On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities

We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d, formulated in terms of the Laplacian Delta and of the fractional powers D^n := (-Delta)^(n/2) with real n >= 0; we review known facts and present novel results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the L^2 case where, for all sufficiently regular f : R^d -> C, the norm || D^j f||_{L^r} is bounded in terms of || f ||_{L^2} and || D^n f ||_{L^2} for 1/r = 1/2 - (theta n - j)/d, and suitable values of j,n,theta (with j,n possibly noninteger). In the special cases theta = 1 and theta = j/n + d/2 n (i.e., r = + infinity), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general L^2 case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the Gagliardo-Nirenberg and Sobolev sharp constants are confined to quite narrow intervals. Several examples are given.Comment: LaTex, 63 pages, 3 tables. In comparison with version v2, just a few corrections to eliminate typo

A time machine for free fall into the past

Inspired by some recent works of Tippett-Tsang and Mallary-Khanna-Price, we present a new spacetime model containing closed timelike curves (CTCs). This model is obtained postulating an ad hoc Lorentzian metric on $\mathbb{R}^4$, which differs from the Minkowski metric only inside a spacetime region bounded by two concentric tori. The resulting spacetime is topologically trivial, free of curvature singularities and is both time and space orientable; besides, the inner region enclosed by the smaller torus is flat and displays geodesic CTCs. Our model shares some similarities with the time machine of Ori and Soen but it has the advantage of a higher symmetry in the metric, allowing for the explicit computation of a class of geodesics. The most remarkable feature emerging from this computation is the presence of future-oriented timelike geodesics starting from a point in the outer Minkowskian region, moving to the inner spacetime region with CTCs, and then returning to the initial spatial position at an earlier time; this means that time travel to the past can be performed by free fall across our time machine. The amount of time travelled into the past is determined quantitatively; this amount can be made arbitrarily large keeping non-large the proper duration of the travel. An important drawback of the model is the violation of the classical energy conditions, a common feature of many time machines. Other problems emerge from our computations of the required (negative) energy densities and of the tidal accelerations; these are small only if the time machine is gigantic.Comment: 40 pages, 10 figures; the final version accepted for publication. In comparison with version v2, some references added (see [4,21,35]) and commented on in the Introductio

On the constants for multiplication in Sobolev spaces

For n > d/2, the Sobolev (Bessel potential) space H^n(R^d, C) is known to be a Banach algebra with its standard norm || ||_n and the pointwise product; so, there is a best constant K_{n d} such that || f g ||_{n} <= K_{n d} || f ||_{n} || g ||_{n} for all f, g in this space. In this paper we derive upper and lower bounds for these constants, for any dimension d and any (possibly noninteger) n > d/2. Our analysis also includes the limit cases n -> (d/2) and n -> + Infinity, for which asymptotic formulas are presented. Both in these limit cases and for intermediate values of n, the lower bounds are fairly close to the upper bounds. Numerical tables are given for d=1,2,3,4, where the lower bounds are always between 75% and 88% of the upper bounds.Comment: LaTeX, 45 page

Quantitative functional calculus in Sobolev spaces

In the framework of Sobolev (Bessel potential) spaces H^n(\reali^d, \reali {or} \complessi), we consider the nonlinear Nemytskij operator sending a function x \in \reali^d \mapsto f(x) into a composite function x \in \reali^d \mapsto G(f(x), x). Assuming sufficient smoothness for $G$, we give a "tame" bound on the $H^n$ norm of this composite function in terms of a linear function of the $H^n$ norm of $f$, with a coefficient depending on $G$ and on the $H^a$ norm of $f$, for all integers $n, a, d$ with $a > d/2$. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the $H^n$ norm of the function $x \mapsto G(f(x),x)$. When applied to the case $G(f(x), x) = f^2(x)$, this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.Comment: LaTex, 37 pages. Final version, differing only by minor typographical changes from the versions of May 23, 2003 and March 8, 200

Smooth solutions of the Euler and Navier-Stokes equations from the a posteriori analysis of approximate solutions

