$BMO$ and gradient estimates for solutions of critical elliptic equations
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| Format: | Preprint |
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2024
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| _version_ | 1866911988004683776 |
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| author | Chen, You-Wei Benson Manfredi, Juan Spector, Daniel |
| author_facet | Chen, You-Wei Benson Manfredi, Juan Spector, Daniel |
| contents | In this paper we explore several applications of the recently introduced spaces of functions of bounded $β$-dimensional mean oscillation for $β\in (0,n]$ to regularity theory of critical exponent elliptic equations.
We first show that functions with gradient in weak-$L^n$ are in $BMO^β$ for any $β\in (0,n]$, improving the classical result $\nabla u\in L^n$ implies $u\in BMO$. We apply this result to the Poisson equation $-Δu = \operatorname*{div} F$ with zero boundary conditions in a bounded $C^1$ domain to show that $u\in BMO^β$ when $F$ is in weak-$L^n$. Next, we consider the $n$-Laplace equation \begin{align*} -\operatorname*{div}( |\nabla U|^{n-2} \nabla U) &= F \text{ in } Ω, \newline
U &=0 \text{ on }\partial Ω. \end{align*} with $F\in L^1(Ω)$ and show that the classical result $u\in BMO$ can be improved to $u\in BMO^β$. Finally, we consider the $n$-Laplace equation in the case when $F \in L^1$, $\operatorname*{div} F=0$ and prove that for smooth domains $Ω$ we have the estimate \begin{align*} \|\nabla U \|_{L^n} \mathbb \leq C \, \|F\|^{1/(n-1)}_{L^1}, \end{align*} where the constant $C$ is independent of $F$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_13884 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $BMO$ and gradient estimates for solutions of critical elliptic equations Chen, You-Wei Benson Manfredi, Juan Spector, Daniel Analysis of PDEs Classical Analysis and ODEs Functional Analysis In this paper we explore several applications of the recently introduced spaces of functions of bounded $β$-dimensional mean oscillation for $β\in (0,n]$ to regularity theory of critical exponent elliptic equations. We first show that functions with gradient in weak-$L^n$ are in $BMO^β$ for any $β\in (0,n]$, improving the classical result $\nabla u\in L^n$ implies $u\in BMO$. We apply this result to the Poisson equation $-Δu = \operatorname*{div} F$ with zero boundary conditions in a bounded $C^1$ domain to show that $u\in BMO^β$ when $F$ is in weak-$L^n$. Next, we consider the $n$-Laplace equation \begin{align*} -\operatorname*{div}( |\nabla U|^{n-2} \nabla U) &= F \text{ in } Ω, \newline U &=0 \text{ on }\partial Ω. \end{align*} with $F\in L^1(Ω)$ and show that the classical result $u\in BMO$ can be improved to $u\in BMO^β$. Finally, we consider the $n$-Laplace equation in the case when $F \in L^1$, $\operatorname*{div} F=0$ and prove that for smooth domains $Ω$ we have the estimate \begin{align*} \|\nabla U \|_{L^n} \mathbb \leq C \, \|F\|^{1/(n-1)}_{L^1}, \end{align*} where the constant $C$ is independent of $F$. |
| title | $BMO$ and gradient estimates for solutions of critical elliptic equations |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2407.13884 |