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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.19934 |
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| _version_ | 1866912159346196480 |
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| author | Koch, Lukas Ruf, Matthias Schäffner, Mathias |
| author_facet | Koch, Lukas Ruf, Matthias Schäffner, Mathias |
| contents | We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form
$$
F: g+W_0^{1,1}(Ω)^m\to\mathbb{R}\cup\{+\infty\},\qquad F(u)=\int_ΩW(x,\mathrm{D} u)\,\mathrm{d}x,
$$
where the boundary datum $g:Ω\subset \mathbb{R}^d\to\mathbb{R}^m$ is sufficiently regular, $ξ\mapsto W(x,ξ)$ is convex and lower semicontinuous, satisfies $p$-growth from below and suitable growth conditions from above. More precisely, if $p\leq d-1$, we assume $q$-growth from above with $q\leq \frac{(d-1)p}{d-1-p}$, while for $p>d-1$ we require essentially no growth conditions from above and allow for unbounded integrands. Concerning the $x$-dependence, we impose a well-known local stability estimate that is redundant in the autonomous setting, but in the general non-autonomous case can further restrict the growth assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19934 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Lavrentiev gap for convex, vectorial integral functionals Koch, Lukas Ruf, Matthias Schäffner, Mathias Analysis of PDEs 49K40 We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form $$ F: g+W_0^{1,1}(Ω)^m\to\mathbb{R}\cup\{+\infty\},\qquad F(u)=\int_ΩW(x,\mathrm{D} u)\,\mathrm{d}x, $$ where the boundary datum $g:Ω\subset \mathbb{R}^d\to\mathbb{R}^m$ is sufficiently regular, $ξ\mapsto W(x,ξ)$ is convex and lower semicontinuous, satisfies $p$-growth from below and suitable growth conditions from above. More precisely, if $p\leq d-1$, we assume $q$-growth from above with $q\leq \frac{(d-1)p}{d-1-p}$, while for $p>d-1$ we require essentially no growth conditions from above and allow for unbounded integrands. Concerning the $x$-dependence, we impose a well-known local stability estimate that is redundant in the autonomous setting, but in the general non-autonomous case can further restrict the growth assumptions. |
| title | On the Lavrentiev gap for convex, vectorial integral functionals |
| topic | Analysis of PDEs 49K40 |
| url | https://arxiv.org/abs/2305.19934 |