We discuss the Γ-convergence, under the appropriate scaling, of the energy functional ‖u‖Hs(Ω) + ∫ Ω W (u) dx, with s ∈ (0, 1), where ‖u‖Hs(Ω) denotes the total contribution from Ω in the Hs norm of u, and W is a double-well potential. When s ∈ [1/2, 1), we show that the energy Γ-converges to the classical minimal surface functional – while, when s ∈ (0, 1/2), it is easy to see that the functio...

We consider the equation ∆u − V (x)u + W (x)u = 0 and its parabolic counterpart in noncompact manifolds. Under some natural conditions on the positive functions V and W , which may only have ‘slow’ or no decay near infinity, we establish existence of positive solutions in both the critical and the subcritical case. This leads to the solutions, in the difficult positive curvature case, of many s...

1. Introduction. Let J be a closed linear interval ao^tSfo. Let r(/) = (#(/), y(t) } z(t)), tSI, represent a vector function whose three components x(t), y(i) t z(t) are of bounded variation and continuous on I. This vector function determines in Euclidean 3-space a curve x~x{t), y~y(t), z=*z(t) whose length we denote by LQç). By convergence in length of a sequence of such vector functions $ n ...

We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Γ-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that t...

We study a special class of solutions to the 3D Navier-Stokes equations ∂tu +∇uνu +∇p = ν∆u , with no-slip boundary condition, on a domain of the form Ω = {(x, y, z) : 0 ≤ z ≤ 1}, dealing with velocity fields of the form u(t, x, y, z) = (v(t, z), w(t, x, z), 0), describing plane-parallel channel flows. We establish results on convergence u → u as ν → 0, where u solves the associated Euler equat...

We study the homogenization of 2D linear transport equations, ut + ~a(~x/ε) · ∇~xu = 0, where ~a is a non-vanishing vector field with integral invariance on the torus T . When the underlying flow on T 2 is ergodic, we derive the efficient equation which is a linear transport equation with constant coefficients and quantify the pointwise convergence rate. This result unifies and illuminates the ...

Abstract We introduce a novel method for the implementation of shape optimization non-parameterized shapes in fluid dynamics applications, where we propose to use derivative determine deformation fields with help $$p-$$ p - Laplacian $$p &gt; 2$$ &gt; <...

The purpose of this paper is to prove strong convergence theorems for common fixed points of two families of weak relatively nonexpansive mappings and a family of equilibrium problems by a new monotone hybrid method in Banach spaces. Because the hybrid method presented in this paper is monotone, so that the method of the proof is different from the original one. We shall give an example which i...

In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale (Wt)t≥0 to prove convergence in probability, mean and almost surely, supercritical general branching processes (also known as Crump-Mode-Jagers processes) counted with characteristic. The terminal value W figures the limits his results. We investigate rate at which martingale, now called Nerman’s converg...

This paper focuses on the relationships between stratified $L$-conver-gence spaces, stratified strong $L$-convergence spaces and stratifiedlevelwise  $L$-convergence spaces. It has been known that: (1) astratified $L$-convergence space is precisely a left-continuousstratified levelwise $L$-convergence space; and (2) a  stratifiedstrong $L$-convergence space is naturally a stratified $L$-converg...