Regularity results for minimizers of irregular integrals with (p,q) growth

Variational integrals with a wide range of anisotropy

“…A natural space for local minimizers is the class of functions u from W 1 p (Ω; R nN ) such that Ω f (∇u) dx < ∞ for each subregion Ω Ω, and so one is interested in the regularity properties of local minimizers u which means that one asks for higher integrability of ∇u, Hölder-continuity of u or even Hölder-continuity of ∇u provided that f satisfies additional smoothness and convexity assumptions. In general, the hope for positive results increases in the scalar case but counterexamples of [Gi2] and (later) of [Ho] show that even for N = 1 unbounded minimizers exist, when q is too big with respect to p. On the contrary, there is a long list of authors investigating the different aspects of the regularity theory, we mention (without being complete) the works of Acerbi and Fucso ( [AF]), Fusco and Sbordone ([FS]), Marcellini ([Ma2]), Choe ([Ch]) and the papers [ELM1], [ELM2] of Esposito, Leonetti and Mingione, where the interested reader can also find further references. Typically, in the above mentioned works either a bound of the form q < c(n)p (1.2) with c(n) > 1, but c(n) → 1 as n → ∞ is required, or a dimensionless restriction like q < p + 2 (1.3) occurs together with the assumption that u is a locally bounded function.…”

Two-dimensional variational problems with a wide range of anisotropy

“…In connection with II. the reader should consult for example the papers of Acerbi and Fusco [AF], Cupini, Guidorzi and Mascolo [CGM], Esposito, Leonetti and Mingione [ELM1,2] and of Passarelli Di Napoli and Siepe [PS] together with the references cited by these authors. We further refer to [BF1,2].…”

Regularity results for minimizers of (2, q)-growth functionals in the Heisenberg Group

“…This obstacle was finally overcome by Esposito, Leonetti and Mingione [13]: Under the assumption q/p < (n + 2)/n, they employed an interpolation argument to prove a priori higher integrability for minimizers in W 1,p . They further extended their argument to non-C 2 functionals and vectorial functionals with Uhlenbeck structure [14], obtaining higher integrability and Lipschitz regularity for minimizers if q/p ≤ (n + 1)/n. We note that while all the bounds on the gap ratio mentioned so far are needed due to technical considerations in the respective papers, they are lower than the minimal gap ratio for which counterexamples to regularity have been found, leaving open the question for the optimal bound on q/p.…”

“…We adapt the (p, q)-growth interpolation approach from [14]: We approximate the original functional by standard q-growth functionals, and use the regularity of their minimizers in a qualitative way to construct quantitative estimates that only rely on the original functional's properties. In particular, we prove the pointwise boundedness of Xu and T u.…”