Asymptotic Analysis - Volume 134, issue 1-2

Authors: Bejaoui, Eya | Ben Belgacem, Faker

Article Type: Research Article

Abstract: We explore the mathematical structure of the solution to an elliptic diffusion problem with point-wise Dirac sources. The conductivity parameter is space-varying, may have jumps and the Dirac sources may be located along the discontinuity curves of that parameter. The variational problem, issued by duality, is proven to be well posed using a sharp elliptic regularity result by Di-Giorgi [Mem. Accad. Sci. Torino, 3, 1957]. The paper is aimed at a key expansion into a split singular/regular contributions. The singular part is calculated by an explicit formula, while the regular correction can be computed as the solution to a standard …variational Poisson problem. The latter can be successfully approximated by most of the numerical methods practiced nowadays. Some analytical examples are discussed at last to assess the minimality of the assumptions we use to establish our theoretical results. Show more

Keywords: Elliptic Problems, point-wise Dirac sources, non-smooth conductivity, singular/regular splitting

DOI: 10.3233/ASY-221824

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 1-23, 2023

Authors: Hinz, Michael | Rozanova-Pierrat, Anna | Teplyaev, Alexander

Article Type: Research Article

Abstract: We study boundary value problems for bounded uniform domains in R n , n ⩾ 2 , with non-Lipschitz, and possibly fractal, boundaries. We prove Poincaré inequalities with uniform constants and trace terms for ( ε , ∞ ) -domains contained in a fixed bounded Lipschitz domain. We introduce generalized Dirichlet, Neumann, and Robin problems for Poisson-type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, the norm convergence of the associated resolvents, and the …convergence of the corresponding eigenvalues and eigenfunctions. We provide compactness results for parametrized classes of admissible domains, energy functionals, and weak solutions. Using these results, we can then prove the existence of optimal shapes in these classes in the sense that they minimize the initially given energy functionals. For the Robin boundary problems, this result is new. Show more

Keywords: Fractal boundaries, Poincaré inequalities, Robin problems, Mosco convergence, norm resolvent convergence, optimal shapes

DOI: 10.3233/ASY-231825

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 25-61, 2023

Authors: Ramos, A.J.A. | Araújo, A.L.A. | Campelo, A.D.S. | Freitas, M.M. | Veras, L.S.

Article Type: Research Article

Abstract: In this paper, we are interested in studying the well-posedness, optimal polynomial stability, and the lack of exponential stability for a class of thermoelastic system of Reissner–Mindlin–Timoshenko plates with structural damping, that is, with the dissipation of Kelvin–Voigt type on the equations for the rotation angles. We also consider the thermal effect with thermal variables described by Fourier’s law of heat conduction.

Keywords: 35B40, 35L51, 74D05, 74F05

DOI: 10.3233/ASY-231826

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 63-84, 2023

Authors: Liang, Shuaishuai | Shi, Shaoyun

Article Type: Research Article

Abstract: In this paper, we study a class of the ( p , q ) Kirchhoff type problems with convolution term in R N . With the appropriate assumptions on potential function V and convolution term f , together with the penalization techniques, Morse iterative method and variational method, the existence and multiplicity of multi-bump solutions are obtained for this problem. In some sense, our results also generalize some known results.

Keywords: Double phase Kirchhoff problems, potential well, multi-bump solutions, convolution term, variational methods

DOI: 10.3233/ASY-231827

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 85-126, 2023

Authors: Chen, Taiyong | Jiang, Yahui | Squassina, Marco | Zhang, Jianjun

Article Type: Research Article

Abstract: In this paper, we are concerned with the coupled nonlinear Schrödinger system − ε 2 Δ u + a ( x ) u = μ 1 u 3 + β v 2 u in  R N , − ε 2 Δ v + b ( x ) v = μ 2 v 3 + β u 2 v …in  R N , where 1 ⩽ N ⩽ 3 , μ 1 , μ 2 , β > 0 , a ( x ) and b ( x ) are nonnegative continuous potentials, and ε > 0 is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as ε → 0 , when a ( x ) and b ( x ) achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary. Show more

Keywords: Nonlinear Schrödinger systems, semiclassical limit, critical frequency, variational methods

