Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset \real^N$ is a bounded domain, and $1 \leq q < 2$. For $q=1$, $|u|^{q-2}u$ will be identified with \sgn(u). We establish a variational principle for least energy nodal solutions, and we investigate their qualitative properties. In particular, we show that they satisfy a unique continuation property (their zero set is Lebesgue-negligible). Moreover, if $\Omega$ is radial, then least energy nodal solutions are foliated Schwarz symmetric, and they are nonradial in case $\Omega$ is a ball. The case $q=1$ requires special treatment since the formally associated energy functional is not differentiable, and many arguments have to be adjusted

The second eigenvalue of the fractional p−p-Laplacian

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\lambda_2(\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem $$\inf \{\lambda_2(\Omega)\,:\,|\Omega|=c\}.$$ We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity.Comment: 38 pages. The test function used in the proof of Theorem 3.1 needed to be slightly modified, in order to be admissible for $1<p<2$. We fixed this issu

The fractional Cheeger problem

Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the nonlocal version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.Comment: 33 pages, 2 figure

Does Bridget Jones watch EastEnders or The Love Boat? Cultural and linguistic issues in the translation of chick-lit novels

Abstract - Chick lit is a genre comprised of books chiefly written by women for women. These books are mainly characterized by the tone, which is very confidential and personal, and often relies on the use of colloquial and slangy (and occasionally obscene) expressions, by the use of humour, and by the numerous references to popular culture. This paper aims at analyzing the strategies used in the Italian translation of one of the best-known examples of the genre, Helen Fielding’s Bridget Jones’s Diary, taking two frames of reference. First of all, the analysis will try to identify in the target text the distinctive universal features observed in computer-based corpus translation (Baker 1996), namely explicitation, simplification and normalization. Secondly, it will consider the translation in terms of Venuti’s dichotomy of translation strategies into domestication and foreignization (1995). Riassunto -Il chick lit è un genere letterario costituito principalmente da romanzi scritti perlopiù da autrici donne per un pubblico femminile. I romanzi chick-lit sono caratterizzati dal tono confidenziale e personale, dall’uso di espressioni colloquiali e slang (e occasionalmente volgari), dalla presenza costante di humour e dai numerosi riferimenti alla pop culture contemporanea. Questo saggio intende analizzare le strategie utilizzate nella traduzione italiana di uno degli esempi più rappresentativi del genere, ovvero Bridget Jones’s Diary di Helen Fielding, utilizzando due modelli di riferimento. Innanzitutto, lo studio tenterà di identificare la presenza degli universali traduttivi osservati nei corpus-based translation studies (Baker 1996), ovvero explicitation, simplification e normalization. Successivamente, si analizzerà la traduzione secondo la teoria delle strategie traduttive proposta da Venuti (1995), ovvero domestication e foreignization