicators were proposed in order to guide the University to the principles proposed by the Global Compact, aiming at reducing environmental impacts, generating awareness towards the preservation of the environment and the implementation of new technologies to reduce environmental impact.