Abstract
We obtain an a-priori \(W_{{\mathrm{loc}}}^{1,\infty }\left( \Omega ; {\mathbb {R}}^{m}\right) \)-bound for weak solutions to the elliptic system
where \(\Omega \) is an open set of \({\mathbb {R}}^{n}\), \(n\ge 2\), u is a vector-valued map \(u:\Omega \subset {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{m}\) . The vector field \(A\left( x,\xi \right) \) has a variational nature in the sense that \(A\left( x,\xi \right) =D_{\xi }f\left( x,\xi \right) \), where \( f=f\left( x,\xi \right) \) is a convex function with respect to \(\xi \in {\mathbb {R}}^{m\times n}\). 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 \(f\left( x,\xi \right) =g\left( x,\left| \xi \right| \right) \), where \( g\left( x,t\right) \) is convex and increasing with respect to the gradient variable \(t\in \left[ 0,\infty \right) \). 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 \(g\left( x,\left| Du\right| \right) =a(x)|Du|^{p(x)}\log (1+|Du|)\) or, when \(n=2,3\), even asymptotic linear growth with energy integrals of the type
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Acknowledgements
The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors thank the referee for having carefully read the manuscript, for her/his patience, remarks and suggestions.
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Di Marco, T., Marcellini, P. A-priori gradient bound for elliptic systems under either slow or fast growth conditions. Calc. Var. 59, 120 (2020). https://doi.org/10.1007/s00526-020-01769-7
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DOI: https://doi.org/10.1007/s00526-020-01769-7
Keywords
- Regularity
- Local Lipschitz continuity
- Nonlinear elliptic differential systems
- Calculus of variations
- p
- q-growth conditions
- General growth conditions
- Slow or fast growth conditions