On the explicit representation of the trace space H^{frac{3}{2}} and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

Pier Domenico Lamberti,Luigi Provenzano

doi:10.1007/s13163-021-00385-z

Pier Domenico Lamberti, Luigi Provenzano

Open Access

https://doi.org/10.1007/s13163-021-00385-z

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Abstract

We consider the problem of describing the traces of functions in H^2(Omega ) on the boundary of a Lipschitz domain Omega of mathbb R^N, Nge 2. We provide a definition of those spaces, in particular of H^{frac{3}{2}}(partial Omega ), by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space H^1(Omega ), based on the classical second order Steklov problem.

Highlights

We consider the trace spaces of functions in H 2( ) when is a bounded Lipschitz domain in RN, briefly is of class C0,1, for N ≥ 2
A relevant problem in the theory of Sobolev Spaces consists in describing the trace spaces γ0(H 2( )), γ1(H 2( )), and the total trace space (H 2( ))
From a historical point of view, this issue finds its origins in [23] where J. Hadamard proposed his famous counterexample pointing out the importance to understand which conditions on the datum g guarantee that the solution v to the Dirichlet problem v = 0, in, v = g, on ∂, has square summable gradient. This problem can be reformulated as the problem of finding necessary and sufficient conditions on g such that g = γ0(u) for some u ∈ H 1( )

Summary

Introduction

We consider the trace spaces of functions in H 2( ) when is a bounded Lipschitz domain in RN , briefly is of class C0,1, for N ≥ 2. It turns out that the analysis of problems (BSμ)-(BSλ) provides further information on the total trace (H 2( )). The definitions in (4.1) and (4.2) are given by means of Fourier series and the coefficients in such expansions need to satisfy certain summability conditions, which are strictly related to the asymptotic behavior of the eigenvalues of (BSλ) and (BSμ). Lipschitz continuity results for the functions μ → λ j (μ) and λ → μ j (λ) and we show that problems (DBS) and (NBS) can be seen as limiting problems for (BSμ) and (BSλ) as μ → −∞ and λ → −∞, respectively. In Appendix A we provide a complete description of problems (BSμ) and (BSλ) on the unit ball for σ = 0. In Appendix C we discuss problems (BSμ) and (BSλ) when μ > 0 and λ > η1

Preliminaries and notation

Multi-parameter Steklov problems

The BS eigenvalue problem

Representation of the solutions to the Dirichlet problem