2025/06/17 by Kim, Hyunseok, Lee, Young-Ran, Ok, Jihoon
Mathematics · #35J15 #35J25 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2506.15005
openalex publication_date 2025/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It has been well known that if Ω is a bounded C1-domain in \Rn, n ≥ 2, then for every Radon measure f on Ω with finite total variation, there exists a unique weak solution u∈ W01,1(Ω) of the Poisson equation -Δu=f in Ω satisfying ∇ u ∈ Ln/(n-1),∞(Ω;\Rn ). In this paper, optimal regularity properties of the solution u are established in Sobolev-Lorentz spaces Lαp,q(Ω) of order α less than but arbitrarily close to 2. More precisely, for any 0 ≤ α<1, we show that u∈ Lα+1p(α),∞(Ω), where p(α)= n/(n-1+α). Moreover, using an embedding result for Sobolev-Lorentz spaces Lαp,q(Ω) into classical Besov spaces Bαp,q(Ω), we deduce that u∈ Bα+1p(α),∞(Ω). Indeed, these regularity results are proved for solutions of the Dirichlet problems for more general linear elliptic equations with nonhomogeneous boundary data. On the other hand, it is known that if Ω is of class C1,1, then for each G∈ L1 (Ω;\Rn ) there exists a unique very weak solution v∈ Ln/(n-1),∞ (Ω) of -Δv= \rm div G in Ω satisfying the boundary condition v=0 in some sense. We prove that v has the optimal regularity property, that is, v∈ Lαp(α),∞(Ω)∩ Bαp(α),∞(Ω) for every 0 ≤ α< 1. This regularity result is also proved for more general equations with nonhomogeneous boundary data.