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
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.
[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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