Georgia Institute of Technology College of Sciences

In this talk, I will describe several models arising in congested transport problems. I will first describe static models which lead to some highly degenerate elliptic PDEs. In the second part of the talk, I will address dynamic models which can be seen as a generalization of the Benamou-Brenier formulation of the quadratic optimal transport problem and will discuss the existence and regularity of the adjoint state. The talk will be based on several joint works with Lorenzo Brasco, Pierre Cardaliaguet, Bruno Nazaret and Filippo Santambrogio.

This is a joint work with F.~Nazarov and A.~Volberg.Let $s\in(1,2)$, and let $\mu$ be a finite positive Borel measure in $\mathbb R^2$ with $\mathcal H^s(\supp\mu)<+\infty$. We prove that if the lower $s$-density of $\mu$ is+equal to zero $\mu$-a.~e. in $\mathbb R^2$, then$\|R\mu\|_{L^\infty(m_2)}=\infty$, where $R\mu=\mu\ast\frac{x}{|x|^{s+1}}$ and $m_2$ is the Lebesque measure in $\mathbb R^2$. Combined with known results of Prat and+Vihtil\"a, this shows that for any noninteger $s\in(0,2)$ and any finite positive Borel measure in $\mathbb R^2$ with $\mathcal H^s(\supp\mu)<+\infty$, we have+$\|R\mu\|_{L^\infty(m_2)}=\infty$.Also I will tell about the resent result of Ben Jaye, as well as about open problems.

Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a result such as a comparison theorem which can deal with only measure theoretic norms (e.g. L-n and L-infinity) of the right hand side of the equation has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We will discuss this estimate in the context of fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])

Guided by direct experiments on many-legged animals, mathematical models and physical models (robots), we postulate a hierarchical family of control loops that necessarily include constraints of the body's mechanics. At the lowest end of this neuromechanical hierarchy, we hypothesize the primacy of mechanical feedback - neural clocks exciting tuned muscles acting through chosen skeletal postures. Control algorithms appear embedded in the form and skeleton of the animal itself. The control potential of muscles must be realized through complex, viscoelastic bodies. Bodies can absorb and redirect energy for transitions. Tails can be used as inertial control devices. On top of this physical layer reside sensory feedback driven reflexes that increase an animal's stability further and, at the highest level, environmental sensing that operates on a stride-to-stride timescale to direct the animal's body. Most importantly, locomotion requires an effective interaction with the environment. Understanding control requires understanding the coupling to environment. Amazing feet permit creatures such as geckos to climb up walls at over meter per second without using claws, glue or suction - just molecular forces using hairy toes. Fundamental principles of animal locomotion have inspired the design of self-clearing dry adhesives and autonomous legged robots such as the Ariel, Mecho-gecko, Sprawl, RHex, RiSE and Stickybot that can aid in search and rescue, inspection, detection and exploration.