[Show abstract][Hide abstract] ABSTRACT: 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 $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$
elementary symmetric polynomial of eigenvalues of the Hessian matrix and the
datum $f$ 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 $\alpha \in \mathbb{N}$
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.
[Show abstract][Hide abstract] ABSTRACT: We study a higher order parabolic partial differential equation that arises
in the context of condensed matter physics. It is a fourth order semilinear
equation whose nonlinearity is the determinant of the Hessian matrix of the
solution. We consider this model in a bounded domain of the real plane and
study its stationary solutions both when the geometry of this domain is
arbitrary and when it is the unit ball and the solution is radially symmetric.
We also consider the initial-boundary value problem for the full parabolic
equation. We summarize our results on existence of solutions in these cases and
propose an open problem related to the existence of self-similar solutions.
[Show abstract][Hide abstract] ABSTRACT: This work presents the construction of the existence theory of radial
solutions to the elliptic equation \begin{equation}\nonumber \Delta^2 u =
(-1)^k S_k[u] + \lambda f(x), \qquad x \in B_1(0) \subset \mathbb{R}^N,
\end{equation} provided either with Dirichlet boundary conditions
\begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0),
\end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u =
\Delta u = 0, \qquad x \in \partial B_1(0), \end{equation} where the
$k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial
of eigenvalues of the Hessian matrix and the datum $f \in L^1(B_1(0))$ while
$\lambda \in \mathbb{R}$. We prove the existence of a Carath\'eodory solution
to these boundary value problems that is unique in a certain neighborhood of
the origin provided $|\lambda|$ is small enough. Moreover, we prove that the
solvability set of $\lambda$ is finite, giving an explicity bound of the
extreme value.
Journal of Differential Equations 03/2015; DOI:10.1016/j.jde.2015.04.001 · 1.68 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: On considère une équation différentielle qui décrit la croissance épitaxiale d'une couche rugueuse de grains. Il s'agit d'une équation parabolique pour laquelle l'évolution est gouvernée par une compétiton entre le déterminant Hessien de la solution et l'opérateur biharmonique. Ce modèle peut présenter une structure de flux gradient suivant les conditions au bord. On étend d'abord des résultats précédents sur l'existence de solutions stationnaires pour ce modèle avec des conditions de Dirichlet. Pour l'équation d'évolution on démontre l'existence locale de solutions pour tout donné initial et l'existence globale pour des donnés suffisamment démontre. En exploitant les conditions au bord et la structure variationnelle de l'équation, suivant la taille de la donné initial on démontre l'explosion en temps fini et/ou la convergence à une solution stationnaire pour les solutions globales.
Journal de Mathématiques Pures et Appliqués 09/2014; 103(4). DOI:10.1016/j.matpur.2014.09.007 · 1.68 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this work we study the stochastic process of two-species coagulation. This
process consists in the aggregation dynamics taking place in a ring. Particles
and clusters of particles are set in this ring and they can move either
clockwise or counterclockwise. They have a probability to aggregate forming
larger clusters when they collide with another particle or cluster. We study
the stochastic process both analytically and numerically. Analytically, we
derive a kinetic theory which approximately describes the process dynamics. One
of our strongest assumptions in this respect is the so called well-stirred
limit, that allows neglecting the appearance of spatial coordinates in the
theory, so this becomes effectively reduced to a zeroth dimensional model. We
determine the long time behavior of such a model, making emphasis in one
special case in which it displays self-similar solutions. In particular these
calculations answer the question of how the system gets ordered, with all
particles and clusters moving in the same direction, in the long time. We
compare our analytical results with direct numerical simulations of the
stochastic process and both corroborate its predictions and check its
limitations. In particular, we numerically confirm the ordering dynamics
predicted by the kinetic theory and explore properties of the realizations of
the stochastic process which are not accessible to our theoretical approach.
