Global Calderón-Zygmund estimates for asymptotically convex fully nonlinear Grad-Mercier type equations
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911028141359104 |
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| author | Zhang, Yao Jin, Xiaofeng Ma, Lingwei Zhang, Zhenqiu |
| author_facet | Zhang, Yao Jin, Xiaofeng Ma, Lingwei Zhang, Zhenqiu |
| contents | In this paper, we consider the following Dirichlet problem for the fully nonlinear elliptic equation of Grad-Mercier type under asymptotic convexity conditions \begin{equation*}
\left\{
\begin{array}{ll}
F(D^2u(x),Du(x),u(x),x)=g(|\{y\in Ω:u(y)\ge u(x)\}|)+f(x) & \text{in } Ω,
u=ψ&\text{on } \partial Ω.
\end{array}
\right. \end{equation*} In order to overcome the non-convexity of the operator $F$ and the nonlocality of the nonhomogeneous term $g$, we apply the compactness methods and frozen technique to prove the existence of the $W^{2,p}$-viscosity solutions and the global $W^{2,p}$ estimate. As an application, we derive a Cordes-Nirenberg type continuous estimate up to boundary. Furthermore, we establish a global BMO estimate for the second derivatives of solutions by using an asymptotic approach, thereby refining the borderline case of Calderón-Zygmund estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23216 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Calderón-Zygmund estimates for asymptotically convex fully nonlinear Grad-Mercier type equations Zhang, Yao Jin, Xiaofeng Ma, Lingwei Zhang, Zhenqiu Analysis of PDEs 35B45, 35R05, 35B65 In this paper, we consider the following Dirichlet problem for the fully nonlinear elliptic equation of Grad-Mercier type under asymptotic convexity conditions \begin{equation*} \left\{ \begin{array}{ll} F(D^2u(x),Du(x),u(x),x)=g(|\{y\in Ω:u(y)\ge u(x)\}|)+f(x) & \text{in } Ω, u=ψ&\text{on } \partial Ω. \end{array} \right. \end{equation*} In order to overcome the non-convexity of the operator $F$ and the nonlocality of the nonhomogeneous term $g$, we apply the compactness methods and frozen technique to prove the existence of the $W^{2,p}$-viscosity solutions and the global $W^{2,p}$ estimate. As an application, we derive a Cordes-Nirenberg type continuous estimate up to boundary. Furthermore, we establish a global BMO estimate for the second derivatives of solutions by using an asymptotic approach, thereby refining the borderline case of Calderón-Zygmund estimates. |
| title | Global Calderón-Zygmund estimates for asymptotically convex fully nonlinear Grad-Mercier type equations |
| topic | Analysis of PDEs 35B45, 35R05, 35B65 |
| url | https://arxiv.org/abs/2506.23216 |