2024/07/18 by You-Wei Benson Chen, Chen, You-Wei Benson, Juan J. Manfredi +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2407.13884
openalex publication_date 2024/07/18 · openalex created_date 2024/09/26 · openalex updated_date 2026/07/28
In this paper we explore several applications of the recently introduced spaces of functions of bounded β-dimensional mean oscillation for β∈ (0,n] to regularity theory of critical exponent elliptic equations. We first show that functions with gradient in weak-Ln are in BMOβ for any β∈ (0,n], improving the classical result ∇ u∈ Ln implies u∈ BMO. We apply this result to the Poisson equation -Δu = \operatorname*div F with zero boundary conditions in a bounded C1 domain to show that u∈ BMOβ when F is in weak-Ln. Next, we consider the n-Laplace equation -\operatorname*div( |∇ U|n-2 ∇ U) amp;= F in Ω, \newline U amp;=0 on ∂ Ω. with F∈ L1(Ω) and show that the classical result u∈ BMO can be improved to u∈ BMOβ. Finally, we consider the n-Laplace equation in the case when F ∈ L1, \operatorname*div F=0 and prove that for smooth domains Ω we have the estimate ‖∇ U ‖Ln \mathbb ≤ C ‖F‖1/(n-1)L1, where the constant C is independent of F.