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- Title
A-priori gradient bound for elliptic systems under either slow or fast growth conditions.
- Authors
Di Marco, Tommaso; Marcellini, Paolo
- Abstract
We obtain an a-priori W loc 1 , ∞ Ω ; R m -bound for weak solutions to the elliptic system div A x , D u = ∑ i = 1 n ∂ ∂ x i a i α x , D u = 0 , α = 1 , 2 , ... , m , where Ω is an open set of R n , n ≥ 2 , u is a vector-valued map u : Ω ⊂ R n → R m . The vector field A x , ξ has a variational nature in the sense that A x , ξ = D ξ f x , ξ , where f = f x , ξ is a convex function with respect to ξ ∈ R m × n . In this context of vector-valued maps and systems, a classical assumption finalized to the everywhere regularity is a modulus-dependence in the energy integrand; i.e., we require that f x , ξ = g x , ξ , where g x , t is convex and increasing with respect to the gradient variable t ∈ 0 , ∞ . We allow x-dependence, which turns out to be a relevant difference with respect to the autonomous case and not only a technical perturbation. Our assumptions allow us to consider both fast and slow growth. We consider fast growth even of exponential type; and slow growth, for instance of Orlicz-type with energy-integrands such as g x , D u = a (x) | D u | p (x) log (1 + | D u |) or, when n = 2 , 3 , even asymptotic linear growth with energy integrals of the type ∫ Ω g x , D u d x = ∫ Ω D u - a x D u d x.
- Subjects
ELLIPTIC equations; CONVEX functions; EXPONENTIAL functions; CALCULUS of variations
- Publication
Calculus of Variations & Partial Differential Equations, 2020, Vol 59, Issue 4, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-020-01769-7