A pointwise differential inequality and second-order regularity for nonlinear elliptic systems

Abstract

A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in R n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known.

Natural second-order regularity for parabolic systems with operators having $$(p,\delta )$$-structure and depending only on the symmetric gradient

“…Here, we focus on the regularity of the quantity in (1.3). Thus, this work can be seen as a natural extension of previous results we have done in the case p ∈ (1,2] for the steady problem in [12] and for the unsteady continuous/discrete in [13]. Note that our approach allows to treat the full range of exponents p ∈ (1, ∞), as in the scalar case, even if we give full details only in the case p > 2, as the case p ∈ (1, 2) is already treated in a different way.…”

“…Note that our approach allows to treat the full range of exponents p ∈ (1, ∞), as in the scalar case, even if we give full details only in the case p > 2, as the case p ∈ (1, 2) is already treated in a different way. Notice that the results in [18] hold only for p > 3 2 , which has been improved in [1], reaching p > 4 − 2 √ 2. The limitation on p > 3/2 was also present in prior results of "natural" regularity in the symmetric gradient case [5], but it has then later removed completely in [11] to the case p > 1.…”

“…where the operator S depends only on the symmetric gradient Du = 1 2 ((∇u) + ∇u) and has ( p, δ)-structure (cf. Definition 2.15).…”

“…The techniques employed for p > 2 are rather different from those previously used in the case for p < 2, where calculations can be more easily justified by approximation of the system by means of adding the term −ε u (and then showing that estimates for a system with leading linear part could be made independent of ε > 0). Anyway, the technique we use can be also employed in the case p ∈ (1,2] to prove in an alternative way the regularity results already known. This requires some technical adjustments which are left for a further investigation, since the technicalities are complex enough already in the case p > 2.…”

A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form

“…The rôle of this structural assumption is in fact the main motivation of this work: indeed, (with the exception of [45,9]), the higher order Calderón-Zygmund theory exposed so far is restricted to equations of the form (1.7) and in this case it can be actually extended to systems (see [17,18,25] and the recently appeared [10]). As the L 2 -theory seems to be the basic step to deal with the general problem (1.6), it is worth investigating to what extent the Uhlenbeck structure is necessary to develop such a theory and whether the general nonlinear problem (1.1) enjoys the same Sobolev regularity for its stress field V = DF (Du).…”