2024/06/19 by Ting Peng, Peng, Ting, Chaochuan Wang +3
Mathematics · #30C70 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2406.13285
openalex publication_date 2024/06/19 · openalex created_date 2024/06/22 · openalex updated_date 2026/07/28
Let A1 and A2 be two circular annuli and let ρ be a radial metric defined in the annuli A2. We study the existence and uniqueness of the extremal problem for weighted combined energy between A1 and A2, and obtain that the extremal mapping is a certain radial mapping. In fact, this extremal mapping generalizes the ρ-harmonic mapping and satisfies equation (2.7) obtained by mean of variation for weighted combined energy. Meanwhile, we get a ρ-Nitsche type inequality. This extends the results of Kalaj (J. Differential Equations, 268(2020)) and YTF (Arch. Math., 122(2024)), where they considered the case ρ=1 and ρ=\frac1|h|2, respectively. Moreover, in the course of proving the extremal problem for weighted combined energy we also investigate the extremal problem for the weighted combined distortion (see Theorem 4.1). This extends the result obtained by Kalaj (J. London Math. Soc., 93(2016)).