Obstructions to regularity in the classical Monge problem

We provide counterexamples to regularity of optimal maps in the classical Monge problem under various assumptions on the initial data. Our construction is based on a variant of the counterexample in \cite{LSW} to Lipschitz regularity of the monotone optimal map between smooth densities supported on convex domains

Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow

We prove that solutions of a mildly regularized Perona-Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.Comment: 19 page

Slow time behavior of the semidiscrete Perona-Malik scheme in dimension one

We consider the long time behavior of the semidiscrete scheme for the Perona-Malik equation in dimension one. We prove that approximated solutions converge, in a slow time scale, to solutions of a limit problem. This limit problem evolves piecewise constant functions by moving their plateaus in the vertical direction according to a system of ordinary differential equations. Our convergence result is global-in-time, and this forces us to face the collision of plateaus when the system singularizes. The proof is based on energy estimates and gradient-flow techniques, according to the general idea that "the limit of the gradient-flows is the gradient-flow of the limit functional". Our main innovations are a uniform H\"{o}lder estimate up to the first collision time included, a well preparation result with a careful analysis of what happens at discrete level during collisions, and renormalizing the functionals after each collision in order to have a nontrivial Gamma-limit for all times.Comment: 42 page

Regularity for general functionals with double phase

We prove sharp regularity results for a general class of functionals of the type $$w \mapsto \int F(x, w, Dw) \, dx\;,$$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;,$$ with $0<\nu \leq b(\cdot)\leq L$. This changes its ellipticity rate according to the geometry of the level set $\{a(x)=0\}$ of the modulating coefficient $a(\cdot)$. We also present new methods and proofs, that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon

A novel mechanism underlying the innate immune response induction upon viral-dependent replication of host cell mRNA: A mistake of +sRNA viruses' replicases

Viruses are lifeless particles designed for setting virus-host interactome assuring a new generation of virions for dissemination. This interactome generates a pressure on host organisms evolving mechanisms to neutralize viral infection, which places the pressure back onto virus, a process known as virus-host cell co-evolution. Positive-single stranded RNA (+sRNA) viruses are an important group of viral agents illustrating this interesting phenomenon. During replication, their genomic +sRNA is employed as template for translation of viral proteins; among them the RNA-dependent RNA polymerase (RdRp) is responsible of viral genome replication originating double-strand RNA molecules (dsRNA) as intermediates, which accumulate representing a potent threat for cellular dsRNA receptors to initiate an antiviral response. A common feature shared by these viruses is their ability to rearrange cellular membranes to serve as platforms for genome replication and assembly of new virions, supporting replication efficiency increase by concentrating critical factors and protecting the viral genome from host anti-viral systems. This review summarizes current knowledge regarding cellular dsRNA receptors and describes prototype viruses developing replication niches inside rearranged membranes. However, for several viral agents it's been observed both, a complex rearrangement of cellular membranes and a strong innate immune antiviral response induction. So, we have included recent data explaining the mechanism by, even though viruses have evolved elegant hideouts, host cells are still able to develop dsRNA receptors-dependent antiviral response.Fil: Delgui, Laura Ruth. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mendoza. Instituto de Histología y Embriología de Mendoza Dr. Mario H. Burgos. Universidad Nacional de Cuyo. Facultad de Cienicas Médicas. Instituto de Histología y Embriología de Mendoza Dr. Mario H. Burgos; ArgentinaFil: Colombo, Maria Isabel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mendoza. Instituto de Histología y Embriología de Mendoza Dr. Mario H. Burgos. Universidad Nacional de Cuyo. Facultad de Cienicas Médicas. Instituto de Histología y Embriología de Mendoza Dr. Mario H. Burgos; Argentin

Existence and uniqueness of maximal regular flows for non-smooth vector fields

In this paper we provide a complete analogy between the Cauchy-Lipschitz and the DiPerna-Lions theories for ODE's, by developing a local version of the DiPerna-Lions theory. More precisely, we prove existence and uniqueness of a maximal regular flow for the DiPerna-Lions theory using only local regularity and summability assumptions on the vector field, in analogy with the classical theory, which uses only local regularity assumptions. We also study the behaviour of the ODE trajectories before the maximal existence time. Unlike the Cauchy-Lipschitz theory, this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field. In particular, a global assumption on the divergence is needed to obtain a proper blow-up of the trajectories.Comment: 38 page