arXiv subjects
Publications and source records attributed to Ana Margarida Ribeiro.
We address a deep study of the convexity notions that arise in the study of weak* lower semicontinuity of supremal functionals as well as those raised by the power-law approximation of such functionals. Our quest is motivated by the knowledge we have on the analogous integral functionals and aims at establishing a solid groundwork to ease any research in the $L^\infty$ context.
We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii) and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimization of free energy functionals.
The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz-Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.
The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem ${\rm inf}{{\rm ess sup}_{x \in Ω} f(\nabla u(x)): u \in u_0 + W^{1,\infty}_0(Ω)}$ when the supremand $f$ is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand $f$ are also investigated.
It is studied the lower semicontinuity of functionals of the type $\int_Ωf(x,u,v, \nabla u)dx$ with respect to the $(W^{1,1}\times L^p)$-weak \ast topology. Moreover in absence of lower semicontinuity, it is also provided an integral representation in $W^{1,1} \times L^p$ for the lower semicontinuous envelope.
We provide a relaxation result in $BV \times L^q$, $1\leq q < +\infty$ as a first step towards the analysis of thermochemical equilibria.