Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers
We establish C∞-partial regularity results for relaxed minimizers of strongly quasiconvex functionals F[u; Ω] := ˆ Ω F(∇u) dx, u: Ω → R N , subject to a q-growth condition |F(z)| 6 c(1 + |z| q ), z ∈ RN×n, and natural p-mean coercivity conditions on F ∈ C∞(RN×n) for the basically optimal exponent ra...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Springer
2024
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| _version_ | 1826314501122686976 |
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| author | Gmeineder, F Kristensen, JAN |
| author_facet | Gmeineder, F Kristensen, JAN |
| author_sort | Gmeineder, F |
| collection | OXFORD |
| description | We establish C∞-partial regularity results for relaxed minimizers of strongly quasiconvex functionals F[u; Ω] := ˆ Ω F(∇u) dx, u: Ω → R N , subject to a q-growth condition |F(z)| 6 c(1 + |z| q ), z ∈ RN×n, and natural p-mean coercivity conditions on F ∈ C∞(RN×n) for the basically optimal exponent range 1 6 p 6 q < min{ np n−1 , p + 1}. With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F, our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from SCHMIDT’s foundational work [127] for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for p = 1. We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations a la F ` ONSECA & MALY´ [67]. |
| first_indexed | 2024-09-25T04:35:03Z |
| format | Journal article |
| id | oxford-uuid:6e10b77f-8944-4787-9527-a440514560bb |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:35:03Z |
| publishDate | 2024 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:6e10b77f-8944-4787-9527-a440514560bb2024-09-16T11:55:28ZQuasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:6e10b77f-8944-4787-9527-a440514560bbEnglishSymplectic ElementsSpringer2024Gmeineder, FKristensen, JANWe establish C∞-partial regularity results for relaxed minimizers of strongly quasiconvex functionals F[u; Ω] := ˆ Ω F(∇u) dx, u: Ω → R N , subject to a q-growth condition |F(z)| 6 c(1 + |z| q ), z ∈ RN×n, and natural p-mean coercivity conditions on F ∈ C∞(RN×n) for the basically optimal exponent range 1 6 p 6 q < min{ np n−1 , p + 1}. With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F, our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from SCHMIDT’s foundational work [127] for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for p = 1. We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations a la F ` ONSECA & MALY´ [67]. |
| spellingShingle | Gmeineder, F Kristensen, JAN Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers |
| title | Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers |
| title_full | Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers |
| title_fullStr | Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers |
| title_full_unstemmed | Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers |
| title_short | Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers |
| title_sort | quasiconvex functionals of p q growth and the partial regularity of relaxed minimizers |
| work_keys_str_mv | AT gmeinederf quasiconvexfunctionalsofpqgrowthandthepartialregularityofrelaxedminimizers AT kristensenjan quasiconvexfunctionalsofpqgrowthandthepartialregularityofrelaxedminimizers |