1,706 research outputs found
Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries
In this work we study the behavior of a family of solutions of a semilinear
elliptic equation, with homogeneous Neumann boundary condition, posed in a
two-dimensional oscillating thin region with reaction terms concentrated in a
neighborhood of the oscillatory boundary. Our main result is concerned with the
upper and lower semicontinuity of the set of solutions. We show that the
solutions of our perturbed equation can be approximated with ones of a
one-dimensional equation, which also captures the effects of all relevant
physical processes that take place in the original problem
The Neumann problem in thin domains with very highly oscillatory boundaries
In this paper we analyze the behavior of solutions of the Neumann problem
posed in a thin domain of the type with and , defined by smooth
functions and , where the function is supposed to be
-periodic in the second variable . The condition implies
that the upper boundary of this thin domain presents a very high oscillatory
behavior. Indeed, we have that the order of its oscillations is larger than the
order of the amplitude and height of given by the small parameter
. We also consider more general and complicated geometries for thin
domains which are not given as the graph of certain smooth functions, but
rather more comb-like domains.Comment: 20 pages, 4 figure
Homogenization of Dirichlet parabolic problems for coefficients and open sets simultaneously variable and applications to optimal design
In a previous paper, we studied the homogenization of a sequence of parabolic linear Dirichlet problems, when the coefficients and the domains vary arbitrarily. Here, we improve the convergence result given in this paper by showing the strong convergence in every time. This is applied to obtain an existence result for control problems in optimal design written in a relaxed form. The control variables are the material and the shape
Stochastic homogenization of -convex gradient flows
In this paper we present a stochastic homogenization result for a class of
Hilbert space evolutionary gradient systems driven by a quadratic dissipation
potential and a -convex energy functional featuring random and rapidly
oscillating coefficients. Specific examples included in the result are
Allen-Cahn type equations and evolutionary equations driven by the -Laplace
operator with . The homogenization procedure we apply is based
on a stochastic two-scale convergence approach. In particular, we define a
stochastic unfolding operator which can be considered as a random counterpart
of the well-established notion of periodic unfolding. The stochastic unfolding
procedure grants a very convenient method for homogenization problems defined
in terms of (-)convex functionals.Comment: arXiv admin note: text overlap with arXiv:1805.0954
Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?
A wide variety of techniques have been developed to homogenize transport equations in multiscale and multiphase systems. This has yielded a rich and diverse field, but has also resulted in the emergence of isolated scientific communities and disconnected bodies of literature. Here, our goal is to bridge the gap between formal multiscale asymptotics and the volume averaging theory. We illustrate the methodologies via a simple example application describing a parabolic transport problem and, in so doing, compare their respective advantages/disadvantages from a practical point of view. This paper is also intended as a pedagogical guide and may be viewed as a tutorial for graduate students as we provide historical context, detail subtle points with great care, and reference many fundamental works
Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets
For a fixed bounded open set Ω ⊂ RN , a sequence of open sets Ωn ⊂ Ω and a sequence of sets Γn ⊂ ∂Ω ∩ ∂Ωn, we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on Ωn, satisfying Neumann boundary conditions on Γn and Dirichlet boundary conditions on ∂Ωn \ Γn. We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on Ωn and Γn locally.Ministerio de Educación y CienciaJunta de AndalucÃ
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