A-priori gradient bound for elliptic systems under either slow or fast growth conditions
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Calculus of Variations
A-priori gradient bound for elliptic systems under either slow or fast growth conditions Tommaso Di Marco1 · Paolo Marcellini1 Received: 13 September 2019 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 1,∞ We obtain an a-priori Wloc (; Rm )-bound for weak solutions to the elliptic system divA (x, Du) =
n ∂ α a (x, Du) = 0, ∂ xi i
α = 1, 2, . . . , m,
i=1
where is an open set of Rn , n ≥ 2, u is a vector-valued map u : ⊂ Rn → Rm . The vector field A (x, ξ ) has a variational nature in the sense that A (x, ξ ) = Dξ f (x, ξ ), where f = f (x, ξ ) is a convex function with respect to ξ ∈ Rm×n . In this context of vector-valued maps and systems, a classical assumption finalized to the everywhere regularity is a modulusdependence 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 xdependence, 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, |Du|) = a(x)|Du| p(x) log(1 + |Du|) or, when n = 2, 3, even asymptotic linear growth with energy integrals of the type
g (x, |Du|) d x =
|Du| − a (x) |Du| d x.
Keywords Regularity · Local Lipschitz continuity · Nonlinear elliptic differential systems · Calculus of variations · p, q-growth conditions · General growth conditions · Slow or fast growth conditions Mathematics Subject Classification 35B45 · 35B65 · 35D30 · 35J20 · 35J60 · 49J45 · 49N60
Communicated by A. Malchiodi.
B
Paolo Marcellini [email protected] Tommaso Di Marco [email protected]
1
Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, Florence, Italy 0123456789().: V,-vol
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T. Di Marco, P. Marcellini
1 Introduction We are interested in the regularity of local minimizers of energy-integrals of the calculus of variations of the form f (x, Du) d x, (1.1) F (u) =
where is an open set of Rn for some n ≥ 2 and Du is the m × n gradient-matrix of a map u : ⊂ Rn → Rm with m ≥ 1. Here f : × Rm×n → R is a convex Carathéodory integrand; i.e., f = f (x, ξ ) is measurable with respect to x ∈ Rn and it is a convex function with respect to ξ ∈ Rm×n . A local minimizer of the energy-functional F in (1.1) is a map u : ⊂ Rn → Rm satisfying the inequality f (x, Du (x)) d x ≤ f (x, Du (x) + Dϕ (x)) d x
for every test function ϕ with compact support in ; i.e., ϕ ∈ C01 (; Rm ). Under some growth conditions on f (see the proof of Lemma 4.1 for details) every local minimizer u of F is a weak solution to the nonlinear elliptic system of m partial differential equations n ∂ α a (x, Du) = 0, ∂ xi i
α = 1, 2, . . . , m,
(1.2)
i=1
∂f where aiα (x, ξ ) = ∂ξ α = f ξ α fo
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