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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Eigenvalues of elliptic boundary value problems with an indefinite weight function


Authors: Jacqueline Fleckinger and Michel L. Lapidus
Journal: Trans. Amer. Math. Soc. 295 (1986), 305-324
MSC: Primary 35P20; Secondary 35J10
MathSciNet review: 831201
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Abstract: We consider general selfadjoint elliptic eigenvalue problems (P)

$\displaystyle \mathcal{A}u = \lambda r(x)u,$

in an open set $ \Omega \subset {{\mathbf{R}}^k}$. Here, the operator $ \mathcal{A}$ is positive and of order $ 2m$ and the "weight" $ r$ is a function which changes sign in $ \Omega $ and is allowed to be discontinuous. A scalar $ \lambda $ is said to be an eigenvalue of $ ({\text{P}})$ if $ \mathcal{A}u = \lambda ru$--in the variational sense--for some nonzero $ u$ satisfying the appropriate growth and boundary conditions. We determine the asymptotic behavior of the eigenvalues of $ ({\text{P}})$, under suitable assumptions. In the case when $ \Omega $ is bounded, we assumed Dirichlet or Neumann boundary conditions. When $ \Omega $ is unbounded, we work with operators of "Schrödinger type"; if we set $ r \pm = \max ( \pm r,0)$, two cases appear naturally: First, if $ \Omega $ is of "weighted finite measure" (i.e., $ \int_\Omega {{{({r_ + })}^{k/2m}} < + \infty \;} {\text{or}}\;\int_\Omega {{{({r_ - })}^{k/2m}} < + \infty } $), we obtain an extension of the well-known Weyl asymptotic formula; secondly, if $ \Omega $ is of "weighted infinite measure" (i.e., $ \int_\Omega {{{({r_ + })}^{k/2m}} = + \infty \;{\text{or}}\;\int_\Omega {{{({r_ - })}^{k/2m}} = + \infty } } $), our results extend the de Wet-Mandl formula (which is classical for Schrödinger operators with weight $ r \equiv 1$). When $ \Omega $ is bounded, we also give lower bounds for the eigenvalues of the Dirichlet problem for the Laplacian.

