901 research outputs found
Supercritical biharmonic equations with power-type nonlinearity
The biharmonic supercritical equation , where and
, is studied in the whole space as well as in a
modified form with as right-hand-side with an additional
eigenvalue parameter in the unit ball, in the latter case together
with Dirichlet boundary conditions. As for entire regular radial solutions we
prove oscillatory behaviour around the explicitly known radial {\it singular}
solution, provided , where
is a further critical exponent, which was introduced in a recent work by
Gazzola and the second author. The third author proved already that these
oscillations do not occur in the complementing case, where .
Concerning the Dirichlet problem we prove existence of at least one singular
solution with corresponding eigenvalue parameter. Moreover, for the extremal
solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue
problem, we prove smoothness as long as
Well-posedness for a class of nonlinear degenerate parabolic equations
In this paper we obtain well-posedness for a class of semilinear weakly
degenerate reaction-diffusion systems with Robin boundary conditions. This
result is obtained through a Gagliardo-Nirenberg interpolation inequality and
some embedding results for weighted Sobolev spaces
On mathematical models for Bose-Einstein condensates in optical lattices (expanded version)
Our aim is to analyze the various energy functionals appearing in the physics
literature and describing the behavior of a Bose-Einstein condensate in an
optical lattice. We want to justify the use of some reduced models. For that
purpose, we will use the semi-classical analysis developed for linear problems
related to the Schr\"odinger operator with periodic potential or multiple wells
potentials. We justify, in some asymptotic regimes, the reduction to low
dimensional problems and analyze the reduced problems
Convergence of Ginzburg-Landau functionals in 3-d superconductivity
In this paper we consider the asymptotic behavior of the Ginzburg- Landau
model for superconductivity in 3-d, in various energy regimes. We rigorously
derive, through an analysis via {\Gamma}-convergence, a reduced model for the
vortex density, and we deduce a curvature equation for the vortex lines. In a
companion paper, we describe further applications to superconductivity and
superfluidity, such as general expressions for the first critical magnetic
field H_{c1}, and the critical angular velocity of rotating Bose-Einstein
condensates.Comment: 45 page
Local regularity for fractional heat equations
We prove the maximal local regularity of weak solutions to the parabolic
problem associated with the fractional Laplacian with homogeneous Dirichlet
boundary conditions on an arbitrary bounded open set
. Proofs combine classical abstract regularity
results for parabolic equations with some new local regularity results for the
associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756
Global attractors for Cahn-Hilliard equations with non constant mobility
We address, in a three-dimensional spatial setting, both the viscous and the
standard Cahn-Hilliard equation with a nonconstant mobility coefficient. As it
was shown in J.W. Barrett and J.W. Blowey, Math. Comp., 68 (1999), 487-517, one
cannot expect uniqueness of the solution to the related initial and boundary
value problems. Nevertheless, referring to J. Ball's theory of generalized
semiflows, we are able to prove existence of compact quasi-invariant global
attractors for the associated dynamical processes settled in the natural
"finite energy" space. A key point in the proof is a careful use of the energy
equality, combined with the derivation of a "local compactness" estimate for
systems with supercritical nonlinearities, which may have an independent
interest. Under growth restrictions on the configuration potential, we also
show existence of a compact global attractor for the semiflow generated by the
(weaker) solutions to the nonviscous equation characterized by a "finite
entropy" condition
Ginzburg-Landau model with small pinning domains
We consider a Ginzburg-Landau type energy with a piecewise constant pinning
term in the potential . The function is different from
1 only on finitely many disjoint domains, called the {\it pinning domains}.
