1,722 research outputs found
On polyharmonic regularizations of Hessian equations: Variational methods
This work is devoted to the study of the boundary value problem
\begin{eqnarray}\nonumber (-1)^\alpha \Delta^\alpha u = (-1)^k S_k[u] + \lambda
f, \qquad x &\in& \Omega \subset \mathbb{R}^N, \\ \nonumber u = \partial_n u =
\partial_n^2 u = \cdots = \partial_n^{\alpha-1} u = 0, \qquad x &\in& \partial
\Omega, \end{eqnarray} where the Hessian is the
elementary symmetric polynomial of eigenvalues of the Hessian matrix and the
datum obeys suitable summability properties. We prove the existence of at
least two solutions, of which at least one is isolated, strictly by means of
variational methods. We look for the optimal values of
that allow the construction of such an existence and multiplicity theory and
also investigate how a weaker definition of the nonlinearity permits improving
these results
Shock wave formation in Rosenau's extended hydrodynamics
We study the extended hydrodynamics proposed by Philip Rosenau [Phys. Rev. A
40, 7193 (1989)] in the context of a regularization of the Chapman-Enskog
expansion. We are able to prove that shock waves appear in finite time in
Rosenau's extended Burgers' equation, and we discuss the physical implications
of this fact and its connection with a possible extension of hydrodynamics to
the short wavelength domain
Addendum to the Reply Comment [Phys. Rev. Lett. 102, 139602 (2009), arXiv:0811.0518] on ''Dynamic Scaling of Non-Euclidean Interfaces''
This is an addendum to the Reply Comment [Phys. Rev. Lett. 102, 139602
(2009), arXiv:0811.0518] to Comment [Phys. Rev. Lett. 102, 139601 (2009),
arXiv:0810.4791] on Letter [Phys. Rev. Lett. 100, 116101 (2008),
arXiv:0804.1898]
Reply to the revised Comment [PRL 102, 139601; arXiv:0810.4791] on ''Dynamic Scaling of Non-Euclidean Interfaces''
This is the Reply to the revised Comment [Phys. Rev. Lett. 102, 139601;
arXiv:0810.4791] by Joachim Krug on our paper: C. Escudero, Phys. Rev. Lett.
100, 116101 (2008); arXiv:0804.1898.Comment: Newer version than the one appeared in PR
A simple comparison between Skorokhod & Russo-Vallois integration for insider trading
We consider a simplified version of the problem of insider trading in a
financial market. We approach it by means of anticipating stochastic calculus
and compare the use of the Skorokhod and the Russo-Vallois forward integrals
within this context. We conclude that, while the forward integral yields
results with a clear financial meaning, the Skorokhod integral does not provide
a suitable formulation for this problem
Stochastic resonance due to internal noise in reaction kinetics
We study a reaction model that presents stochastic resonance purely due to
internal noise. This means that the only source of fluctuations comes from the
discrete character of the reactants, and no more noises enter into the system.
Our analysis reveals that the phenomenon is highly complex, and that is
generated by the interplay of different stochasticity at the three fixed points
of a bistable system
Stochastic growth of radial clusters: weak convergence to the asymptotic profile and implications for morphogenesis
The asymptotic shape of randomly growing radial clusters is studied. We pose
the problem in terms of the dynamics of stochastic partial differential
equations. We concentrate on the properties of the realizations of the
stochastic growth process and in particular on the interface fluctuations. Our
goal is unveiling under which conditions the developing radial cluster
asymptotically weakly converges to the concentrically propagating spherically
symmetric profile or either to a symmetry breaking shape. We demonstrate that
the long range correlations of the surface fluctuations obey a self-affine
scaling and that scale invariance is achieved by means of the introduction of
three critical exponents. These are able to characterize the large scale
dynamics and to describe those regimes dominated by system size evolution. The
connection of these results with mathematical morphogenetic problems is also
outlined
Origins of scaling relations in nonequilibrium growth
Scaling and hyperscaling laws provide exact relations among critical
exponents describing the behavior of a system at criticality. For
nonequilibrium growth models with a conserved drift there exist few of them.
One such relation is , found to be inexact in a renormalization
group calculation for several classical models in this field. Herein we focus
on the two-dimensional case and show that it is possible to construct conserved
surface growth equations for which the relation is exact in the
renormalization group sense. We explain the presence of this scaling law in
terms of the existence of geometric principles dominating the dynamics
Some fourth order nonlinear elliptic problems related to epitaxial growth
This paper deals with some mathematical models arising in the theory of
epitaxial growth of crystal. We focalize the study on a stationary problem
which presents some analytical difficulties. We study the existence of
solutions. The central model in this work is given by the following fourth
order elliptic equation, The framework to
study the problem deeply depends on the boundary conditions
It\^o vs Stratonovich in the presence of absorbing states
It is widely assumed that there exists a simple transformation from the It\^o
interpretation to the one by Stratonovich and back for any stochastic
differential equation of applied interest. While this transformation exists
under suitable conditions, and transforms one interpretation into the other at
the price of modifying the drift of the equation, it cannot be considered
universal. We show that a class of stochastic differential equations,
characterized by the presence of absorbing states and of interest in
applications, does not admit such a transformation. In particular, formally
applying this transformation may lead to the disappearance of some absorbing
states. In turn, this modifies the long-time, and even the intermediate-time,
behavior of the solutions. The number of solutions can also be modified by the
unjustified application of the mentioned transformation, as well as by a change
in the interpretation of the noise. We discuss how these facts affect the
classical debate on the It\^o vs Stratonovich dilemma.Comment: Accepted in the Journal of Mathematical Physic
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