Optimal regularity results in Sobolev-Lorentz spaces for linear elliptic equations with $L^1$- or measure data
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arXiv
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| Natura: | Preprint |
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2025
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| author | Kim, Hyunseok Lee, Young-Ran Ok, Jihoon |
| author_facet | Kim, Hyunseok Lee, Young-Ran Ok, Jihoon |
| contents | It has been well known that if $Ω$ is a bounded $C^1$-domain in $\R^n,\ n \ge 2$, then for every Radon measure $f$ on $Ω$ with finite total variation, there exists a unique weak solution $u\in W_0^{1,1}(Ω)$ of the Poisson equation $-Δu=f$ in $Ω$ satisfying $\nabla u \in L^{n/(n-1),\infty}(Ω;\R^n )$. In this paper, optimal regularity properties of the solution $u$ are established in Sobolev-Lorentz spaces $L_α^{p,q}(Ω)$ of order $ α$ less than but arbitrarily close to $2$. More precisely, for any $0 \le α<1$, we show that $u\in L_{α+1}^{p(α),\infty}(Ω)$, where $p(α)= n/(n-1+α)$. Moreover, using an embedding result for Sobolev-Lorentz spaces $L_α^{p,q}(Ω)$ into classical Besov spaces $B_α^{p,q}(Ω)$, we deduce that $u\in B_{α+1}^{p(α),\infty}(Ω)$. 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 $C^{1,1}$, then for each $G\in L^1 (Ω;\R^n )$ there exists a unique very weak solution $v\in L^{n/(n-1),\infty} (Ω)$ 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, $v\in L_α^{p(α),\infty}(Ω)\cap B_α^{p(α),\infty}(Ω)$ for every $0 \le α< 1$. This regularity result is also proved for more general equations with nonhomogeneous boundary data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal regularity results in Sobolev-Lorentz spaces for linear elliptic equations with $L^1$- or measure data Kim, Hyunseok Lee, Young-Ran Ok, Jihoon Analysis of PDEs 35J15, 35J25 It has been well known that if $Ω$ is a bounded $C^1$-domain in $\R^n,\ n \ge 2$, then for every Radon measure $f$ on $Ω$ with finite total variation, there exists a unique weak solution $u\in W_0^{1,1}(Ω)$ of the Poisson equation $-Δu=f$ in $Ω$ satisfying $\nabla u \in L^{n/(n-1),\infty}(Ω;\R^n )$. In this paper, optimal regularity properties of the solution $u$ are established in Sobolev-Lorentz spaces $L_α^{p,q}(Ω)$ of order $ α$ less than but arbitrarily close to $2$. More precisely, for any $0 \le α<1$, we show that $u\in L_{α+1}^{p(α),\infty}(Ω)$, where $p(α)= n/(n-1+α)$. Moreover, using an embedding result for Sobolev-Lorentz spaces $L_α^{p,q}(Ω)$ into classical Besov spaces $B_α^{p,q}(Ω)$, we deduce that $u\in B_{α+1}^{p(α),\infty}(Ω)$. 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 $C^{1,1}$, then for each $G\in L^1 (Ω;\R^n )$ there exists a unique very weak solution $v\in L^{n/(n-1),\infty} (Ω)$ 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, $v\in L_α^{p(α),\infty}(Ω)\cap B_α^{p(α),\infty}(Ω)$ for every $0 \le α< 1$. This regularity result is also proved for more general equations with nonhomogeneous boundary data. |
| title | Optimal regularity results in Sobolev-Lorentz spaces for linear elliptic equations with $L^1$- or measure data |
| topic | Analysis of PDEs 35J15, 35J25 |
| url | https://arxiv.org/abs/2506.15005 |