Gradient estimates for semilinear elliptic systems and other related results

A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville Theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove in some particular cases the Liouville Theorem. Finally, we give an alternative form of the stress-energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.Comment: 16 page

Regulatory Barriers to Entry in Industrial Sectors

The entry of new competitors operates as a balancing force against high levels of industrial concentration and the abuse of dominant position by firms with large market shares. Entry increases supply, lowers prices, intensifies innovation and brings equilibrium to the markets that don’t operate in a socially desirable manner. This paper examines the impact of regulatory restrictions to the entry of new competitors in industrial sectors. It provides a short description of the 13 most important sources of regulatory barriers and assesses their role and importance as entry barriers. The conclusion is that regulatory restrictions can be a very important, almost insurmountable barrier to the entry of new competitors, but their role is not always socially harmful. The use of certain sources of regulatory barriers is effective in protecting social welfare instead of harming it. Barriers that promote new competition or are applied in order to protect consumer welfare are socially useful, while barriers that restrict competition and limit new competitor entry, in cases other than natural monopolies, are socially harmful.entry, competition, industry, barriers

Stochastic mirror descent dynamics and their convergence in monotone variational inequalities

We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system's controllable parameters are two variable weight sequences that respectively pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence of the process. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around this average.Comment: 23 pages; updated proofs in Section 3 and Section