The main result of [C. Morosi and L. Pizzocchero, Nonlinear Analysis, 2012] is presented in a variant, based on a C^infinity formulation of the Cauchy problem; in this approach, the a posteriori analysis of an approximate solution gives a bound on the Sobolev distance of any order between the exact and the approximate solution.Comment: Author's note. Some overlaps with our previous works arXiv:1402.0487, arXiv:1310.5642, arXiv:1304.2972, arXiv:1203.6865, arXiv:1104.3832, arXiv:1009.2051, arXiv:1007.4412, arXiv:0909.3707, arXiv:0709.1670; these overlaps aim to make the paper self-contained and do not involve the main results. Final version to appear in Nonlinear Analysi

Local zeta regularization and the scalar Casimir effect III. The case with a background harmonic potential

Applying the general framework for local zeta regularization proposed in Part I of this series of papers, we renormalize the vacuum expectation value of the stress-energy tensor (and of the total energy) for a scalar field in presence of an external harmonic potential.Comment: Some overlaps with our works arXiv:1104.4330, arXiv:1505.00711, arXiv:1505.01044, arXiv:1505.03276. These overlaps aim to make the present paper self-contained, and do not involve the main result

Local Casimir Effect for a Scalar Field in Presence of a Point Impurity

The Casimir effect for a scalar field in presence of delta-type potentials has been investigated for a long time in the case of surface delta functions, modelling semi-transparent boundaries. More recently Albeverio, Cacciapuoti, Cognola, Spreafico and Zerbini [9,10,51] have considered some configurations involving delta-type potentials concentrated at points of $\mathbb{R}^3$; in particular, the case with an isolated point singularity at the origin can be formulated as a field theory on $\mathbb{R}^3\setminus \{\mathbf{0}\}$, with self-adjoint boundary conditions at the origin for the Laplacian. However, the above authors have discussed only global aspects of the Casimir effect, focusing their attention on the vacuum expectation value (VEV) of the total energy. In the present paper we analyze the local Casimir effect with a point delta-type potential, computing the renormalized VEV of the stress-energy tensor at any point of $\mathbb{R}^3\setminus \{\mathbf{0}\}$; to this purpose we follow the zeta regularization approach, in the formulation already employed for different configurations in previous works of ours (see [29-31] and references therein).Comment: 20 pages, 6 figures; the final version accepted for publication. In the initial part of the paper, possible text overlaps with our previous works arXiv:1104.4330, arXiv:1505.00711, arXiv:1505.01044, arXiv:1505.01651, arXiv:1505.03276. These overlaps aim to make the present paper self-contained, and do not involve the main result

Local zeta regularization and the scalar Casimir effect IV. The case of a rectangular box

Applying the general framework for local zeta regularization proposed in Part I of this series of papers, we compute the renormalized vacuum expectation value of several observables (in particular, of the stress-energy tensor and of the total energy) for a massless scalar field confined within a rectangular box of arbitrary dimension.Comment: Some overlaps with our works arXiv:1104.4330, arXiv:1505.00711, arXiv:1505.01044, arXiv:1505.01651. These overlaps aim to make the present paper self-contained, and do not involve the main results. In comparison with version v3, reference [26] adde

On the expansion of the Kummer function in terms of incomplete Gamma functions

The expansion of Kummer's hypergeometric function as a series of incomplete Gamma functions is discussed, for real values of the parameters and of the variable. The error performed approximating the Kummer function with a finite sum of Gammas is evaluated analytically. Bounds for it are derived, both pointwisely and uniformly in the variable; these characterize the convergence rate of the series, both pointwisely and in appropriate sup norms. The same analysis shows that finite sums of very few Gammas are sufficiently close to the Kummer function. The combination of these results with the known approximation methods for the incomplete Gammas allows to construct upper and lower approximants for the Kummer function using only exponentials, real powers and rational functions. Illustrative examples are provided.Comment: 21 pages, 6 figures. To appear in "Archives of Inequalities and Applications