1,790 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]
Focusing beams with widely varying current using fixed strenght quadrupoles for heavy-ion inertial fusion
A differential method for bounding the ground state energy
For a wide class of Hamiltonians, a novel method to obtain lower and upper
bounds for the lowest energy is presented. Unlike perturbative or variational
techniques, this method does not involve the computation of any integral (a
normalisation factor or a matrix element). It just requires the determination
of the absolute minimum and maximum in the whole configuration space of the
local energy associated with a normalisable trial function (the calculation of
the norm is not needed). After a general introduction, the method is applied to
three non-integrable systems: the asymmetric annular billiard, the many-body
spinless Coulombian problem, the hydrogen atom in a constant and uniform
magnetic field. Being more sensitive than the variational methods to any local
perturbation of the trial function, this method can used to systematically
improve the energy bounds with a local skilled analysis; an algorithm relying
on this method can therefore be constructed and an explicit example for a
one-dimensional problem is given.Comment: Accepted for publication in Journal of Physics
Recommended from our members
L-Infinity variational problems of maps and the Aronsson PDE system
By employing Aronsson's absolute minimizers of L â functionals, we prove that absolutely minimizing maps u:RnâRN solve a "tangential" Aronsson PDE system. By following Sheffield and Smart (2012) [24], we derive ÎŽ â with respect to the dual operator norm and show that such maps miss information along a hyperplane when compared to tight maps. We recover the lost term which causes non-uniqueness and derive the complete Aronsson system which has discontinuous coefficients. In particular, the Euclidean â-Laplacian is ÎŽ âu=DuâDu:D 2u+|Du| 2[Du] â„ÎŽu where [Du] â„ is the projection on the null space of Du â€. We demonstrate C â solutions having interfaces along which the rank of their gradient is discontinuous and propose a modification with C 0 coefficients which admits varifold solutions. Away from the interfaces, Aronsson maps satisfy a structural property of local splitting to 2 phases, a horizontal and a vertical; horizontally they possess gradient flows similar to the scalar case and vertically solve a linear system coupled by a scalar Hamilton Jacobi PDE. We also construct singular â-harmonic local C 1 diffeomorphisms and singular Aronsson maps. © 2012 Elsevier Inc
Explicit singular viscosity solutions of the Aronsson equation
We establish that when nâ„2 and HâC1(Rn) is a Hamiltonian such that some level set contains a line segment, the Aronsson equation D2u:Hp(Du)âHp(Du)=0 admits explicit entire viscosity solutions. They are superpositions of a linear part plus a Lipschitz continuous singular part which in general is non-C1 and nowhere twice differentiable. In particular, we supplement the C1 regularity result of Wang and Yu (2008) [11] by deducing that strict level convexity is necessary for C1 regularity of solutions. © 2011 AcadĂ©mie des sciences
On the Path Integral in Imaginary Lobachevsky Space
The path integral on the single-sheeted hyperboloid, i.e.\ in -dimensional
imaginary Lobachevsky space, is evaluated. A potential problem which we call
``Kepler-problem'', and the case of a constant magnetic field are also
discussed.Comment: 16 pages, LATEX, DESY 93-14
The heart of a convex body
We investigate some basic properties of the {\it heart}
of a convex set It is a subset of
whose definition is based on mirror reflections of euclidean
space, and is a non-local object. The main motivation of our interest for
is that this gives an estimate of the location of the
hot spot in a convex heat conductor with boundary temperature grounded at zero.
Here, we investigate on the relation between and the
mirror symmetries of we show that
contains many (geometrically and phisically) relevant points of
we prove a simple geometrical lower estimate for the diameter of
we also prove an upper estimate for the area of
when is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic
and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6
The Tychonoff uniqueness theorem for the G-heat equation
In this paper, we obtain the Tychonoff uniqueness theorem for the G-heat
equation
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