Local boundedness of vectorial minimizers of non-convex functionals

We prove a local boundedness result for local minimizers of a class of non-convex functionals, under special structure assumptions on the energy density. The proof follows the lines of that in [CupLeoMas17], where a similar result is proved under slightly stronger assumptions on the energy density

Limitatezza locale di minimi vettoriali di funzionali non convessi

We prove a local boundedness result for local minimizers of a class of non-convex functionals, under special structure assumptions on the energy density. The proof follows the lines of that in [CupLeoMas17], where a similar result is proved under slightly stronger assumptions on the energy density.Dimostriamo un risultato di limitatezza locale per minimi locali di una classe di funzionali non convessi, con particolari ipotesi di struttura sulla densità di energia. La dimostrazione procede come quella in [CupLeoMas17], dove un risultato simile è dimostrato con ipotesi leggermente più forti sulla densità di energia

On the H\uf6lder continuity for a class of vectorial problems

In this paper we prove local H\uf6lder continuity of vectorial local minimizers of special classes of integral functionals with rank-one and polyconvex integrands. The energy densities satisfy suitable structure assumptions and may have neither radial nor quasi-diagonal structure. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the H\uf6lder continuity. In the final section, we provide some non-trivial applications of our results

Local boundedness for solutions of a class of nonlinear elliptic systems

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of m equations in divergence form, satisfying p growth from below and q growth from above, with p <= q; this case is known as p, q-growth conditions. Well known counterexamples, even in the simpler case p = q, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component u(alpha) of the solution u = (u(1),..., u(m)) satisfies an improved Caccioppoli's inequality and we get the boundedness of u(alpha) by applying De Giorgi's iteration method, provided the two exponents p and q are not too far apart. Let us remark that, in dimension n = 3 and when p = q, our result works for 3/2 < p <= 3, thus it complements the one of Bjorn whose technique allowed her to deal with p <= 2 only. In the final section, we provide applications of our result

Butterfly support for o diagonal coeficients and boundedness of solutions to quasilinear elliptic systems

We consider quasilinear elliptic systems in divergence form. In general,we cannot expect thatweak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coeficients have a "butterfly support": this allows us to prove local boundedness of weak solutions.publishe

Minimizers of Nonlocal Polyconvex Energies in Nonlocal Hyperelasticity

We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient

Minimizers of Nonlocal Polyconvex Energies in Nonlocal Hyperelasticity

We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz' fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola's identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient. Contrary to classical elasticity, this existence result is compatible with cavitation and fracture