46 research outputs found
Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates
We consider homogenization for weakly coupled systems of Hamilton--Jacobi
equations with fast switching rates. The fast switching rate terms force the
solutions converge to the same limit, which is a solution of the effective
equation. We discover the appearance of the initial layers, which appear
naturally when we consider the systems with different initial data and analyze
them rigorously. In particular, we obtain matched asymptotic solutions of the
systems and rate of convergence. We also investigate properties of the
effective Hamiltonian of weakly coupled systems and show some examples which do
not appear in the context of single equations.Comment: final version, to appear in Arch. Ration. Mech. Ana
Aubry sets for weakly coupled systems of Hamilton--Jacobi equations
We introduce a notion of Aubry set for weakly coupled systems of
Hamilton--Jacobi equations on the torus and characterize it as the region where
the obstruction to the existence of globally strict critical subsolutions
concentrates. As in the case of a single equation, we prove the existence of
critical subsolutions which are strict and smooth outside the Aubry set. This
allows us to derive in a simple way a comparison result among critical sub and
supersolutions with respect to their boundary data on the Aubry set, showing in
particular that the latter is a uniqueness set for the critical system. We also
highlight some rigidity phenomena taking place on the Aubry set.Comment: 35 pages v.2 the introduction has been rewritten and shortened. Some
proofs simplified. Corrections and references added. Corollary 5.3 added
stating antisymmetry of the Ma\~n\'e matrix on points of the Aubry set.
Section 6 contains a new example
Stochastic homogenization for functionals with anisotropic rescaling and non-coercive Hamilton-Jacobi equations
We study the stochastic homogenization for a Cauchy problem for a first-order
Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient
variable. We look at Hamiltonians like where
is a matrix associated to a Carnot group. The rescaling considered
is consistent with the underlying Carnot group structure, thus anisotropic. We
will prove that under suitable assumptions for the Hamiltonian, the solutions
of the -problem converge to a deterministic function which can be
characterized as the unique (viscosity) solution of a suitable deterministic
Hamilton-Jacobi problem
International Conference on Nonlinear Differential Equations and Applications
Dear Participants, Colleagues and Friends
It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA).
This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics.
The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area
Recommended from our members
Variational Methods for Evolution
The workshop brought together researchers from geometry, nonlinear functional analysis, calculus of variations, partial differential equations, and stochastics around a common topic: systems whose evolution is driven by variational principles such as gradient or Hamiltonian systems.
The talks covered a wide range of topics, including variational tools such as incremental minimization approximations, Gamma convergence, and optimal transport, reaction-diffusion systems, singular perturbation and homogenization, rate-independent models for visco-plasticity and fracture, Hamiltonian and hyperbolic systems, stochastic models and new gradient structures for Markov processes or variational large-deviation principles
Stochastic homogenization of Hamilton--Jacobi--Bellman equations on continuum percolation clusters
We prove homogenization properties of random Hamilton--Jacobi--Bellman (HJB) equations on continuum percolation clusters, almost surely w.r.t. the law of the environment when the origin belongs to the unbounded component in the continuum. Here, the viscosity term carries a degenerate matrix, the Hamiltonian is convex and coercive w.r.t. the degenerate matrix and the underlying environment is non-elliptic and its law is non-stationary w.r.t. the translation group. We do not assume uniform ellipticity inside the percolation cluster, nor any finite-range dependence (i.i.d.) assumption on the percolation models and the effective Hamiltonian admits a variational formula which reflects some key properties of percolation. The proof is inspired by a method of Kosygina--Rezakhanlou--Varadhan developed for the case of HJB equations with constant viscosity and uniformly coercive Hamiltonian in a stationary, ergodic and elliptic random environment. In the non-stationary and non-elliptic set up, we leverage the coercivity property of the underlying Hamiltonian as well as a relative entropy structure (both being intrinsic properties of HJB, in any framework) and make use of the random geometry of continuum percolation