Kinetic and Related Models 04/2014; 7(2). DOI:10.3934/krm.2014.7.253 · 1.13 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The existence of stationary radial solutions to a partial differential
equation arising in the theory of epitaxial growth is studied. Our results
depend on the size of a parameter that plays the role of the velocity at which
mass is introduced into the system. For small values of this parameter we prove
existence of solutions to this boundary value problem. For large values of the
same parameter we prove nonexistence of solutions. We also provide rigorous
bounds for the values of this parameter which separate existence from
nonexistence. The proofs come as a combination of several differential
inequalities and the method of upper and lower functions.
Mathematical Methods in the Applied Sciences 04/2014; To appear(6). DOI:10.1002/mma.2836 · 0.92 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Recently, a variational approach has been introduced for the paradigmatic
Kardar--Parisi--Zhang (KPZ) equation. Here we review that approach, together
with the functional Taylor expansion that the KPZ nonequilibrium potential
(NEP) admits. Such expansion becomes naturally truncated at third order, giving
rise to a nonlinear stochastic partial differential equation to be regarded as
a gradient-flow counterpart to the KPZ equation. A dynamic renormalization
group analysis at one-loop order of this new mesoscopic model yields the KPZ
scaling relation alpha+z=2, as a consequence of the exact cancelation of the
different contributions to vertex renormalization. This result is quite
remarkable, considering the lower degree of symmetry of this equation, which is
in particular not Galilean invariant. In addition, this scheme is exploited to
inquire about the dynamical behavior of the KPZ equation through a
path-integral approach. Each of these aspects offers novel points of view and
sheds light on particular aspects of the dynamics of the KPZ equation.
[Show abstract][Hide abstract] ABSTRACT: We present the formal geometric derivation of a nonequilibrium growth model
that takes the form of a parabolic partial differential equation. Subsequently,
we study its stationary radial solutions by means of variational techniques.
Our results depend on the size of a parameter that plays the role of the
strength of forcing. For small forcing we prove the existence and multiplicity
of solutions to the elliptic problem. We discuss our results in the context of
nonequilibrium statistical mechanics.
European Journal of Applied Mathematics 09/2013; 24(03). DOI:10.1017/S0956792512000484 · 0.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We discuss a tentative path-integral approach to numerically follow the scaling properties of the mean rugosity (and other typical averages) of an interface whose growth is described by the Kardar-Parisi-Zhang equation. It resorts to functional minimization and a cellular automata-like algorithm, and can be regarded as a kind of importance-sampling approach. This method is intended to predict the crossover time as a function of the coefficient of the nonlinear term, through the comparison of the weight of the different terms in the "stochastic action".
Acta Physica Polonica Series B 05/2013; 44(5):889. DOI:10.5506/APhysPolB.44.889 · 0.85 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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, Delta(2)u = det(D(2)u) + lambda f, x is an element of Omega subset of R-2, conditions on partial derivative Omega. The framework to study the problem deeply depends on the boundary conditions. (c) 2013 Elsevier Inc. All rights reserved.
[Show abstract][Hide abstract] ABSTRACT: The effect of a uniform dilation of space on stochastically driven nonlinear
field theories is examined. This theoretical question serves as a model problem
for examining the properties of nonlinear field theories embedded in expanding
Euclidean Friedmann-Lema\^{\i}tre-Robertson-Walker metrics in the context of
cosmology, as well as different systems in the disciplines of statistical
mechanics and condensed matter physics. Field theories are characterized by the
speed at which they propagate correlations within themselves. We show that for
linear field theories correlations stop propagating if and only if the speed at
which the space dilates is higher than the speed at which correlations
propagate. The situation is in general different for nonlinear field theories.
In this case correlations might stop propagating even if the velocity at which
space dilates is lower than the velocity at which correlations propagate. In
particular, these results imply that it is not possible to characterize the
dynamics of a nonlinear field theory during homogeneous spatial dilation {\it a
priori}. We illustrate our findings with the nonlinear Kardar-Parisi-Zhang
equation.
