On very weak solutions of certain elliptic systems with double phase growth
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917244332670976 |
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| author | Kaiho, Yoshiki |
| author_facet | Kaiho, Yoshiki |
| contents | In this paper, we prove a higher integrability result for very weak solutions of higher-order elliptic systems involving a double phase operator as the principal part. As a model case, we consider \begin{equation} \int_Ω \left( |D^m u|^{p-2}D^m u + a(x)|D^m u|^{q-2}D^m u \right) \cdot D^m φ= 0 \quad \text{for any } φ\in C_c^{\infty}(Ω), \end{equation} where $n,m \in \mathbb{N},\ n\ge 2,\,1 < p \le q < \infty,\,Ω\subset \mathbb{R}^n$ is an open set and $a:Ω\rightarrow [0,\infty)$ is a measurable function. The proof is based on a construction of an appropriate test function by the Lipschitz truncation technique, a deduction of a reverse Hölder inequality and an application of Gehring's lemma. Our contributions include estimates for weighted mean value polynomials and sharp Sobolev--Poincaré-type inequalities for the double phase operator. Our result can be viewed as a generalization with respect to the derivative order, the coefficient function and the growth conditions of the recent paper by Baasandorj, Byun and Kim (Trans. Amer. Math. Soc. 376:8733-8768,2023). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03044 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On very weak solutions of certain elliptic systems with double phase growth Kaiho, Yoshiki Analysis of PDEs 35D30 (primary), 35J48, 35J60 (secondary) In this paper, we prove a higher integrability result for very weak solutions of higher-order elliptic systems involving a double phase operator as the principal part. As a model case, we consider \begin{equation} \int_Ω \left( |D^m u|^{p-2}D^m u + a(x)|D^m u|^{q-2}D^m u \right) \cdot D^m φ= 0 \quad \text{for any } φ\in C_c^{\infty}(Ω), \end{equation} where $n,m \in \mathbb{N},\ n\ge 2,\,1 < p \le q < \infty,\,Ω\subset \mathbb{R}^n$ is an open set and $a:Ω\rightarrow [0,\infty)$ is a measurable function. The proof is based on a construction of an appropriate test function by the Lipschitz truncation technique, a deduction of a reverse Hölder inequality and an application of Gehring's lemma. Our contributions include estimates for weighted mean value polynomials and sharp Sobolev--Poincaré-type inequalities for the double phase operator. Our result can be viewed as a generalization with respect to the derivative order, the coefficient function and the growth conditions of the recent paper by Baasandorj, Byun and Kim (Trans. Amer. Math. Soc. 376:8733-8768,2023). |
| title | On very weak solutions of certain elliptic systems with double phase growth |
| topic | Analysis of PDEs 35D30 (primary), 35J48, 35J60 (secondary) |
| url | https://arxiv.org/abs/2602.03044 |