53 research outputs found
(Almost) Everything You Always Wanted to Know About Deterministic Control Problems in Stratified Domains
We revisit the pioneering work of Bressan \& Hong on deterministic control
problems in stratified domains, i.e. control problems for which the dynamic and
the cost may have discontinuities on submanifolds of R N . By using slightly
different methods, involving more partial differential equations arguments, we
(i) slightly improve the assumptions on the dynamic and the cost; (ii) obtain a
comparison result for general semi-continuous sub and supersolutions (without
any continuity assumptions on the value function nor on the
sub/supersolutions); (iii) provide a general framework in which a stability
result holds
Large Deviations estimates for some non-local equations I. Fast decaying kernels and explicit bounds
We study large deviations for some non-local parabolic type equations. We
show that, under some assumptions on the non-local term, problems defined in a
bounded domain converge with an exponential rate to the solution of the problem
defined in the whole space. We compute this rate in different examples, with
different kernels defining the non-local term, and it turns out that the
estimate of convergence depends strongly on the decay at infinity of that
kernel
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 in
in the case where the idea is to first estimate .
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
On nonlocal quasilinear equations and their local limits
We introduce a new class of quasilinear nonlocal operators and study
equations involving these operators. The operators are degenerate elliptic and
may have arbitrary growth in the gradient. Included are new nonlocal versions
of p-Laplace, -Laplace, mean curvature of graph, and even strongly
degenerate operators, in addition to some nonlocal quasilinear operators
appearing in the existing literature. Our main results are comparison,
uniqueness, and existence results for viscosity solutions of linear and fully
nonlinear equations involving these operators. Because of the structure of our
operators, especially the existence proof is highly non-trivial and
non-standard. We also identify the conditions under which the nonlocal
operators converge to local quasilinear operators, and show that the solutions
of the corresponding nonlocal equations converge to the solutions of the local
limit equations. Finally, we give a (formal) stochastic representation formula
for the solutions and provide many examples
Fundamental solutions and singular shocks in scalar conservation laws
We study the existence and non-existence of fundamental solutions for the scalar conservation laws ut + f(u)x = 0; related to convexity assumptions on f. We also study the limits of those solutions as the initial mass goes to infinity. We especially prove the existence of so-called Friendly Giants and Infinite Shock Solutions according to the convexity of f, which generalize the explicit power case f(u) = um. We introduce an extended notion of solution and entropy criterion to allow infinite shocks in the theory, and the initial data also has to be understood in a generalized sense, since locally infinite measures appear.We study the existence and non-existence of fundamental solutions for the scalar conservation laws ut + f (u)x = 0, related to convexity assumptions on f . We also study the limits of those solu tions as the initial mass goes to infinity. We especially prove the existence of so-called Friend ly Giants and Infinite Shock Solutions according to the convex ity of f , which generalize the explicit power case f (u) = um . We introduce an extended notion of solution and entropy criterion to allow infinite shocks in the theory, and the initial data also has to be understood in a generalized sense, since locally infinite measures appear. 2000 Mathematics Sub ject Classification: 35L60, 35L67
A nonlocal two phase Stefan problem
We study a nonlocal version of the two-phase Stefan problem, which models a
phase transition problem between two distinct phases evolving to distinct heat
equations. Mathematically speaking, this consists in deriving a theory for
sign-changing solutions of the equation, ut = J * v - v, v = {\Gamma}(u), where
the monotone graph is given by {\Gamma}(s) = sign(s)(|s|-1)+ . We give general
results of existence, uniqueness and comparison, in the spirit of [2]. Then we
focus on the study of the asymptotic behaviour for sign-changing solutions,
which present challenging difficulties due to the non-monotone evolution of
each phase
A Bellman approach for two-domains optimal control problems in
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
. 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
The Dirichlet problem for some nonlocal diffusion equations
International audienceWe study the Dirichlet problem for the non-local diffusion equation u_t=\int\{u(x+z,t)-u(x,t)\}\dmu(z), where is a function and on '' has to be understood in a non-classical sense. We prove existence and uniqueness results of solutions in this setting. Moreover, we prove that our solutions coincide with those obtained through the standard ``vanishing viscosity method'', but show that a boundary layer occurs: the solution does not take the boundary data in the classical sense on , a phenomenon related to the non-local character of the equation. Finally, we show that in a bounded domain, some regularization may occur, contrary to what happens in the whole space
Unbounded solutions of the nonlocal heat equation
We consider the Cauchy problem posed in the whole space for the following
nonlocal heat equation: u_t = J * u - u, where J is a symmetric continuous
probability density. Depending on the tail of J, we give a rather complete
picture of the problem in optimal classes of data by: (i) estimating the
initial trace of (possibly unbounded) solutions; (ii) showing existence and
uniqueness results in a suitable class; (iii) giving explicit unbounded
polynomial solutions
- …