These pinning domains model small impurities in a homogeneous superconductor
and shrink to single points in the limit ; here, \v is the inverse of
the Ginzburg-Landau parameter. We study the energy minimization in a smooth
simply connected domain with Dirichlet boundary
condition on \d \O, with topological degree {\rm deg}_{\d \O} (g) = d
>0. Our main result is that, for small \v, minimizers have distinct
zeros (vortices) which are inside the pinning domains and they have a degree
equal to 1. The question of finding the locations of the pinning domains with
vortices is reduced to a discrete minimization problem for a finite-dimensional
functional of renormalized energy. We also find the position of the vortices
inside the pinning domains and show that, asymptotically, this position is
determined by {\it local renormalized energy} which does not depend on the
external boundary conditions.Comment: 39 page
A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space
We consider the task of computing an approximate minimizer of the sum of a
smooth and non-smooth convex functional, respectively, in Banach space.
Motivated by the classical forward-backward splitting method for the
subgradients in Hilbert space, we propose a generalization which involves the
iterative solution of simpler subproblems. Descent and convergence properties
of this new algorithm are studied. Furthermore, the results are applied to the
minimization of Tikhonov-functionals associated with linear inverse problems
and semi-norm penalization in Banach spaces. With the help of
Bregman-Taylor-distance estimates, rates of convergence for the
forward-backward splitting procedure are obtained. Examples which demonstrate
the applicability are given, in particular, a generalization of the iterative
soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as
well as total-variation based image restoration in higher dimensions are
presented
Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case
We study the following nonlinear Schr\"{o}dinger system which is related to
Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1
u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in
\Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1}
u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in \om},\quad
u=v=0 \,\,\hbox{on \partial\om}.{cases}{displaymath} Here \om\subset \R^N
is a smooth bounded domain, is the Sobolev critical
exponent, -\la_1(\om)0 and , where
\lambda_1(\om) is the first eigenvalue of with the Dirichlet
boundary condition. When \bb=0, this is just the well-known Brezis-Nirenberg
problem. The special case N=4 was studied by the authors in (Arch. Ration.
Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher
dimensional case }. It is interesting that we can prove the existence
of a positive least energy solution (u_\bb, v_\bb) {\it for any } (which can not hold in the special case N=4). We also study the limit
behavior of (u_\bb, v_\bb) as and phase separation is
expected. In particular, u_\bb-v_\bb will converge to {\it sign-changing
solutions} of the Brezis-Nirenberg problem, provided . In case
\la_1=\la_2, the classification of the least energy solutions is also
studied. It turns out that some quite different phenomena appear comparing to
the special case N=4.Comment: 48 pages. This is a revised version of arXiv:1209.2522v1 [math.AP
Choices change the temporal weighting of decision evidence
Many decisions result from the accumulation of decision-relevant information (evidence) over time. Even when maximizing decision accuracy requires weighting all the evidence equally, decision-makers often give stronger weight to evidence occurring early or late in the evidence stream. Here, we show changes in such temporal biases within participants as a function of intermittent judgments about parts of the evidence stream. Human participants performed a decision task that required a continuous estimation of the mean evidence at the end of the stream. The evidence was either perceptual (noisy random dot motion) or symbolic (variable sequences of numbers). Participants also reported a categorical judgment of the preceding evidence half-way through the stream in one condition or executed an evidence-independent motor response in another condition. The relative impact of early versus late evidence on the final estimation flipped between these two conditions. In particular, participants’ sensitivity to late evidence after the intermittent judgment, but not the simple motor response, was decreased. Both the intermittent response as well as the final estimation reports were accompanied by nonluminance-mediated increases of pupil diameter. These pupil dilations were bigger during intermittent judgments than simple motor responses and bigger during estimation when the late evidence was consistent than inconsistent with the initial judgment. In sum, decisions activate pupil-linked arousal systems and alter the temporal weighting of decision evidence. Our results are consistent with the idea that categorical choices in the face of uncertainty induce a change in the state of the neural circuits underlying decision-making. NEW & NOTEWORTHY The psychology and neuroscience of decision-making have extensively studied the accumulation of decision-relevant information toward a categorical choice. Much fewer studies have assessed the impact of a choice on the processing of subsequent information. Here, we show that intermittent choices during a protracted stream of input reduce the sensitivity to subsequent decision information and transiently boost arousal. Choices might trigger a state change in the neural machinery for decision-making
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