Journal of Physics A Mathematical and Theoretical 08/2012; 46(35). DOI:10.1088/1751-8113/46/35/355403 · 1.58 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: To this day, computer models for stromatolite formation have made substantial use of the Kardar-Parisi-Zhang (KPZ) equation. Oddly enough, these studies yielded mutually exclusive conclusions about the biotic or abiotic origin of such structures. We show in this paper that, at our current state of knowledge, a purely biotic origin for stromatolites can neither be proved nor disproved by means of a KPZ-based model. What can be shown, however, is that whatever their (biotic or abiotic) origin might be, some morphologies found in actual stromatolite structures (e.g. overhangs) cannot be formed as a consequence of a process modelled exclusively in terms of the KPZ equation and acting over sufficiently large times. This suggests the need to search for alternative mathematical approaches to model these structures, some of which are discussed in this paper.
Journal of The Royal Society Interface 10/2011; 9(70):1051-62. DOI:10.1098/rsif.2011.0516 · 3.92 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scale-dependent fractality, and suggests the advent of a self-similar fractal dimension. The center-of-mass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the long-range surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilution-free universality class.
[Show abstract][Hide abstract] ABSTRACT: Starting from a variational formulation of the Kardar-Parisi-Zhang (KPZ) equation, we point out some strong constraints and consistency tests, to be fulfilled by real-space discretization schemes. In the light of these findings, the mainstream opinion on the relevance of Galilean invariance and the fluctuation--dissipation theorem (peculiar of 1D) is challenged.
[Show abstract][Hide abstract] ABSTRACT: The stochastic nonlinear partial differential equation known as the Kardar-Parisi-Zhang (KPZ) equation is a highly successful phenomenological mesoscopic model of surface and interface growth processes. Its suitability for analytical work, its explicit symmetries and its prediction of an exact dynamic scaling relation for a one-dimensional substratum led people to adopt it as a 'standard' model in the field during the last quarter of a century. At the same time, several conjectures deserving closer scrutiny were established as dogmas throughout the community. Among these, we find the beliefs that 'genuine' non-equilibrium processes are non-variational in essence, and that the exactness of the dynamic scaling relation owes its existence to a Galilean symmetry. Additionally, the equivalence among planar and radial interface profiles has been generally assumed in the literature throughout the years. Here--among other topics--we introduce a variational formulation of the KPZ equation, remark on the importance of consistency in discretization and challenge the mainstream view on the necessity for scaling of both Galilean symmetry and the one-dimensional fluctuation-dissipation theorem. We also derive the KPZ equation on a growing domain as a first approximation to radial growth, and outline the differences with respect to the classical case that arises in this new situation.
Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 01/2011; 369(1935):396-411. DOI:10.1098/rsta.2010.0259 · 2.15 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.
[Show abstract][Hide abstract] ABSTRACT: We explore the self-organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole. To this end, we employ the widespread example of coagulation kinetics, and characterize which interaction types lead to consensus formation and which do not, as well as the corresponding different macroscopic patterns. The crucial technical point is extending the usual one species coagulation dynamics to the two species one. This is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning. The corresponding solutions are calculated in the long time limit, in which consensus may or may not be reached. The lack of consensus is characterized by means of scaling limits of the solutions. The possible applications of our results to some topics in which consensus reaching is fundamental, such as collective animal motion and opinion spreading dynamics, are also outlined.
[Show abstract][Hide abstract] ABSTRACT: In order to perform numerical simulations of the Kardar-Parisi-Zhang (KPZ) equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a Hopf-Cole transformation applied to a diffusion equation (with multiplicative noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on space and the Hopf-Cole transformation is local both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between real-space and pseudospectral discrete representations. In addition we discuss the relevance of the Galilean-invariance violation in these consistent discretization schemes and the alleged conflict of standard discretization with the fluctuation-dissipation theorem, peculiar of one dimension.