Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917265185701888 |
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| author | Ambrosio, Pasquale Cupini, Giovanni Mascolo, Elvira |
| author_facet | Ambrosio, Pasquale Cupini, Giovanni Mascolo, Elvira |
| contents | We establish the local boundedness of the local minimizers $u:Ω\rightarrow\mathbb{R}^{m}$ of non-uniformly elliptic integrals of the form $\int_Ωf(x,Dv)\,dx$, where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ ($n\geq2)$ and the integrand satisfies anisotropic growth conditions of the type \[ \sum_{i=1}^{n}λ_{i}(x)|ξ_{i}|^{p_{i}}\le f(x,ξ)\leμ(x)\left\{ 1+|ξ|^{q}\right\} \] for some exponents $q\geq p_{i}>1$ and with non-negative functions $λ_{i},μ$ fulfilling suitable summability assumptions. The main novelties here are the degenerate and anisotropic behaviour of the integrand and the fact that we also address the case of vectorial minimizers ($m>1$). Our proof is based on the celebrated Moser iteration technique and employs an embedding result for anisotropic Sobolev spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals Ambrosio, Pasquale Cupini, Giovanni Mascolo, Elvira Analysis of PDEs 49N60, 49J40, 35J60, 35A23 We establish the local boundedness of the local minimizers $u:Ω\rightarrow\mathbb{R}^{m}$ of non-uniformly elliptic integrals of the form $\int_Ωf(x,Dv)\,dx$, where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ ($n\geq2)$ and the integrand satisfies anisotropic growth conditions of the type \[ \sum_{i=1}^{n}λ_{i}(x)|ξ_{i}|^{p_{i}}\le f(x,ξ)\leμ(x)\left\{ 1+|ξ|^{q}\right\} \] for some exponents $q\geq p_{i}>1$ and with non-negative functions $λ_{i},μ$ fulfilling suitable summability assumptions. The main novelties here are the degenerate and anisotropic behaviour of the integrand and the fact that we also address the case of vectorial minimizers ($m>1$). Our proof is based on the celebrated Moser iteration technique and employs an embedding result for anisotropic Sobolev spaces. |
| title | Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals |
| topic | Analysis of PDEs 49N60, 49J40, 35J60, 35A23 |
| url | https://arxiv.org/abs/2503.18917 |