On the Lavrentiev phenomenon for multiple integral scalar variational problems
We prove the non-occurrence of Lavrentiev gaps between Lipschitz and Sobolev functions for functionals of the formI(u)=∫ΩF(u,∇u),u|∂Ω=ϕ when ϕ:Rn→R is Lipschitz and Ω belongs to a wide class of open bounded sets in Rn containing Lipschitz domains. The Lagrangian F is assumed to be either convex in b...
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| Veröffentlicht in: | Journal of functional analysis 2014-05, Vol.266 (9), p.5921-5954 |
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| container_title | Journal of functional analysis |
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| creator | Bousquet, Pierre Mariconda, Carlo Treu, Giulia |
| description | We prove the non-occurrence of Lavrentiev gaps between Lipschitz and Sobolev functions for functionals of the formI(u)=∫ΩF(u,∇u),u|∂Ω=ϕ when ϕ:Rn→R is Lipschitz and Ω belongs to a wide class of open bounded sets in Rn containing Lipschitz domains. The Lagrangian F is assumed to be either convex in both variables or a sum of functions F(s,ξ)=a(s)g(ξ)+b(s) with g convex and s↦a(s)g(0)+b(s) satisfying a non-oscillatory condition at infinity. We thus derive the non-occurrence of the Lavrentiev phenomenon for unnecessarily convex functionals of the gradient. No growth conditions are assumed. |
| doi_str_mv | 10.1016/j.jfa.2013.12.020 |
| format | Article |
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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Analysis of PDEs Lavrentiev Lavrentiev gap Lavrentiev phenomenon Lipschitz approximation Mathematics Regularity Star-shaped domain |
| title | On the Lavrentiev phenomenon for multiple integral scalar variational problems |
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