Quantitative isoperimetric inequalities for log-convex probability measures on the line

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdr\'e). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) to any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type

On isoperimetric inequalities with respect to infinite measures

We study isoperimetric problems with respect to infinite measures on $R ^n$. In the case of the measure $\mu$ defined by $d\mu = e^{c|x|^2} dx$, $c\geq 0$, we prove that, among all sets with given $\mu-$measure, the ball centered at the origin has the smallest (weighted) $\mu-$perimeter. Our results are then applied to obtain Polya-Szego-type inequalities, Sobolev embeddings theorems and a comparison result for elliptic boundary value problems.Comment: 25 page

The isoperimetric problem for a class of non-radial weights and applications

We study a class of isoperimetric problems on R+ N where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type |x|kxN α. Our results imply some sharp functional inequalities, like for instance, Caffarelli-Kohn-Nirenberg type inequalities

Half-space Gaussian symmetrization: applications to semilinear elliptic problems

Abstract We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in ℝ N without growth restrictions at infinity. A comparison result in terms of the half-space Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type

Robot-mediated therapy for paretic upper limb of chronic patients following neurological injury

Objective: To evaluate the effectiveness of robot-mediated therapy targeted at the motor recovery of the upper limb in chronic patients following neurological injury

Upper Limb Spasticity Reduction Following Active Training: A Robot-Mediated Study In Patients With Chronic Hemiparesis

sion of the arm. A 3-month follow-up was performed. Results: Statistically significant improvements were found in both groups after treatment. Some differences were found in elbow motor improvement between the 2 groups. Conclusion: Comparison between groups confirms that active movement training does not result in increased hypertonia, but results in spasticity reduction in antagonist muscles by activating the reciprocal inhibition mechanism. Furthermore, robot-mediated therapy contributes to a decrease in motor impairment of the upper limbs in subjects with chronic hemiparesis, resulting in a reduction in shoulder pain

Systematic review of outcome measures of walking training using electromechanical and robotic devices in patients with stroke

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