Homogenization of oscillating boundaries and applications to thin films

We prove a homogenization result for integral functionals in domains with oscillating boundaries, showing that the limit is defined on a degenerate Sobolev space. We apply this result to the description of the asymptotic behaviour of thin films with fast-oscillating profile, proving that they can be described by first applying the homogenization result above and subsequently a dimension-reduction technique.Comment: 31 pages, 7 figure

Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation

We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density $u$. In case of \emph{fast-decay} mobilities, namely mobilities functions under a Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process. We show that the rescaled density $\rho$ is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density $\rho$ allow us to motivate the aforementioned change of variable and to state the results in terms of the original density $u$ without prescribing any boundary conditions

The Neumann sieve problem and dimensional reduction: a multiscale approach

We perform a multiscale analysis for the elastic energy of a $n$-dimensional bilayer thin film of thickness $2\delta$ whose layers are connected through an $\epsilon$-periodically distributed contact zone. Describing the contact zone as a union of $(n-1)$-dimensional balls of radius $r\ll \epsilon$ (the holes of the sieve) and assuming that $\delta \ll \epsilon$, we show that the asymptotic memory of the sieve (as $\epsilon \to 0$) is witnessed by the presence of an extra interfacial energy term. Moreover we find three different limit behaviors (or regimes) depending on the mutual vanishing rate of $\delta$ and $r$. We also give an explicit nonlinear capacitary-type formula for the interfacial energy density in each regime.Comment: 43 pages, 4 figure

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 $R^\epsilon = \{(x_1,x_2) \in \R^2 \; | \; x_1 \in (0,1), \, - \, \epsilon \, b(x_1) < x_2 < \epsilon \, G(x_1, x_1/\epsilon^\alpha) \}$ with $\alpha>1$ and $\epsilon > 0$, defined by smooth functions $b(x)$ and $G(x,y)$, where the function $G$ is supposed to be $l(x)$-periodic in the second variable $y$. The condition $\alpha > 1$ 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 $R^\epsilon$ given by the small parameter $\epsilon$. 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

Sketchy

Sketchy is the story of Charlie, a 30-year-old underachiever working in the IT department of an office building. Despite his best efforts, he has slowly moved up the corporate ladder that he never intended to climb. In the beginning of the film, Charlie’s long term girlfriend Julie is thrilled to hear Charlie has been offered a management position with the company. She takes it as a sign that now is the perfect time to take their relationship to the next level and get married. Charlie, on the other hand, sees this as a different sign. He has decided that he wants to quit his job in order to go back to school and explore his passion for illustration. When he tells Julie of his desires, she becomes infuriated with him and they break out into a fight. Afraid of loosing her, Charlie ultimately gives into Julie’s wishes, and reluctantly agrees to propose to Julie at a party in front of all of their friends. After taking the promotion, Charlie finds that the new job is even more monotonous and unfulfilling than ever. Charlie develops a friendship with Lila, a spunky and free spirited waitress. He envy’s her ability to just follow her passions without being weighed down with the pressure of responsibility. During their planned “spontaneous” proposal party, Charlie finally breaks down and leaves Julie in front of everyone. When Julie confronts him, Charlie states that even though he loves Julie and wants to be with her, following his dream is something that he needs to do. Julie eventually comes around, and supports Charlie in his journey

Non-Newtonian thin films with normal stresses: dynamics and spreading

The dynamics of thin films on a horizontal solid substrate is investigated in the case of non-Newtonian fluids exhibiting normal stress differences, the rheology of which is strongly non-linear. Two coupled equations of evolution for the thickness of the film and the shear rate are proposed within the lubrication approximation. This framework is applied to the motion of an advancing contact line. The apparent dynamic contact angle is found to depend logarithmically on a lengthscale determined solely by the rheological properties of the fluid and the velocity of the contact line

Minimizing movements for oscillating energies:The critical regime

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter with τ describing the time step and the frequency of the oscillations being proportional to 1/. The extreme cases of fast time scales τ â and slow time scales â τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio /τ &gt; 0 is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.</p