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The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.
Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
Authors: Fernandez, L. | Novotny, A.A. | Prakash, R.
Article Type: Research Article
Abstract: In this paper, we deal with the topological asymptotic analysis of an optimal control problem modeled by a coupled system. The control is a geometrical object and the cost is given by the misfit between a target function and the state, solution of the Helmholtz–Laplace coupled system. Higher-order topological derivatives are used to devise a non-iterative algorithm to compute the optimal control for the problem of interest. Numerical examples are presented in order to demonstrate the effectiveness of the proposed algorithm.
Keywords: Control problem, Helmholtz–Laplace coupled system, topological asymptotic analysis, topological derivatives, topology optimization
DOI: 10.3233/ASY-181465
Citation: Asymptotic Analysis, vol. 109, no. 1-2, pp. 1-26, 2018
Authors: Chang, Yong-Kui | N’Guérékata, G.M. | Zhao, Zhi-Han
Article Type: Research Article
Abstract: In this paper, we consider recurrence of bounded solutions to semilinear stochastic integro-differential equations via uniformly exponentially stable resolvent operators family. Concretely, we first establish the existence and uniqueness of the L 2 -bounded solution to a semilinear stochastic integro-differential equations driven by Lévy processes, and then we show this kind of L 2 -bounded solution is almost automorphic in distribution or weighted pseudo almost automorphic in distribution under suitable conditions respectively.
Keywords: Stochastic integro-differential equations, almost automorphy in distribution, weighted pseudo almost automorphy in distribution, Lévy noise
DOI: 10.3233/ASY-181466
Citation: Asymptotic Analysis, vol. 109, no. 1-2, pp. 27-52, 2018
Authors: Bruneau, Vincent | Raikov, Georgi
Article Type: Research Article
Abstract: We consider harmonic Toeplitz operators T V = P V : H ( Ω ) → H ( Ω ) where P : L 2 ( Ω ) → H ( Ω ) is the orthogonal projection onto H ( Ω ) = { u ∈ L 2 ( Ω ) ∣ Δ u = 0 in Ω } , Ω ⊂ R d , d ⩾ 2 , is a bounded domain with boundary …∂ Ω ∈ C ∞ , and V : Ω → C is an appropriate multiplier. First, we complement the known criteria which guarantee that T V is in the p th Schatten–von Neumann class S p , by simple sufficient conditions which imply T V ∈ S p , w , the weak counterpart of S p . Next, we consider symbols V ⩾ 0 which have a regular power-like decay of rate γ > 0 at ∂ Ω , and we show that T V is unitarily equivalent to a classical pseudo-differential operator of order − γ , self-adjoint in L 2 ( ∂ Ω ) . Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for T V , and establish a sharp remainder estimate. Further, we assume that Ω is the unit ball in R d , and V = V ‾ is compactly supported in Ω, and investigate the eigenvalue asymptotics of the Toeplitz operator T V . Finally, we introduce the Krein Laplacian K , self-adjoint in L 2 ( Ω ) , perturb it by a multiplier V ∈ C ( Ω ‾ ; R ) , and show that σ ess ( K + V ) = V ( ∂ Ω ) . Assuming that V ⩾ 0 and V | ∂ Ω = 0 , we study the asymptotic distribution of the discrete spectrum of K ± V near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator T V . Show more
Keywords: Harmonic Toeplitz operators, Krein Laplacian, eigenvalue asymptotics, effective Hamiltonian
DOI: 10.3233/ASY-181467
Citation: Asymptotic Analysis, vol. 109, no. 1-2, pp. 53-74, 2018
Authors: Cao, Dat | Ibraguimov, Akif | Nazarov, Alexander I.
Article Type: Research Article
Abstract: We investigate the qualitative properties of solutions to the Zaremba type problem in unbounded domains for non-divergence elliptic equation with possible degeneration at infinity. The main result is a Phragmén–Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the “thickness” of its Dirichlet portion. The result is formulated in terms of the so-called s -capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain “admissibility” condition in the sequence of layers converging to infinity.
Keywords: Non-divergence elliptic equations, mixed boundary value problems, growth lemma, Phragmén–Lindelöf theorem, Zaremba type problem
DOI: 10.3233/ASY-181469
Citation: Asymptotic Analysis, vol. 109, no. 1-2, pp. 75-90, 2018
Authors: Khoutir, Sofiane | Chen, Haibo
Article Type: Research Article
Abstract: In this paper, we study the following Schrödinger–Poisson system − Δ u + u + λ ϕ u = f ( u ) + u 3 , x ∈ R 4 , − Δ ϕ = u 2 , x ∈ R 4 , where λ > 0 is a parameter, f ∈ C ( R ) and the nonlinear growth …of u 3 reaches the Sobolev critical exponent since 2 ∗ = 4 in dimension 4. Under some suitable assumptions on f , we establish the existence of a positive radial and non-radial ground state solutions for the above system by using variational methods. We also discuss the asymptotic behavior of solutions with respect to the parameter λ . Show more
Keywords: Schrödinger–Poisson system, variational method, critical exponent, ground state solution
DOI: 10.3233/ASY-181471
Citation: Asymptotic Analysis, vol. 109, no. 1-2, pp. 91-109, 2018
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