DOI: 10.3233/ASY-231828

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 127-154, 2023

Authors: Destuynder, Philippe

Article Type: Research Article

Abstract: There is a narrow but hidden link between optimal control theory and the so-called Tikhonov regularization method. In fact, the small coefficient representing the marginal cost of the control can be interpreted as the regularization parameter in a Tikhonov method as far as there exists an exact control. This strategy enables one to adjust the cost function in the optimal control model in order to define the exact control which minimizes a given functional involving both the control but also the state variables during the control process. The goal of this paper is to suggest a method which gives a …simple way to characterize and compute the exact control corresponding to the minimum of a given cost functional as said above. It appears as an extension of the phase control which is a finite dimensional version of the HUM control of J.L. Lions but for partial differential equations. Show more

Keywords: Optimal control, asymptotic methods, Tikhonov regularization, exact control, dynamical systems

DOI: 10.3233/ASY-231829

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 155-181, 2023

Authors: Papageorgiou, Nikolaos S. | Vetro, Calogero | Vetro, Francesca

Article Type: Research Article

Abstract: We consider a double phase (unbalanced growth) Dirichlet problem with a Carathéodory reaction f ( z , x ) which is superlinear in x but without satisfying the AR-condition. Using the symmetric mountain pass theorem, we produce a whole sequence of distinct bounded solutions which diverge to infinity.

Keywords: Double phase operator, generalized Orlicz spaces, Z2-mountain pass theorem, C-condition, AR-condition

DOI: 10.3233/ASY-231830

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 183-192, 2023

Authors: Dalla Riva, Matteo | Luzzini, Paolo | Musolino, Paolo

Article Type: Research Article

Abstract: We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number ϵ > 0 , proportional to the radius of the holes, and a map ϕ , which models the shape of the holes. So, if g denotes the Dirichlet boundary datum and f the Poisson datum, we have a solution for each quadruple ( ϵ , ϕ , g , f ) . Our aim is to study how the solution depends on ( ϵ , ϕ …, g , f ) , especially when ϵ is very small and the holes narrow to points. In contrast with previous works, we do not introduce the assumption that f has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for ϵ close to 0. We show that, when the dimension n of the ambient space is greater than or equal to 3, a suitable restriction of the solution can be represented with an analytic map of the quadruple ( ϵ , ϕ , g , f ) multiplied by the factor 1 / ϵ n − 2 . In case of dimension n = 2 , we have to add log ϵ times the integral of f / 2 π . Show more

Keywords: Dirichlet problem, integral equation method, Poisson equation, periodically perforated domain, singularly perturbed domain, real analytic continuation in Banach spaces

DOI: 10.3233/ASY-231831

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 193-212, 2023

Authors: Sbai, Abdelaaziz | El Hadfi, Youssef | El Ouardy, Mounim

Article Type: Research Article

Abstract: We consider the following non-linear singular elliptic problem (1) − div ( M ( x ) | ∇ u | p − 2 ∇ u ) + b | u | r − 2 u = a u p − 1 | x | p + f u γ in  Ω u > 0 in  Ω u = 0 …on  ∂ Ω , where 1 < p < N ; Ω ⊂ R N is a bounded regular domain containing the origin and 0 < γ < 1 , a ⩾ 0 , b > 0 , 0 ⩽ f ∈ L m ( Ω )  and  1 < m < N p . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side. Show more

Keywords: Singular non-linearity, Hardy Potential, p-Laplacian, Lower Order Terms, Regularizing Effect

DOI: 10.3233/ASY-231832

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 213-225, 2023

Authors: Ambrosio, Vincenzo

Article Type: Research Article

Abstract: Let s ∈ ( 0 , 1 ) , N > 2 s and D s , 2 ( R N ) : = { u ∈ L 2 s ∗ ( R N ) : ‖ u ‖ D s , 2 ( R N ) : = ( C N , s 2 …∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ) 1 2 < ∞ } , where 2 s ∗ : = 2 N N − 2 s is the fractional critical exponent and C N , s is a positive constant. We consider functionals J : D s , 2 ( R N ) → R of the type J ( u ) : = 1 2 ‖ u ‖ D s , 2 ( R N ) 2 − ∫ R N b ( x ) G ( u ) d x , where G ( t ) : = ∫ 0 t g ( τ ) d τ , g : R → R is a continuous function with subcritical growth at infinity, and b : R N → R is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace V s : = { u ∈ D s , 2 ( R N ) : u ∈ C ( R N )  with  sup x ∈ R N ( 1 + | x | N − 2 s ) | u ( x ) | < ∞ } must be a local minimizer of J in the D s , 2 ( R N ) -topology. Show more

Keywords: Fractional operators, variational methods, L∞-estimate

DOI: 10.3233/ASY-231833

Citation: Asymptotic Analysis, vol. 134, no. 1-2, pp. 227-239, 2023