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

Abstract

We prove global $$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$ W 1 , q ( Ω , R N ) -regularity for minimisers of $${{\mathscr {F}}(u)=\int _\Omega F(x,Du)\,{\mathrm{d}}x}$$ F ( u ) = ∫ Ω F ( x , D u ) d x satisfying $$u\ge \psi $$ u ≥ ψ for a given Sobolev obstacle $$\psi $$ ψ . $$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$ W 1 , q ( Ω , R N ) 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 natural (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n . In the autonomous case $$F\equiv F(z)$$ F ≡ F ( z ) we can improve the gap to $$q<\min \left( \frac{np}{n-1},p+1\right) $$ q < min np n - 1 , p + 1 , a result new even in the unconstrained case.

Gradient regularity in mixed local and nonlocal problems

“…This first leads to Hölder regularity of solutions with every exponent (Theorem 3) and then to the same kind of estimates globally (Theorem 4); combining these ingredients with a priori regularity estimates from the classical local theory, leads to Theorem 5. We mention that, due to the assumption p = γ , functionals as in (1.1)-(1.3) connect to a large family of problems featuring anisotropic operators and integrands with socalled nonstandard growth conditions [26,28,29,32,40,54,66], and to some other classes of anisotropic nonlocal problems [16-20, 33, 67, 73]. We mention that a further connection has been established in [31], where a class of mixed functionals has been used to approximate local functionals with ( p, q)-growth in order to prove higher integrability of minimizers.…”