402 research outputs found
An overview of Viscosity Solutions of Path-Dependent PDEs
This paper provides an overview of the recently developed notion of viscosity
solutions of path-dependent partial di erential equations. We start by a quick
review of the Crandall- Ishii notion of viscosity solutions, so as to motivate
the relevance of our de nition in the path-dependent case. We focus on the
wellposedness theory of such equations. In partic- ular, we provide a simple
presentation of the current existence and uniqueness arguments in the
semilinear case. We also review the stability property of this notion of
solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme
approximation method. Our results rely crucially on the theory of optimal
stopping under nonlinear expectation. In the dominated case, we provide a
self-contained presentation of all required results. The fully nonlinear case
is more involved and is addressed in [12]
Existence, uniqueness and structure of second order absolute minimisers
Let ⊆ Rn be a bounded open C1,1 set. In this paper we prove the existence
of a unique second order absolute minimiser u∞ of the functional
E∞(u, O) := F(·, u)L∞(O), O ⊆ measurable,
with prescribed boundary conditions for u and Du on ∂ and under natural assumptions
on F. We also show that u∞ is partially smooth and there exists a harmonic
function f∞ ∈ L1() such that
F(x, u∞(x)) = e∞ sgn
f∞(x)
for all x ∈ { f∞ = 0}, where e∞ is the infimum of the global energy
Global Continua of Positive Equilibria for some Quasilinear Parabolic Equation with a Nonlocal Initial Condition
This paper is concerned with a quaslinear parabolic equation including a
nonlinear nonlocal initial condition. The problem arises as equilibrium
equation in population dynamics with nonlinear diffusion. We make use of global
bifurcation theory to prove existence of an unbounded continuum of positive
solutions
Necrotic tumor growth: an analytic approach
The present paper deals with a free boundary problem modeling the growth
process of necrotic multi-layer tumors. We prove the existence of flat
stationary solutions and determine the linearization of our model at such an
equilibrium. Finally, we compute the solutions of the stationary linearized
problem and comment on bifurcation.Comment: 14 pages, 3 figure
Energy solutions to one-dimensional singular parabolic problems with data are viscosity solutions
We study one-dimensional very singular parabolic equations with periodic
boundary conditions and initial data in , which is the energy space. We
show existence of solutions in this energy space and then we prove that they
are viscosity solutions in the sense of Giga-Giga.Comment: 15 page
Evolutionary game of coalition building under external pressure
We study the fragmentation-coagulation (or merging and splitting)
evolutionary control model as introduced recently by one of the authors, where
small players can form coalitions to resist to the pressure exerted by the
principal. It is a Markov chain in continuous time and the players have a
common reward to optimize. We study the behavior as grows and show that the
problem converges to a (one player) deterministic optimization problem in
continuous time, in the infinite dimensional state space
Tightness for a stochastic Allen--Cahn equation
We study an Allen-Cahn equation perturbed by a multiplicative stochastic
noise which is white in time and correlated in space. Formally this equation
approximates a stochastically forced mean curvature flow. We derive uniform
energy bounds and prove tightness of of solutions in the sharp interface limit,
and show convergence to phase-indicator functions.Comment: 27 pages, final Version to appear in "Stochastic Partial Differential
Equations: Analysis and Computations". In Version 4, Proposition 6.3 is new.
It replaces and simplifies the old propositions 6.4-6.
On the dynamics of WKB wave functions whose phase are weak KAM solutions of H-J equation
In the framework of toroidal Pseudodifferential operators on the flat torus
we begin by proving the closure under
composition for the class of Weyl operators with
simbols . Subsequently, we
consider when where and we exhibit the toroidal version of the
equation for the Wigner transform of the solution of the Schr\"odinger
equation. Moreover, we prove the convergence (in a weak sense) of the Wigner
transform of the solution of the Schr\"odinger equation to the solution of the
Liouville equation on written in the measure sense.
These results are applied to the study of some WKB type wave functions in the
Sobolev space with phase functions in the class
of Lipschitz continuous weak KAM solutions (of positive and negative type) of
the Hamilton-Jacobi equation for with , and to the study of the
backward and forward time propagation of the related Wigner measures supported
on the graph of
On the upstream mobility scheme for two-phase flow in porous media
When neglecting capillarity, two-phase incompressible flow in porous media is
modelled as a scalar nonlinear hyperbolic conservation law. A change in the
rock type results in a change of the flux function. Discretizing in
one-dimensional with a finite volume method, we investigate two numerical
fluxes, an extension of the Godunov flux and the upstream mobility flux, the
latter being widely used in hydrogeology and petroleum engineering. Then, in
the case of a changing rock type, one can give examples when the upstream
mobility flux does not give the right answer.Comment: A preprint to be published in Computational Geoscience
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