On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations

We give a simplified proof of regularizing effects for first-order Hamilton-Jacobi Equations of the form $u\_t+H(x,t,Du)=0$ in $\R^N\times(0,+\infty)$ in the case where the idea is to first estimate $u\_t$. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an H\''older regularizing effect in space following a result of L. C. Evans and M. R. James

Survey of the galactic disk from 1 = -150 deg to 1 = 82 deg in the submillimeter range

The first almost complete survey of the galactic disk from 1 = -150 deg to 1 = 82 deg in the submillimeter range (effective wavelength = 380 microns), performed with the AGLAE balloon-borne instrument modified to include a submillimeter channel, is reported. The instrumentation and observational procedures are described, as are the signal processing and calibration. The results are presented as a profile of the submillimeter brightness of the galactic disk displayed as a function of the galactic longitude. This profile exhibits diffuse emission all along the disk with bright peaks associated with resolved sources. The averaged galactic spectrum is in agreement with a temperature distribution of the interstellar cold dust

Value function for regional control problems via dynamic programming and Pontryagin maximum principle

In this paper we focus on regional deterministic optimal control problems, i.e., problems where the dynamics and the cost functional may be different in several regions of the state space and present discontinuities at their interface. Under the assumption that optimal trajectories have a locally finite number of switchings (no Zeno phenomenon), we use the duplication technique to show that the value function of the regional optimal control problem is the minimum over all possible structures of trajectories of value functions associated with classical optimal control problems settled over fixed structures, each of them being the restriction to some submanifold of the value function of a classical optimal control problem in higher dimension.The lifting duplication technique is thus seen as a kind of desingularization of the value function of the regional optimal control problem. In turn, we extend to regional optimal control problems the classical sensitivity relations and we prove that the regularity of this value function is the same (i.e., is not more degenerate) than the one of the higher-dimensional classical optimal control problem that lifts the problem

A Bellman approach for two-domains optimal control problems in RN\R^N

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space $\R^N$. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness

H\^older continuity of solutions of second-order non-linear elliptic integro-differential equations

This paper is concerned with H\"older regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth conditions on the equation. These results are concerned with a large class of integro-differential operators whose singular measures depend on $x$ and also a large class of equations, including Bellman-Isaacs Equations

A review of test protocols for assessing coating performance of water ballast tank coatings

Concerns on corrosion and effective coating protection of double hull tankers and bulk carriers in service have been raised especially in water ballast tanks (WBTs). Test protocols/methodologies specifically that which is incorporated in the International Maritime Organisation (IMO), Performance Standard for Protective Coatings for Dedicated Sea Water ballast tanks (PSPC) are being used to assess and evaluate the performance of the coatings for type approval prior to their application in WBTs. However, some of the type approved coatings may be applied as very thick films to less than ideally prepared steel substrates in the WBT. As such films experience hygrothermal cycling from operating and environmental conditions, they become embrittled which may ultimately result in cracking. This embrittlement of the coatings is identified as an undesirable feature in the PSPC but is not mentioned in the test protocols within it. There is therefore renewed industrial research aimed at understanding this issue in order to eliminate cracking and achieve the intended coating lifespan of 15 years in good condition. This paper will critically review test protocols currently used for assessing and evaluating coating performance, particularly the IMO PSPC

An Illustrated Guide of the Modern Approches of Hamilton-Jacobi Equations and Control Problems with Discontinuities

This manuscript is a project of book on Hamilton-Jacobi Equations and Control Problems with discontinuities.For the time being, it is composed of 4 parts: Part I is a toolbox with key results which are used in all the other parts. Part II describes several approaches which were introduced recently to treat problems involving co-dimension 1 discontinuities or networks. Part III concerns stratified problems in $R^N$, i.e. problems with discontinuities of any co-dimensions. Part IV addresses the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions.This second version is still imperfect and we will welcome any (fair) comment on it. In this second version, besides of improving few pointsin the first one, we have added applications to KPP (Kolmogorov-Petrovsky-Piskunov) type problems and started to develop the use of stratified solutions to treat problems with boundary conditions (Dirichlet, Neumann and mixed boundary conditions),where both the boundary may be non-smooth and the data may present discontinuities. Of course, for all these questions, we only provide examples of what can be done since developing a whole theory would be too long, out of the scope of this book and probably a little bit beyond what we are able to do up to now.In these two directions, we address, in particular, a new question which is interesting when considering applications: under which conditions can one prove that Ishii's viscosity solutions are stratified solutions? The interest of this question is clear: when these conditions hold, one can take advantages of both the nice stability properties of classical viscosity solutions and the general comparison result for stratified solutions. A priori the third version (next release planned in june 2020) will probably be the last one: besides of improving the second version, we plan to address Large Deviations and homogenization problems. This will be the occasion to test the results we have presented so far and to improve them

Commercial shipping in the Arctic : new perspectives, challenges and regulations

Maritime traffic is increasing in Arctic seas in the context of climate change. The rapid melting of sea ice led to the widespread belief that traffic was set to rapidly expand, challenging Canadian and Russian-claimed sovereignties over their respective Arctic passage, and underlining the risk posed by such a traffic in a risky but fragile environment. If projections on potential traffic for the medium term are probably exaggerated, the increasing traffic nevertheless challenges the adequacy of the regulatory framework