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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2603.10797 |
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| _version_ | 1866914386359091200 |
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| author | Liang, Lichun |
| author_facet | Liang, Lichun |
| contents | In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u,x)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ has the periodicity in $x$. Under the assumption that the oscillation of $F(M,x)$ in $x$ is ``small", we establish the existence and Liouville type results for quadratic growth solutions, which can be expressed into the sum of a quadratic polynomial and a periodic function. Consequently, these results are generalization of the existing results for linear elliptic equations $a_{ij}D_{ij}u=0$ and fully nonlinear elliptic equations $F(D^2u)=f$ with the periodic data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Liouville theorem for fully nonlinear elliptic equations with the small oscillation and the periodicity in $x$ and the periodic right hand term Liang, Lichun Analysis of PDEs In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u,x)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ has the periodicity in $x$. Under the assumption that the oscillation of $F(M,x)$ in $x$ is ``small", we establish the existence and Liouville type results for quadratic growth solutions, which can be expressed into the sum of a quadratic polynomial and a periodic function. Consequently, these results are generalization of the existing results for linear elliptic equations $a_{ij}D_{ij}u=0$ and fully nonlinear elliptic equations $F(D^2u)=f$ with the periodic data. |
| title | Liouville theorem for fully nonlinear elliptic equations with the small oscillation and the periodicity in $x$ and the periodic right hand term |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.10797 |