**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, 1–3 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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