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].


[1] H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris 1983; (English Translation) Functional Analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York 2011. MR0697382;   MR2759829
[2] M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum 57 (1998), 136. MR1621852
[3] S. Fornaro, G. Metafune, D. Pallara and J. Prüss, \(L^p\)-theory for some elliptic and parabolic problems with first order degeneracy at the boundary, J. Math. Pures Appl. 87 (2007), 367393. MR2317339
[4] S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, One-dimensional degenerate operators in \(L^p\)-spaces, J. Math. Anal. Appl. 402 (2013), 308318. MR3023260
[5] S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Second order elliptic operators in \(L^2\) with first order degeneration at the boundary and outward poiting drift, Commun. Pure Appl. Anal. 14 (2015), 407419. MR3311733
[6] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York 1966; (Reprint of the 2nd edition) Classics in Mathematics, Springer-Verlag, Berlin 1995. MR0203473;   MR1335452
[7] T. Kato, Singular perturbation and semigroup theory, Lecture Notes in Math., 565, Springer, Berlin 1976, 104112. MR0458244
[8] N. Okazawa, Approximation of linear \(m\)-accretive operators in a Hilbert space, Osaka J. Math. 14 (1977), 8594. MR0451024
[9] N. Okazawa, Singular perturbations of \(m\)-accretive operators, J. Math. Soc. Japan 32 (1980), 1944. MR0554513
[10] N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Proc. Amer. Math. Soc. 113 (1991), 701706. MR1072347
[11] Z. Schuss, Singular perturbations and the transition from thin plate to membrane, Proc. Amer. Math. Soc. 58 (1976), 139147. MR0412571

Home Riv.Mat.Univ.Parma