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This article is divided into two chapters. The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in the first chapter. Following a Fourier approach, we discuss some of the basic issues of the subject: main convergence theorem, Bloch approximation, estimates on second order derivatives, correctors for the medium, and so on. The second chapter is devoted to the discussion of some non-classical behaviour of vibration problems of periodic...

Fourier approach to homogenization problems

Carlos Conca, M. Vanninathan (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Fourier transform, oscillatory multipliers and evolution equations in rearrangement invariant function spaces

Luca Brandolini, Leonardo Colzani (1996)

Colloquium Mathematicae

Four-parameter bifurcation for a $p$ -Laplacian system.

Fleckinger, Jacqueline, Pardo, Rosa, de Thélin, François (2001)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Free boundary problems and transonic shocks for the Euler equations in unbounded domains

Gui-Qiang Chen, Mikhail Feldman (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We establish the existence and stability of multidimensional transonic shocks (hyperbolic-elliptic shocks), which are not nearly orthogonal to the flow direction, for the Euler equations for steady compressible potential fluids in unbounded domains in $ℝ^{n}, n \geq 3$ . The Euler equations can be written as a second order nonlinear equation of mixed hyperbolic-elliptic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the...

Free boundary problems arising in tumor models

Avner Friedman (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider several simple models of tumor growth, described by systems of PDEs, and describe results on existence of solutions and on their asymptotic behavior. The boundary of the tumor region is a free boundary. In §1 the model assumes three types of cells, proliferating, quiescent and necrotic, and the corresponding PDE system consists of elliptic, parabolic and hyperbolic equations. The model in §2 assumes that the tumor has only proliferating cells. Finally in §3 we consider a model for treatment...

Free decay of solutions to wave equations on a curved background

Serge Alinhac (2005)

Bulletin de la Société Mathématique de France

We investigate for which metric $g$ (close to the standard metric $g_{0}$ ) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (i.e., non integrable) decay conditions on $g - g_{0}$ ; in particular, $g - g_{0}$ decays like $t^{- \frac{1}{2} - ε}$ along wave cones.

Free vibrations for the equation of a rectangular thin plate

Eduard Feireisl (1988)

Aplikace matematiky

In the paper, we deal with the equation of a rectangular thin plate with a simply supported boundary. The restoring force being an odd superlinear function of the vertical displacement, the existence of infinitely many nonzero time-periodic solutions is proved.

Free-energy-dissipative schemes for the Oldroyd-B model

Sébastien Boyaval, Tony Lelièvre, Claude Mangoubi (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman in [J. Non-Newtonian...

Frentes de explosión en ecuaciones de Hamilton Jacobi por argumentos de control determinista.

G. Díaz (1989)

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales

Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation

Nicola Garofalo, Ermanno Lanconelli (1990)

Annales de l'institut Fourier

A recent result of Bahouri shows that continuation from an open set fails in general for solutions of $ℒ u = V u$ where $V \in C^{\infty}$ and $ℒ = \sum_{j = 1}^{N - 1} X_{j}^{2}$ is a (nonelliptic) operator in $R^{N}$ satisfying Hörmander’s condition for hypoellipticity. In this paper we study the model case when $ℒ$ is the subelliptic Laplacian on the Heisenberg group and $V$ is a zero order term which is allowed to be unbounded. We provide a sufficient condition, involving a first order differential inequality, for nontrivial solutions of $ℒ u = V u$ to have a finite order...

Frequent oscillatory behavior of delay partial difference equations with positive and negative coefficients.

Xu, Li Hua, Yang, Jun (2010)

Advances in Difference Equations [electronic only]

Front propagation for reaction-diffusion equations of bistable type

G. Barles, L. Bronsard, P. E. Souganidis (1992)

Annales de l'I.H.P. Analyse non linéaire

Fully nonlinear second order elliptic equations with large zeroth order coefficient

L. C. Evans, Pierre-Louis Lions (1981)

Annales de l'institut Fourier

We prove the existence of classical solutions to certain fully non-linear second order elliptic equations with large zeroth order coefficient. The principal tool is an a priori estimate asserting that the $C^{2, α}$ -norm of the solution cannot lie in a certain interval of the positive real axis.

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