Journal of the Ramanujan Mathematical Society

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On The Extrema of the Fundamental Eigenvalue of a Family of Schrodinger Operators


Affiliations
1 Department of Mathematics, University of Mumbai, Mumbai - 400098, India
2 K. C. College, Churchgate, Mumbai - 400020, India
     

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Let D be an open regular polygon of n sides in ℝ2. Let ℘0 ⊂ D be an open regular polygon of n sides having the same center of mass and circumscribed by a circle C contained in D. We fix D and vary ℘0 by rotating it in C about its center of mass. Let ℘t (t ∈ ℝ) be the family of polygons obtained in this fashion. Let χ℘t denote the indicator function of the subset ℘t of D. For any non-zero constant α ∈ ℝ it is shown that the Fundamental Eigenvalue of the Schr¨odinger operators −Δ+ αχ℘t attains its extremum when the axes of symmetry of ℘0 coincide with those of D.
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  • Ahmad EL Soufi and Rola Kiwan, Extremal first dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry, SIAM J. Math. Anal., 39 (2007) no. 4, 1112–1119.

  • A. R. Aithal and Rajesh Raut, On the extrema of Dirichlet’s first eigenvalue of a family of punctured regular polygons in two dimensional space forms, Proc. Indian Acad. Sci. (Math. Sc.), 122 (2012) no. 2, pp 257–281.

  • T. Aubin, Nonlinear analysis on manifolds-Monge-Ampere equations, SpringerVerlag (1982).

  • B. Folland Gerald, Introduction to partial differential equations, Prentice Hall of India, Second Edition (2001).

  • W. M. Gidas, B. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun, Math. Phys., 68 (1979) 209–243.

  • P. Grisvard, Singularities in boundary value problems, recherches en mathematiques appliquees, Masson, Paris; Springer-Verlag, Berlin, 22 (1992).

  • E.M. Harell II, P. Kroger and K. Kurata, On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue, Siam J. Math. Anal., 33 no. 1, 240–259.

  • Henrot Antoine, Extremum problems for eigenvalues of elliptic operators, Birkhauser Verlag, Springer (2006).

  • Kreyszig Erwin, Introductory Functional Analysis with Applications, John Wiley & Sons (1978).

  • M. Reed and B. Simon, Methods of modern mathematical physics IV: analysis of operators, Academic Press (1978).

  • J. Sokolowski and J. P. Zolesio, Introduction to shape optimization-shape sensitivity analysis, Springer-Verlag (1992).


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  • On The Extrema of the Fundamental Eigenvalue of a Family of Schrodinger Operators

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Authors

A. R. Aithal
Department of Mathematics, University of Mumbai, Mumbai - 400098, India
Pratiksha M. Kadam
K. C. College, Churchgate, Mumbai - 400020, India

Abstract


Let D be an open regular polygon of n sides in ℝ2. Let ℘0 ⊂ D be an open regular polygon of n sides having the same center of mass and circumscribed by a circle C contained in D. We fix D and vary ℘0 by rotating it in C about its center of mass. Let ℘t (t ∈ ℝ) be the family of polygons obtained in this fashion. Let χ℘t denote the indicator function of the subset ℘t of D. For any non-zero constant α ∈ ℝ it is shown that the Fundamental Eigenvalue of the Schr¨odinger operators −Δ+ αχ℘t attains its extremum when the axes of symmetry of ℘0 coincide with those of D.

References