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| Format: | Preprint |
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2018
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| Online Access: | https://arxiv.org/abs/1802.03066 |
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| _version_ | 1866912175594930176 |
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| author | Maor, Cy |
| author_facet | Maor, Cy |
| contents | Verifying lower-semicontinuity of integral functionals in the weak topology of Sobolev spaces is a central theme in the calculus of variations. For integral functionals with $p$-growth, quasiconvexity is a necessary condition for weak lower-semicontinuity in $W^{1,p}$, but is only sufficient if some additional conditions are met.The standard functional showing the necessity of additional conditions is $f\mapsto \int_Ω\det \nabla f$, which fails to be weakly lower-semicontinuous. However, the common examples showing this failure are non-injective and have a lot of shear. The aim of this short note is to point out that a known sequence of conformal diffeomorphisms of the $d$-dimensional unit ball that converges weakly to a constant in $W^{1,d}$, exemplifies the weak discontinuity of this functional even when restricting a space to functions which are "as nice as possible". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_03066 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A simple example of the weak discontinuity of $f\mapsto \int \det \nabla f$ Maor, Cy Analysis of PDEs 49J45 Verifying lower-semicontinuity of integral functionals in the weak topology of Sobolev spaces is a central theme in the calculus of variations. For integral functionals with $p$-growth, quasiconvexity is a necessary condition for weak lower-semicontinuity in $W^{1,p}$, but is only sufficient if some additional conditions are met.The standard functional showing the necessity of additional conditions is $f\mapsto \int_Ω\det \nabla f$, which fails to be weakly lower-semicontinuous. However, the common examples showing this failure are non-injective and have a lot of shear. The aim of this short note is to point out that a known sequence of conformal diffeomorphisms of the $d$-dimensional unit ball that converges weakly to a constant in $W^{1,d}$, exemplifies the weak discontinuity of this functional even when restricting a space to functions which are "as nice as possible". |
| title | A simple example of the weak discontinuity of $f\mapsto \int \det \nabla f$ |
| topic | Analysis of PDEs 49J45 |
| url | https://arxiv.org/abs/1802.03066 |