2026/01/09 by Chérif Amrouche, Mohand Moussaoui
#math.AP
In this work, we revisit the following estimate due to Dahlberg \citeDahl. Let x0 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 x0, then C-1 \Vert u \VertL2(Γ) ≤ (∫Ω\varrho\vert ∇ u \vert2)1/2 ≤ C \Vert u \VertL2(Γ), 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: (\mathscrLD0) -Δu = f in Ω and u = 0 on Γ, in a framework of fractional Sobolev spaces Hs(Ω), when Ω is a polygon or a polyhedron domain and 1/2 ≤ s ≤ 2. Thanks to these regularity results and an explicit function given by Ne\mathrm\checkcas, 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 H1/200(Ω)\hookrightarrow E(∇; Ω) \hookrightarrow H1/2(Ω) and the trace operator γ0 from E(∇; Ω) into L2(Γ) is well-defined and continuous. This leads to an alternative to the functions H1/2(Ω), non necessarily harmonic, for having a trace in L2(Γ) and also to a new characterization of H1/200(Ω) as the kernel of this operator. However, we show that if the domain Ω is of class \mathscrC1, 1, then the above inequalities are valid.