Superposition operator in Sobolev spaces on domains

Superposition with subunitary powers in Sobolev spaces

Let 0 < a < 1 and set Φ(t) = |t|, ∀ t ∈ R. Let 1 < p < ∞ and n ≥ 1. We prove that the superposition operator u 7→ Φ(u) maps the Sobolev space W (R) into the fractional Sobolev space W (R). We also investigate the case of more general nonlinearities. Résumé. Superposition avec des puissances sousunitaires dans les espaces de Sobolev. Pour 0 < a < 1, soit Φ(t) = |t|, ∀ t ∈ R. Soient 1 < p < ∞ et ...

Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains

Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains

Extending recent work for the linear Poisson problem for the Laplacian in the framework of Sobolev-Besov spaces on Lipschitz domains by Jerison and Kenig [16], Fabes, Mendez and Mitrea [9], and Mitrea and Taylor [30], here we take up the task of developing a similar sharp theory for semilinear problems of the type ∆u − N(x, u) = F (x), equipped with Dirichlet and Neumann boundary conditions.

Sobolev Spaces and Regularity for Polyhedral Domains

We prove a regularity result on polyhedral domains P ⊂ R using the weighted Sobolev spaces Ka (P). In particular, we show that there is no loss of Ka –regularity for solutions of strongly elliptic systems with smooth coefficients. In the proof, we identify Ka (P) with the Sobolev spaces on P associated to the metric r P gE , where gE is the Euclidean metric and rP(x) is a smoothing of the Eucli...

Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions

A famous theorem of E. Gagliardo gives the characterization of traces for Sobolev spaces W 1, p (Ω) for 1 ≤ p < ∞ when Ω ⊂ R is a Lipschitz domain. The extension of this result to W m, p (Ω) for m ≥ 2 and 1 < p < ∞ is now well-known when Ω is a smooth domain. The situation is more complicated for polygonal and polyhedral domains since the characterization is given only in terms of local compati...