Riv. Mat. Univ. Parma, Vol. 7, No. 2, 2016

Angelo Favini[a], Noboru Okazawa[b] and Jan Prüss[c]

Singular perturbation approach to Legendre type operators

Pages: 309-319
Received: 15 January 2016
Accepted in revised form: 10 October 2016
Mathematics Subject Classification (2010): Primary: 35J70, Secondary: 47H06.
Keywords: Selfadjointness, Legendre type operators, degeneration at the boundary, singular perturbation.
Author address:
[a]: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
[b]: Department of Mathematics, Tokyo University of Science, 13 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
[c]: Fachbereich Mathematik und Informatik, Institut für Analysis, Martin-Luther Universtät Halle-Wittenberg, Theodor-Lieser-Strasse 506120, D-06099 Halle (Saale), Germany

N. Okazawa partially supported by Grant-in-Aid for Scientific Research (C), No.25400182

Abstract: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with compact smooth boundary (\(N \in \mathbb{N}\)). Then this paper is concerned with the nonnegative selfadjointness in \(L^2(\Omega)\) of the maximal realization \(T_{2}\) of \(N\)-dimensional second-order differential operators in divergence form with diffusion coefficients vanishing on the boundary \(\Gamma = \partial\Omega\). The operators may be called Legendre type operators over \(\Omega\). The key to the proof is a singular perturbation argument developed in [9] In particular, the resolvent of \(T_{2}\) is given as the uniform limit of \((\xi + n^{-1}(-\Delta) + T_{2})^{-1}\) as \(n \to \infty\), for every \(\xi > 0\), where \(-\Delta\) is the Neumann-Laplacian in \(L^2(\Omega)\). It should be noted that if \(N=1\) then \((\xi + n^{-1}(-\Delta) + T_{p})^{-1}\) converges strongly to \((\xi + T_{p})^{-1}\) in \(L^{p}(I)\), where \(T_{p}\) is the one-dimensional analog constructed by Campiti, Metafune and Pallara [2].

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