Global higher integrability for minimisers of convex functionals with (p,q)-growth

Abstract

We prove global $$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) -regularity for minimisers of convex functionals of the form $${\mathscr {F}}(u)=\int _{\varOmega } F(x,Du)\,{\mathrm{d}}x$$ F ( u ) = ∫ Ω F ( x , D u ) d x .$$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and controlled (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n .

On the global regularity for minimizers of variational integrals: splitting-type problems in 2D and extensions to the general anisotropic setting

“…recalling the bounds (29), (44), ( 46), (48), (57) we are led to the lower bound for the exponent . Assuming the validity of (58) and returning to (43) we now have shown that for i sufficiently close to −1∕2 the right-hand side of (43) can be splitted into terms which either can be absorbed in the left-hand side of (43) or stay uniformly bounded, hence for a finite constant c independent of m. Passing to the limit m → ∞ in (59) our claim (11) follows. Obviously ( 12) is a consequence of (58) and the definition of (55)…”

“…Even more recently the contribution of Koch [11] addresses the global higher integrability of the gradient of solutions of variational problems with (p, q)-growth, which to our knowledge is the first result to improve global integrabilty of the gradient allowing full anisotropy with different growth rates with respect to different partial derivatives. The boundary data are supposed to belong to some fractional order spaces (see, e.g., [12] or [13]) and roughly speaking are handled like an additional x-dependence following ideas as outlined in, e.g., [14] for the interior situation.…”

“…In the case q i > 5 , i = 1 , 2, we can replace (13) by the expression which means that in the case of higher initial integrability we can decrease the exponent t in (11) leading to a stronger result. We also note that for the choice q i > 5 , i = 1 , 2, the condition ( 9) can be replaced by the requirement u 0 ∈ W 1,q min (Ω)…”

On the validity of the Euler-Lagrange system without growth assumptions

“…The reverse question is considerably more difficult and the answer is delicate already in the case of convex, real-valued integrands satisfying p-growth. We do not discuss the issue of integrands with (p, q)-growth further here but refer to [23,12,18,19,20,13,14] for further discussion and references. The question for convex integrands without growth conditions has been studied in [6].…”

Lipschitz bounds for integral functionals with $(p,q)$-growth conditions

“…Currently, regularity theory for non-autonomous integrands with non-standard growth, e.g. p(x)-Laplacian or double phase functionals, are a very active field of research, see, e.g., the recent papers [4,9,12,18,16,17,19,20,21,23,28,27,30,35] and [2,11] for related results about the Lavrentiev phenomena.…”