The Calder\'on-Zygmund theory for elliptic problems with measure data

We consider non-linear elliptic equations having a measure in the right hand side, of the type \divo a(x,Du)=\mu, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calder\'on-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates

The current crisis of intensive work regimes and the question of social exclusion in industrialized countries

The aim of this article is to analyze the difficulties currently being faced by regimes of social regulation of economic life and the ways in which they are being transformed. In order to address this complex question, transformations in the labour market are examined. Emphasis is given in particular to the fact that the decline in life-time jobs has had a destabilizing impact on employment systems. This is true not so much in terms of reduced employment. In fact, numerous job opportunities have been created in the tertiary sector, though insecure and badly paid, but these jobs do not reflect the traditional standards of social regulation and therefore entail a weakening of the mechanisms of social integration and a growing risk of exclusion. At this point the theme of the heterogeneity and polarization of working careers in service society is introduced, highlighting in particular the variety of regulatory forms with their different and changing mixes of family, state and market. These policy mixes form the basis for the various models of welfare capitalism identified here. Finally, consideration is given to the two main responses to this transitional phase in industrialized countries. On the one hand, in English-speaking countries which are characterised by intensive deregulation and the spread of flexible forms of work; on the other hand, in the countries of continental Europe where the redistributive modes of traditional welfare programmes have been preserved. Neither of these strategies, however, has produced new and lasting prospects for synergies between the economy and the society. Potential regulatory innovations would presuppose a reappraisal of modes of activity that are at present 'invisible' such as production for own consumption, family care, volunteer and charitable work and the creation of social capital. Political steps in this direction could lead to a new balance between state, market and family that would secure the level of cooperation needed for socially embedded economic life. -- Der Aufsatz analysiert die gegenwärtigen Schwierigkeiten, mit denen verschiedene Regimes sozialer Regulierung der Wirtschaft konfrontiert sind, einschließlich ihrer Transformationspfade. Um dieser komplexen Fragestellung gerecht zu werden, werden zunächst die allgemeinen Veränderungsprozesse auf dem Arbeitsmarkt dargestellt. Besonderes Gewicht wird dabei auf die Tatsache gelegt, daß der Abbau von lebenslangen Arbeitsverhältnissen das Beschäftigungssystem massiv destabilisiert. Das bezieht sich nicht ausschließlich oder gar vorrangig auf die rückläufigen Beschäftigungsmöglichkeiten. Gleichzeitig sind nämlich im Dienstleistungssektor zahlreiche neue Arbeitsplätze entstanden - wenn auch in der Regel unsichere und schlecht bezahlte. Sie spiegeln ein geringeres Maß gesellschaftlicher Regulierung und implizieren damit eine schwächere soziale Integration sowie ein höheres Risiko sozialer Ausgrenzung. In diesem Zusammenhang werden die Trends zur Heterogenisierung und Polarisierung der Berufsbiographien in der Dienstleistungsgesellschaft thematisiert und dabei besonders die Vielfalt der länderspezifischen Mischungen von Zuständigkeiten von Familie, Staat und Markt dargelegt. Diese bilden die Grundlage für die in diesem Beitrag identifizierten Modelle wohlfahrtsstaatlicher Kapitalismen. Es lassen sich zwei verschiedene Reaktionsmuster der Industriestaaten auf diese Veränderungsprozesse unterscheiden. Auf der einen Seite stehen USA und Großbritannien mit ihren ausgeprägten Deregulierungspolitiken und der Ausweitung von flexiblen Arbeitsverhältnissen; auf der anderen Seite die Länder Kontinentaleuropas, die auf die Beibehaltung der Transferorientierung traditioneller Wohlfahrtsprogramme setzen. Keine dieser Strategien führte jedoch zu neuen und dauerhaften Synergie-Effekten zwischen Wirtschaft und Gesellschaft. Die Möglichkeiten für innovative Regulierungen - so die hier vertretene These - sind daran gebunden, daß bislang 'unsichtbare' Produktionen aufgewertet werden, wie beispielsweise Produktion zum Eigenverbrauch, unbezahlte Tätigkeiten wie Familienarbeit, Solidarität und die Bildung von sozialem Kapital. Ein solcher Ansatz könnte richtungsweisend sein für ein neues regulatives Gleichgewicht, das den Mindestanforderungen an Kooperation Rechnung trägt und damit die Voraussetzung für ein sozial integriertes Wirtschaftsleben gewährleistet.

Optimal Lipschitz criteria and local estimates for non-uniformly elliptic problems

We report on new techniques and results in the regularity theory of general non-uniformly elliptic variational integrals. By means of a new potential theoretic approach we reproduce, in the non-uniformly elliptic setting, the optimal criteria for Lipschitz continuity known in the uniformly elliptic one and provide a unified approach between non-uniformly and uniformly elliptic problems

Two populations mean-field monomer-dimer model

A two populations mean-field monomer-dimer model including both hard-core and attractive interactions between dimers is considered. The pressure density in the thermodynamic limit is proved to satisfy a three-dimensional variational principle. A detailed analysis is made in the limit in which one population is much smaller than the other and a ferromagnetic mean-field phase transition is found.Comment: 12 pages, 2 figure

Manifold constrained non-uniformly elliptic problems

We consider the problem of minimizing variational integrals defined on \cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand exhibiting different polynomial growth conditions and no homogeneity. We develop a few intrinsic methods aimed at proving partial regularity of minima and providing techniques for treating larger classes of similar constrained non-uniformly elliptic variational problems. In order to give estimates for the singular sets we use a general family of Hausdorff type measures following the local geometry of the integrand. A suitable comparison is provided with respect to the naturally associated capacities.Comment: 50 page

On the regularity of minima of non-autonomous functionals

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-energy conditions are satisfied. These are related to the occurrence of the so-called Lavrentiev phenomenon that that non-autonomous functionals might exhibit, and which is a natural obstruction to regularity. In the case of vector valued problems we concentrate on higher gradient integrability of minima. Instead, in the scalar case, we prove local Lipschitz estimates. We also present an approach via a variant of Moser's iteration technique that allows to reduce the analysis of several non-uniformly elliptic problems to that for uniformly elliptic ones.Comment: 32 page

Factorization properties in d-dimensional spin glasses. Rigorous results and some perspectives

In this paper we show that d-dimensional Gaussian spin glass models are strongly stochastically stable, fulfill the Ghirlanda-Guerra identities in distribution and the ultrametricity property.Comment: To appear in Journal of Statistical Physic

Borderline gradient continuity of minima

The gradient of any local minimiser of functionals of the type $$w \mapsto \int_\Omega f(x,w,Dw)\,dx+\int_\Omega w\mu\,dx,$$ where $f$ has $p$-growth, $p>1$, and $\Omega \subset \mathbb R^n$, is continuous provided the optimal Lorentz space condition $\mu \in L(n,1)$ is satisfied and $x\to f(x, \cdot)$ is suitably Dini-continuous.Comment: 30 page

Regularity for general functionals with double phase

We prove sharp regularity results for a general class of functionals of the type $$w \mapsto \int F(x, w, Dw) \, dx\;,$$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;,$$ with $0<\nu \leq b(\cdot)\leq L$. This changes its ellipticity rate according to the geometry of the level set $\{a(x)=0\}$ of the modulating coefficient $a(\cdot)$. We also present new methods and proofs, that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon