On the traces of harmonic functions $H^{1/2}$ and $H^{3/2}$ in Lipschitz domains
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908754835931136 |
|---|---|
| author | Amrouche, Chérif Moussaoui, Mohand |
| author_facet | Amrouche, Chérif Moussaoui, Mohand |
| contents | In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $Ω$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in $Ω$ and vanishes at $\textit{\textbf x}_0$, then \begin{equation*} C^{-1} \Vert u \Vert_{L^2(Γ)} \leq \Big(\int_Ω\varrho\vert \nabla u \vert^2\Big)^{1/2} \leq C \Vert u \Vert_{L^2(Γ)}, \end{equation*} where $\varrho$ is the distance to the boundary of $Ω$. Using Grisvard's work and interpolation theory for subspaces, we complete the solvability of the inhomogeneous Dirichlet problem: $$ (\mathscr{L}_D^0)\ \ \ \ -Δu = f\quad \ \mbox{in}\ Ω\quad \mbox{and } \quad u = 0 \ \ \mbox{on }Γ, $$ in a framework of fractional Sobolev spaces $H^s(Ω)$, when $Ω$ is a polygon or a polyhedron domain and $1/2 \leq s \leq 2$. Thanks to these regularity results and an explicit function given by Ne$\mathrm{\check{c}}$as, we show that the above inequalities cannot be valid in their current form. On the other hand, we identify a functional space which satisfies the embeddings $H^{1/2}_{00}(Ω)\hookrightarrow E(\nabla;\, Ω) \hookrightarrow H^{1/2}(Ω)$ and the trace operator $γ_0$ from $E(\nabla;\, Ω)$ into $L^2(Γ)$ is well-defined and continuous. This leads to an alternative to the functions $H^{1/2}(Ω)$, non necessarily harmonic, for having a trace in $L^2(Γ)$ and also to a new characterization of $H^{1/2}_{00}(Ω)$ as the kernel of this operator. However, we show that if the domain $Ω$ is of class $\mathscr{C}^{1, 1}$, then the above inequalities are valid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05610 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the traces of harmonic functions $H^{1/2}$ and $H^{3/2}$ in Lipschitz domains Amrouche, Chérif Moussaoui, Mohand Analysis of PDEs 35B45, 35B65, 35J05, 35J25, 35J47 In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $Ω$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in $Ω$ and vanishes at $\textit{\textbf x}_0$, then \begin{equation*} C^{-1} \Vert u \Vert_{L^2(Γ)} \leq \Big(\int_Ω\varrho\vert \nabla u \vert^2\Big)^{1/2} \leq C \Vert u \Vert_{L^2(Γ)}, \end{equation*} where $\varrho$ is the distance to the boundary of $Ω$. Using Grisvard's work and interpolation theory for subspaces, we complete the solvability of the inhomogeneous Dirichlet problem: $$ (\mathscr{L}_D^0)\ \ \ \ -Δu = f\quad \ \mbox{in}\ Ω\quad \mbox{and } \quad u = 0 \ \ \mbox{on }Γ, $$ in a framework of fractional Sobolev spaces $H^s(Ω)$, when $Ω$ is a polygon or a polyhedron domain and $1/2 \leq s \leq 2$. Thanks to these regularity results and an explicit function given by Ne$\mathrm{\check{c}}$as, we show that the above inequalities cannot be valid in their current form. On the other hand, we identify a functional space which satisfies the embeddings $H^{1/2}_{00}(Ω)\hookrightarrow E(\nabla;\, Ω) \hookrightarrow H^{1/2}(Ω)$ and the trace operator $γ_0$ from $E(\nabla;\, Ω)$ into $L^2(Γ)$ is well-defined and continuous. This leads to an alternative to the functions $H^{1/2}(Ω)$, non necessarily harmonic, for having a trace in $L^2(Γ)$ and also to a new characterization of $H^{1/2}_{00}(Ω)$ as the kernel of this operator. However, we show that if the domain $Ω$ is of class $\mathscr{C}^{1, 1}$, then the above inequalities are valid. |
| title | On the traces of harmonic functions $H^{1/2}$ and $H^{3/2}$ in Lipschitz domains |
| topic | Analysis of PDEs 35B45, 35B65, 35J05, 35J25, 35J47 |
| url | https://arxiv.org/abs/2601.05610 |