Asymptotic Analysis

We study existence of multiple positive solutions for the nonlinear eigenvalue problem − div ( ϕ ( | ∇ u | ) ∇ u ) = λ f ( u ) in Ω, u = 0 on ∂ Ω , where Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω , ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) is a suitable C 1 -function, λ > 0 is a parameter and f : [ 0 , ∞ ) → R is a sign-changing continuous function. We show existence of a finite number of solutions in the case f changes sign a finite number of times and existence of infinitely many solutions in the case f changes sign an infinite number of times. We employ variational arguments, regularity results, a strong maximum principle by Pucci and Serrin and a general result on lower and upper solutions. Our research was motivated by the work of Hess for the case of the Laplacian and Loc and Schmitt for the case of the p-Laplacian and we were able to extend the major results by Loc and Schmitt.

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix a ε that is periodic with characteristic length scale ε; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions u ε well by a non-dispersive wave equation on fixed time intervals ( 0 , T ) . Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive equation with homogeneous coefficients such that its solutions w ε describe u ε well on time intervals ( 0 , T ε − 2 ) . More precisely, we provide a norm and uniform error estimates of the form ∥ u ε ( t ) − w ε ( t ) ∥ ⩽ C ε for t ∈ ( 0 , T ε − 2 ) . They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an ε-independent equation of third order that describes dispersion along rays and we present numerical examples.

We study the asymptotic behavior of parabolic p ( x ) -Laplacian problems of the form ∂ u λ ∂ t − div ( D λ | ∇ u λ | p ( x ) − 2 ∇ u λ ) + a | u λ | p ( x ) − 2 u λ = B ( u λ ) in L 2 ( R n ) , where n ⩾ 1 , p ∈ L ∞ ( R n ) such that 2 < p − : = ess inf p ( x ) ⩽ p ( x ) ⩽ p + : = ess sup p ( x ) , D λ ∈ L ∞ ( R n ) , ∞ > M ⩾ D λ ( x ) ⩾ σ > 0 a.e. in R n , λ ∈ [ 0 , ∞ ) , B : L 2 ( R n ) → L 2 ( R n ) is a globally Lipschitz map and a : R n → R is a non-negative continuous function such that there exists R 1 > 0 with { x ∈ R n ; a ( x ) = 0 } ⊂ B R 1 ( 0 ) , inf x ∈ R n ∖ B R 1 ( 0 ) a ( x ) > 0 , and ∫ R n ∖ B R 1 ( 0 ) 1 a ( x ) 2 / ( p ( x ) − 2 ) d x < + ∞ . We also study the sensitivity of the problem according to the variation of the diffusion coefficients.

We study the stability, with respect to the G-convergence, of the W 0 1 , 1 distributional solutions of a degenerate elliptic equation.

We consider the Ginzburg–Landau functional with a variable applied magnetic field in a bounded and smooth two-dimensional domain. The applied magnetic field varies smoothly and is allowed to vanish non-degenerately along a curve. Assuming that the strength of the applied magnetic field varies between two characteristic scales, and the Ginzburg–Landau parameter tends to + ∞ , we determine an accurate asymptotic formula for the minimizing energy and show that the energy minimizers have vortices. The new aspect in the presence of a variable magnetic field is that the density of vortices in the sample is not uniform.

The present paper is devoted to the study of a zero-Mach number system with heat conduction but no viscosity. We work in the framework of general non-homogeneous Besov spaces B p , r s ( R d ) , with p ∈ [ 2 , 4 ] and for any d ⩾ 2 , which can be embedded into the class of globally Lipschitz functions.

We prove a local in time well-posedness result in these classes and we are also able to show a continuation criterion and a lower bound for the lifespan of the solutions.

The proof of the results relies on Littlewood–Paley decomposition and paradifferential calculus, and on refined commutator estimates in Chemin–Lerner spaces.

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients.

In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the p ( x ) -Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration–compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain pass theorem.