In the general vector-valued case N≥ 1 , we prove the Lipschitz continuity of local minimizers to some integrals of the calculus of variations of the form ∫Ωg(x,|Du|)dx, with p, q-growth conditions only for | Du| → + ∞ and without further structure conditions on the integrand g= g(x, | Du|). We apply the regularity results to weak solutions to nonlinear elliptic systems of the form ∑i=1n∂∂xiaiα(x,Du)=0, α= 1 , 2 , … , N

AbstractIntegrals of the Calculus of Variations with {p,q}-growth may have not smooth minimizers, not even bounded, for general {p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand {f(x,\xi)} with dependence on the modulus of the gradient, i.e. {f(x,\xi)=g(x,|\xi|)}. Without imposing structure conditions, we prove that if {\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.

We consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. The model was recently introduced in [11] where existence of weak solutions was proved in three space dimensions. Here, we aim to study the properties of solutions in the two-dimensional case. In particular, we can show existence of global in time solutions satisfying a stronger formulation of the model with respect to the one considered in [11]

We establish the higher differentiability of integer and fractional order of the solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra (integer or fractional) differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the formwhere A is a p-harmonic type operator, ψ ∈ W 1,p (Ω) is a fixed function called obstacle ande. in Ω} is the class of the admissible functions. We prove that an extra differentiability assumption on the gradient of the obstacle transfers to Du with no losses in the natural exponent of integrability, provided the partial map x → A(x, ξ) possesses a suitable differentiability property measured or in the scale of the Sobolev space W 1,n or in that of the critical Besov-Lipschitz spaces B α n α ,q , for a suitable 1 ≤ q ≤ +∞.2000 Mathematics Subject Classification. 35J87, 49J40; 47J20. Key words and phrases. Variational inequalities, obstacle problems, higher differentiability. The work of Michela Eleuteri is supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), through the projects GNAMPA 2016 "Regolarità e comportamento asintotico di soluzioni di equazioni paraboliche" (coord. Prof. S. Polidoro) and GNAMPA 2017 "Regolarità per problemi variazionali d'ostacolo e liberi" (coord. Prof. M. Focardi). The work of Antonia Passarelli di Napoli is supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), through the projects GNAMPA 2016 "Problemi di Regolaritá nel Calcolo delle Variazioni e di Approssimazione" (coord. Prof. M. Carozza) and GNAMPA 2017 "Approssimazione con operatori discreti e problemi di minimo per funzionali del calcolo delle variazioni con applicazioni all'imaging" (coord. Dott. D. Costarelli). The work of the authors is also supported by the University of Modena and Reggio Emilia through the project FAR2015 "Equazioni differenziali: problemi evolutivi, variazionali ed applicazioni" (coord. Prof. S. Polidoro). This reaserch started while A. Passarelli di Napoli was visiting the University of Modena and Reggio Emilia . The hospitality of this Institution is warmly aknowledged.[42] G. Stampacchia: Formes bilineaires coercivitives sur les ensembles convexes,

A regularity result for integrals of the Calculus of Variations with variable exponents is presented. Precisely, we prove that any vector-valued minimizer of an energy integral over an open set W H R n , with variable exponent pðxÞ in the Sobolev class W 1; r loc ðWÞ for some r > n, is locally Lipschitz continuous in W and an a priori estimate holds.