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Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Honghui Yin, Zuodong Yang (2011)

Annales Polonici Mathematici

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ - Δ p u + | u | p - 2 u = f 1 λ ( x ) | u | q - 2 u + 2 α / ( α + β ) g μ | u | α - 2 u | v | β , x ∈ Ω, ⎨ - Δ p v + | v | p - 2 v = f 2 λ ( x ) | v | q - 2 v + 2 β / ( α + β ) g μ | u | α | v | β - 2 v , x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and Ω N is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and f i λ i ( x ) = λ i f i + ( x ) + f i - ( x ) (i = 1,2) are sign-changing functions, where f i ± ( x ) = m a x ± f i ( x ) , 0 , g μ ( x ) = a ( x ) + μ b ( x ) , and Δ p u = d i v ( | u | p - 2 u ) denotes the p-Laplace operator. We use variational methods.

Multiplicity results for the prescribed scalar curvature on low spheres

Mohamed Ben Ayed, Mohameden Ould Ahmedou (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar curvature on standard spheres 𝕊 3 , 𝕊 4 . Under generic conditions we establish someMorse Inequalities at Infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of its critical points at Infinityto the difference of topology between the level sets of the associated Euler-Lagrange functional. As a by-product of our arguments we derive a new existence...

Multipliers and weighted ∂ operator estimates.

Joaquim Ortega-Cerdà (2002)

Revista Matemática Iberoamericana

We study estimates for the solution of the equation du=f in one variable. The new ingredient is the use of holomorphic functions with precise growth restrictions in the construction of explicit solution to the equation.

Multiscale convergence and reiterated homogenization of parabolic problems

Anders Holmbom, Nils Svanstedt, Niklas Wellander (2005)

Applications of Mathematics

Reiterated homogenization is studied for divergence structure parabolic problems of the form u ε / t - div a x , x / ε , x / ε 2 , t , t / ε k u ε = f . It is shown that under standard assumptions on the function a ( x , y 1 , y 2 , t , τ ) the sequence { u ϵ } of solutions converges weakly in L 2 ( 0 , T ; H 0 1 ( Ω ) ) to the solution u of the homogenized problem u / t - div ( b ( x , t ) u ) = f .

Multiscale expansion and numerical approximation for surface defects⋆

V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial (2011)

ESAIM: Proceedings

This paper is a survey of articles [5, 6, 8, 9, 13, 17, 18]. We are interested in the influence of small geometrical perturbations on the solution of elliptic problems. The cases of a single inclusion or several well-separated inclusions have been deeply studied. We recall here techniques to construct an asymptotic expansion. Then we consider moderately close inclusions, i.e. the distance between the inclusions tends to zero more slowly than their characteristic size. We provide a complete asymptotic...

Multiscale Finite Element approach for “weakly” random problems and related issues

Claude Le Bris, Frédéric Legoll, Florian Thomines (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale...

Multiscale homogenization of nonlinear hyperbolic-parabolic equations

Abdelhakim Dehamnia, Hamid Haddadou (2023)

Applications of Mathematics

The main purpose of the present paper is to study the asymptotic behavior (when ε 0 ) of the solution related to a nonlinear hyperbolic-parabolic problem given in a periodically heterogeneous domain with multiple spatial scales and one temporal scale. Under certain assumptions on the problem’s coefficients and based on a priori estimates and compactness results, we establish homogenization results by using the multiscale convergence method.

Multiscale modelling of sound propagation through the lung parenchyma

Paul Cazeaux, Jan S. Hesthaven (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency-dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale ε and use two-scale homogenization techniques to derive effective...

Multiscale stochastic homogenization of convection-diffusion equations

Nils Svanstedt (2008)

Applications of Mathematics

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form u ε ω / t + 1 / ϵ 3 𝒞 T 3 ( x / ε 3 ) ω 3 · u ε ω - div α T 1 ( x / ε 1 ) ω 1 , T 2 ( x / ε 2 ) ω 2 , t u ε ω = f . It is shown, under certain structure assumptions on the random vector field 𝒞 ( ω 3 ) and the random map α ( ω 1 , ω 2 , t ) , that the sequence { u ϵ ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem u / t - div ( ( t ) u ) = f .

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