1,105 research outputs found

### Series expansion for L^p Hardy inequalities

We consider a general class of sharp $L^p$ Hardy inequalities in $\R^N$
involving distance from a surface of general codimension $1\leq k\leq N$. We
show that we can succesively improve them by adding to the right hand side a
lower order term with optimal weight and best constant. This leads to an
infinite series improvement of $L^p$ Hardy inequalities.Comment: 16 pages, to appear in the Indiana Univ. Math.

### Critical Hardy--Sobolev Inequalities

We consider Hardy inequalities in $I R^n$, $n \geq 3$, with best constant
that involve either distance to the boundary or distance to a surface of
co-dimension $k<n$, and we show that they can still be improved by adding a
multiple of a whole range of critical norms that at the extreme case become
precisely the critical Sobolev norm.Comment: 22 page

### Augmenting Ship Propulsion in Waves Using Flapping Foils Initially Designed for Roll Stabilization

AbstractBiomimetic flapping foils attached on ship hull are studied for wave-energy extraction, stored in ship motions, and direct conversion to propulsive power. We examine a pair of roll-stabilization wings located at the side of the hull, in horizontal arrangement. The fins gain their linear oscillation (heaving) from ship pitching and heaving responses in irregular waves, while wing's rotation (pitching) is properly controlled with respect to its vertical motion history. A method, developed in our previous works, is applied for that purpose. The system operates in realistic sea conditions modeled by using parametric spectra, taking into account the coupling between ship responses and the hydrodynamics of the lifting appendages. We present numerical calculations concerning the operation of the augmenting system indicating that, although the present arrangement is designed for roll reduction purposes it can obtain significant amount of thrust. Also it is illustrated that, after reposition of the fins near the bow, the performance of the system can be further enhanced

### Sharp two-sided heat kernel estimates for critical Schr\"odinger operators on bounded domains

On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger
operators -\Delta -V, with V being either the critical borderline potential
V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under
Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates
on the corresponding heat kernels. To this end we transform the Scr\"odinger
operators into suitable degenerate operators, for which we prove a new
parabolic Harnack inequality up to the boundary. To derive the Harnack
inequality we have established a serier of new inequalities such as improved
Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincar\'e. As a
byproduct of our technique we are able to answer positively to a conjecture of
E.B.Davies.Comment: 40 page

### Universality in Blow-Up for Nonlinear Heat Equations

We consider the classical problem of the blowing-up of solutions of the
nonlinear heat equation. We show that there exist infinitely many profiles
around the blow-up point, and for each integer $k$, we construct a set of
codimension $2k$ in the space of initial data giving rise to solutions that
blow-up according to the given profile.Comment: 38 page

### Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya
inequalities with best Hardy constants, for domains satisfying suitable
geometric assumptions such as mean convexity or convexity. We then use them to
produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants
for various fractional Laplacians. In the case where the domain is the half
space our results cover the full range of the exponent $s \in (0,1)$ of the
fractional Laplacians. We answer in particular an open problem raised by Frank
and Seiringer \cite{FS}.Comment: 42 page

### Renormalizing Partial Differential Equations

In this review paper, we explain how to apply Renormalization Group ideas to
the analysis of the long-time asymptotics of solutions of partial differential
equations. We illustrate the method on several examples of nonlinear parabolic
equations. We discuss many applications, including the stability of profiles
and fronts in the Ginzburg-Landau equation, anomalous scaling laws in
reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]

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