Complementing maps, continuation and global bifurcation

We state, and indicate some of the consequences of, a theorem whose sole assumption is the nonvanishing of the Leray- Schauder degree of a compact vector field, and whose conclusions yield multidimensional existence, continuation and bifurcation result

An indefinite concave-convex equation under a Neumann boundary condition II

We proceed with the investigation of the problem $(P_\lambda):$ -\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $\lambda \in \mathbb{R}$, and $a,b \in C^\alpha(\overline{\Omega})$ with $0<\alpha<1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_\lambda)$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn's topological method.Comment: 15 pages, 3 figure

On the manifold structure of the set of unparameterized embeddings with low regularity

Given manifolds $M$ and $N$, with $M$ compact, we study the geometrical structure of the space of embeddings of $M$ into $N$, having less regularity than $\mathcal C^\infty$, quotiented by the group of diffeomorphisms of $M$.Comment: To appear in the Bulletin of the Brazilian Mathematical Societ

Weakly-nonlinear analysis of the Rayleigh–Taylor instability in a vertically vibrated, large aspect ratio container

We consider a horizontal liquid layer supported by air in a wide (as compared to depth) container, which is vertically vibrated with an appropriately large frequency, intending to counterbalance the Rayleigh-Taylor instability of the fíat, rigid-body vibrating state. We apply a long-wave, weakly-nonlinear analysis that yields a generalized Cahn-Hilliard equation for the evolution of the fluid interface, with appropriate boundary conditions obtained by a boundary layer analysis. This equation shows that the stabilizing effect of vibration is like that of surface tensión, and is used to analyze the linear stability of the fíat state, and the local bifurcation at the instability threshold