Higher Integrability for Minimizers of Integral Functionals With (p, q) Growth

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

“…We insert this into (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), and obtain an estimate only involving horizontal second derivatives as well as some first-order terms:…”

“…In this step, we estimate the terms on the r.h.s. of (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) in a way that allows us to apply Lemma 2.8 at the end. The expression ϑ(ρ) in the lemma corresponds to the integrals on the l.h.s of (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) over the ball with radius ρ, that is,…”

“…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.…”

Variational integrals of splitting-type: higher integrability under general growth conditions

“…If we pass to the vector case, then there are strong regularity results due to Marcellini [Ma3] and Marcellini and Papi [MP] for integrands of the special form F = F (|Z|), whereas Esposito, Leonetti and Mingione [ELM1] studied more general densities F and proved ∇u ∈ L q loc (Ω; R nM ) (1.4)…”

Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals

“…Note that in [ELM1] actually the case of degenerate ellipticity is considered. Note also that obviously (3) implies (2).…”