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Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems Patrizia Pucci1

· Enzo Vitillaro1

Received: 15 April 2020 / Accepted: 21 April 2020 © Unione Matematica Italiana 2020

Abstract The paper deals with a nontrivial density result for C m (Ω) functions, with m ∈ N ∪ {∞}, in the space   W k,, p (Ω; Γ ) = u ∈ W k, p (Ω) : u |Γ ∈ W , p (Γ ) , endowed with the norm of (u, u |Γ ) in W k, p (Ω) × W , p (Γ ), where Ω is a bounded open subset of R N , N ≥ 2, with boundary Γ of class C m , k ≤  ≤ m and 1 ≤ p < ∞. Such a result is of interest when dealing with doubly elliptic problems involving two elliptic operators, one in Ω and the other on Γ . Moreover we shall also consider the case when a Dirichlet homogeneous boundary condition is imposed on a relatively open part of Γ and, N as a preliminary step, we shall prove an analogous result when either Ω = R N or Ω = R+ N and Γ = ∂R+ . Keywords Density results · Sobolev spaces · Smooth functions · The Laplace–Beltrami operator

To the memory of our dear friend Professor Domenico Candeloro with high feelings of admiration for his notable contributions in Mathematics. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM – GNAMPA Projects Equazioni alle derivate parziali: problemi e modelli (Prot_U-UFMBAZ-2020-000761). The first author was also partly supported by the Fondo Ricerca di Base di Ateneo – Esercizio 2017–2019 of the University of Perugia, named PDEs and Nonlinear Analysis, while the second author by the Progetto Equazione delle onde con condizioni acustiche, finanziato con il Fondo Ricerca di Base, 2019, della Università degli Studi di Perugia and by Progetti Equazioni delle onde con condizioni iperboliche ed acustiche al bordo, finanziati con i Fondi Ricerca di Base 2017 and 2018, della Università degli Studi di Perugia.

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Patrizia Pucci [email protected] Enzo Vitillaro [email protected]

1

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy

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P. Pucci, E. Vitillaro

Mathematics Subject Classification 46E35 · 46.38 · 46M35

1 Introduction and main results Density results for smooth functions in Sobolev spaces constitute a cornerstone in the classical theory of these spaces and in their applications in PDEs theory. Actually every textbook dealing with Sobolev spaces or PDEs devotes some attention to this subject, see for example [1,2,5,11–14,16,17]. The paper deals with Sobolev spaces of integer nonnegative order, which are the most classical ones, but the density subject is standard also when working with Sobolev spaces of fractional order, also known as Sobolev – Slobodeckij spaces, and with Besov spaces and Bessel – potential ones. See [17]. When considering Sobolev spaces in R N , N ≥ 1, it is well–known that the space of compactly supported smooth functions