Nonlinear elliptic systems with variable boundary data

Bors, Dorota; Walczak, S.

doi:10.1016/s0362-546x(02)00179-7

Cited by 8 publications

(4 citation statements)

References 4 publications

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“…Moreover, we choose some special family of the solutions characterized by the limit at minus infinity, not the whole set of possible solutions. The continuous dependence on parameters of the whole set of solutions for elliptic equations was established among others in [4,5,6,7]. One should point out that the passage to the singular limit was rigorously verified both for the related Navier-Stokes-Fourier-Poisson system by Laurençot and Feireisl in [15] while Golse and Saint-Raymond in [18] dealt with celebrated Navier-Stokes and Boltzmann equations.…”

Section: Introductionmentioning

confidence: 88%

“…in the form (5) with the specific dependence on the temperature θ, the density ρ and the dimension of the ambient space d reading p(θ, ρ) = θ d/2+1 P (ρθ −d/2 ) with some given P function (we drop dependence on η) is threefold. First of all one can for zH (z) = P (z) with z = ρθ −d/2 establish the entropy formula W = B(0,1)…”

Section: Appendixmentioning

confidence: 99%

“…The functional is coercive and can be decomposed into compact and continuous part and lower-semicontinuous and convex part thus making the direct approach feasible to yield the existence of minimizer. It seems that the results of [4,5,6,7] can be used to get the continuity of the set of minimizers at least for sufficiently small mass M > 0. The only obstacle is that the limiting functional is defined over the space of ρ log ρ integrable functions as η → 0 + .…”

Section: Open Problems and Possible Extensionsmentioning

confidence: 99%

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Dynamical system modeling fermionic limit

Bors¹,

Stańczy²

2018

Discrete &Amp; Continuous Dynamical Systems - B

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The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate values of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero. arXiv:1612.05442v1 [math.DS]

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Section: Introductionmentioning

confidence: 88%

Section: Appendixmentioning

confidence: 99%

Section: Open Problems and Possible Extensionsmentioning

confidence: 99%

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Dynamical system modeling fermionic limit

Bors¹,

Stańczy²

2018

Discrete &Amp; Continuous Dynamical Systems - B

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“…In this paper we use the Dirichlet fractional Laplacian set in the spectral framework. The problems governed by the Dirichlet fractional Laplacian can be seen as a natural extensions of the problems discussed in [9,10,29] involving the standard Laplace operator. Specifically, we focus our attention on the of the solutions on the functional parameters and then on the existence of the optimal solutions minimizing some cost functional.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of systems governed by fractional Laplacian in the minimax framework

Bors

2019

International Journal of Control

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We consider optimal control problem governed by a class of partial differential equations with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes are proved. The proof of the main result relies on variational structure of the problem. To show that partial differential equations with the Dirichlet fractional Laplacian have a weak solution we employ the renowned Ky Fan Theorem.

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Existence and continuous dependence on parameters of radially symmetric solutions to astrophysical model of self‐gravitating particles

Bors

Stańczy

2019

Math Methods in App Sciences

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In this paper, we obtain the existence of a radial solution for some elliptic nonlocal problem with constraints. The problem is described as stationary state of some evolutionary models provided the pressure function conveys some form arising from statistical mechanics. To be more specific, we consider the following forms of the statistics: Michie-King, Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein, and polytrope. The most recent models were suggested by H. J.de Vega at al. and P. H. Chavanis et al.. We prove the continuous dependence of stationary solutions on parameters for the given statistics. KEYWORDS Fermi-Dirac, Michie-King statistics, nonlocal elliptic equation MSC CLASSIFICATION 35Q85; 47H10; 85A05 ASTROPHYSICAL MOTIVATIONAs it is well established, in the universe, there is vast preponderance of the dark matter and the dark energy. To describe it at the galactic level, numerous models were proposed.The classical Michie-King model was introduced by Michie 1 and King 2 as the natural extension of the Maxwell-Boltzmann case. The classical Michie-King model is a modification of the Maxwell-Boltzmann distribution that takes into account the escape of high-energy stars in a stellar system.Following a long list of works on the subject, de Vega et al 3-5 modeled dark matter halos as self-gravitating gas of fermions at finite temperature and confronted it with observations that correspond to warm dark matter, eg, composed of sterile neutrinos. In the approach by de Vega et al, 3-5 the system does not confine within a box, and halos extend to infinity and have infinite mass. They do not need to compute the total mass in their study, 3-5 so they do not have to deal with this problem.The study that considers self-gravitating fermions in the box is Chavanis, 6 which is concerned with phase transition in the self-gravitating Fermi gas. If we want to study phase transitions between different equilibrium states, it is necessary to have configurations with a finite mass, and this is why Chavanis 6 had to close the system in the artificial box. In more recent papers, Chavanis et al 7,8 studied phase transitions of self-gravitating fermions without the artifice of the material box by using the fermionic King distribution. This approach alleviates the problem by extension of the model with the pressure governed by the Fermi-Dirac statistics to the fermionic Michie-King model. The ad hoc considerations of Ruffini and Stella 9 can be also justified by the statistical mechanics and the kinetic theory once the evaporation of the high energy particles is taken into account as was done by Chavanis et al. 7,8 Although the fermionic King model was introduced heuristically by Ruffini and Stella, 9 a derivation from kinetic theory was first made by Chavanis. 10 To alleviate drawbacks Math Meth Appl Sci. 2019;42:7381-7394.wileyonlinelibrary.com/journal/mma

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Nonlinear elliptic systems with variable boundary data

Cited by 8 publications

References 4 publications

Dynamical system modeling fermionic limit

Dynamical system modeling fermionic limit

Optimal control of systems governed by fractional Laplacian in the minimax framework

Existence and continuous dependence on parameters of radially symmetric solutions to astrophysical model of self‐gravitating particles

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