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  • [Ad] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957 (56 #9247)
  • [Ag] Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0178246 (31 #2504)
  • [Be] Richard Beals, Indefinite Sturm-Liouville problems and half-range completeness, J. Differential Equations 56 (1985), no. 3, 391–407. MR 780497 (86i:34032), http://dx.doi.org/10.1016/0022-0396(85)90085-3
  • [BS 1] M. S. Birman and M. Z. Solomjak, Spectral asymptotics of nonsmooth elliptic operators. II, Trans. Moscow Math. Soc. 28 (1973), 1-32.
  • [BS 2] -, Asymptotics of the spectrum of variational problems on solutions of elliptic equations, Siberian Math. J. 20 (1979), 1-15.
  • [BS 3] -, Asymptotic behavior of the spectrum of the differential equations, J. Soviet Math. 12 (1979), 247-282.
  • [BS 4] M. Š. Birman and M. Z. Solomjak, Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory, American Mathematical Society Translations, Series 2, vol. 114, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by F. A. Cezus. MR 562305 (80m:46026)
  • [Bo] M. Bôcher, Boundary problems in one dimension, Proc. Fifth Internat. Congress Math. (Cambridge, 1912), Vol. I, Cambridge Univ. Press, New York, 1913, pp. 163-195.
  • [Br] Félix Browder, Le problème des vibrations pour un opérateur aux dérivées partielles self-adjoint et du type elliptique, à coefficients variables, C. R. Acad. Sci. Paris 236 (1953), 2140–2142 (French). MR 0058089 (15,320a)
  • [CH] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391 (16,426a)
  • [DGHa] I. Dee Chang, G. W. Grube and E. Y. Harper, On the breakup of accelerating liquid drops, J. Fluid Mech. 52 (1972), 565-591.
  • [dF] Djairo Guedes de Figueiredo, Positive solutions of semilinear elliptic problems, Differential equations (S\ ao Paulo, 1981) Lecture Notes in Math., vol. 957, Springer, Berlin-New York, 1982, pp. 34–87. MR 679140 (84k:35067)
  • [Fl] J. Fleckinger, Estimate of the number of eigenvalues for an operator of Schrödinger type, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), no. 3-4, 355–361. MR 635770 (83f:35085), http://dx.doi.org/10.1017/S0308210500020357
  • [FlF 1] J. Fleckinger and M. El Fetnassi, Asymptotics of eigenvalues of variational elliptic problems with indefinite weight function, Spectral Theory of Sturm-Liouville Differential Operators (Hans G. Kaper and A. Zettl, eds.), ANL-84-73, Argonne National Laboratory, Argonne, Illinois, 1984, pp. 107-118.
  • [FlF 2] Jacqueline Fleckinger-Pellé and Mohamed El Fetnassi, Comportement asymptotique des valeurs propres de problèmes elliptiques “non définis à droite”, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 13, 599–602 (French, with English summary). MR 771357 (86a:35113)
  • [FlMe] Jacqueline Fleckinger and Guy Métivier, Théorie spectrale des opérateurs uniformément elliptiques sur quelques ouverts irréguliers, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A913–A916 (French). MR 0320550 (47 #9087)
  • [Ga] Lars Gårding, On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differential operators, Math. Scand. 1 (1953), 237–255. MR 0064980 (16,366b)
  • [GN] W. N. Gill and R. J. Nunge, Analysis of heat or mass transfer in some countercurrent flows, Internat. J. Heat Mass Transfer 8 (1965), 873-886.
  • [GoK] I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. MR 0264447 (41 #9041)
  • [Gu] David Gurarie, Resolvent and heat kernels for operators of Schrödinger type with applications to spectral theory, Differential equations (Birmingham, Ala., 1983) North-Holland Math. Stud., vol. 92, North-Holland, Amsterdam, 1984, pp. 249–256. MR 799355 (87a:35137), http://dx.doi.org/10.1016/S0304-0208(08)73700-9
  • [H] Peter Hess, On bifurcation from infinity for positive solutions of second order elliptic eigenvalue problems, Nonlinear phenomena in mathematical sciences (Arlington, Tex., 1980), Academic Press, New York, 1982, pp. 537–544. MR 728015 (85c:35006)
  • [HK] Peter Hess and Tosio Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), no. 10, 999–1030. MR 588690 (81m:35102), http://dx.doi.org/10.1080/03605308008820162
  • [Hi] David Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea Publishing Company, New York, N.Y., 1953 (German). MR 0056184 (15,37b)
  • [Ho] E. Holmgren, Über Randwertaufgaben bei einer linearen Differentialgleichung zweiter Ordnung, Ark. Mat., Astro och Fysik 1 (1904), 401-417.
  • [Kc] Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR 0201237 (34 #1121)
  • [KKLeZ] Hans G. Kaper, Man Kam Kwong, C. G. Lekkerkerker, and A. Zettl, Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights, Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), no. 1-2, 69–88. MR 765489 (86j:47067), http://dx.doi.org/10.1017/S0308210500025567
  • [KZ] H. G. Kaper and A. Zettl (eds.), Proc. May-June 1984 Workshop "Spectral Theory of Sturm-Liouville Differential Operators", ANL-84-73, Argonne National Laboratory, Argonne, Illinois, 1984.
  • [La 1] Michel L. Lapidus, Valeurs propres du laplacien avec un poids qui change de signe, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 12, 265–268 (French, with English summary). MR 745319 (85j:35139)
  • [La 2] -, Spectral theory of elliptic problems with indefinite weights, Spectral Theory of Sturm-Liouville Differential Operators (Hans G. Kaper and A. Zettl, eds.), ANL-84-73, Argonne National Laboratory, Argonne, Illinois, 1984, pp. 159-168.
  • [LiY] Peter Li and Shing Tung Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), no. 3, 309–318. MR 701919 (84k:58225)
  • [Lb] Elliott H. Lieb, The number of bound states of one-body Schroedinger operators and the Weyl problem, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 241–252. MR 573436 (82i:35134)
  • [LM] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177 (50 #2670)
  • [LoS] Lynn H. Loomis and Shlomo Sternberg, Advanced calculus, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0227327 (37 #2912)
  • [LuR] G. S. S. Ludford and R. A. Robertson, Fully diffused regions, SIAM J. Appl. Math. 25 (1973), 693–703. MR 0342024 (49 #6770)
  • [LuW] G. S. S. Ludford and Sharon S. Wilson, Subcharacteristic reversal, SIAM J. Appl. Math. 27 (1974), 430–440. MR 0359537 (50 #11990)
  • [MM] Adele Manes and Anna Maria Micheletti, Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4) 7 (1973), 285–301 (Italian, with English summary). MR 0344663 (49 #9402)
  • [Me] Guy Métivier, Valeurs propres de problèmes aux limites elliptiques irrégulières, Bull. Soc. Math. France Suppl. Mém. 51–52 (1977), 125–219 (French). MR 0473578 (57 #13244)
  • [Pe] Ivan N. Pesin, Classical and modern integration theories, Translated from the Russian and edited by Samuel Kotz. Probability and Mathematical Statistics, No. 8, Academic Press, New York-London, 1970. MR 0264015 (41 #8614)
  • [Pl] Å. Pleijel, Sur la distribution des valeurs propres de problèmes régis par l'équation $ \Delta u + \lambda k(x,y)u = 0$, Ark. Mat., Astr. och Fysik 29 B (1942), 1-8.
  • [RS] Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421 (58 #12429c)
  • [Ri] R. G. D. Richardson, Contributions to the Study of Oscillation Properties of the Solutions of Linear Differential Equations of the Second Order, Amer. J. Math. 40 (1918), no. 3, 283–316. MR 1506360, http://dx.doi.org/10.2307/2370485
  • [Ro 1] G. V. Rozenbljum, Distribution of the discrete spectrum of singular differential operators, Dokl. Akad. Nauk SSSR 202 (1972), 1012–1015 (Russian). MR 0295148 (45 #4216)
  • [Ro 2] -, Asymptotics of the eigenvalues of the Schrödinger operator, Math. USSR-Sb. 22 (1974), 349-371.
  • [Ro 3] -, Asymptotics of the eigenvalues of the Schrödinger operator, Problemy Mat. Anal. Slož. Sistem. 5 (1975), 152-166. (Russian)
  • [T] E. C. Tichmarsh, Eigenfunction expansions associated with second-order differential equations, Part II, Oxford Univ. Press, London, 1958.
  • [Wn] Hans F. Weinberger, Variational methods for eigenvalue approximation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Based on a series of lectures presented at the NSF-CBMS Regional Conference on Approximation of Eigenvalues of Differential Operators, Vanderbilt University, Nashville, Tenn., June 26–30, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15. MR 0400004 (53 #3842)
  • [We] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1911), 441-469.
  • [Y] Shing Tung Yau, Survey on partial differential equations in differential geometry, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 3–71. MR 645729 (83i:53003)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0831201-3
PII: S 0002-9947(1986)0831201-3
Keywords: Elliptic boundary value problems, indefinite weight function, asymptotic behavior of eigenvalues, lower bounds of eigenvalues, spectral theory, variational methods, operators of Schrödinger type
Article copyright: © Copyright 1986 American Mathematical Society