簡易檢索 / 詳目顯示

研究生: 林景隆
Lin, Ching-Lung
論文名稱: 對某些彈性系統的唯一延拓性和反問題
Strong unique continuation and inverse problems in some elasticity systems
指導教授: 陳若淳
Chen, Roger
王振男
Wang, Jenn-Nan
學位類別: 博士
Doctor
系所名稱: 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年: 2004
畢業學年度: 92
語文別: 英文
論文頁數: 80
中文關鍵詞: 唯一延拓性
外文關鍵詞: Strong unique continuation
相關次數: 點閱:98下載:11
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報
  • 在這篇文章,我們考慮某些彈性系統的唯一延拓性和反問題。

    In this thesis, we are devoted to prove the strong unique continuation property and
    identify the density for some elasticity systems. These systems contain the
    isotropic elasticity system and elasticity systems with residual stress.

    (1)Introduction to the theory of elasticity(4) (1.1)Preliminaries(4) (1.2)Stress(6) (1.3)Strain(8) (1.4)An isotropic elastic body with residual stress(9) (2)Introduction to unique continuation properties(10) (2.1)Introduction(10) (2.2)Review(11) (2.3)Main Results(13) (3)Strong unique continuation for the Lam(e) system with Lipschitz coefficients(17) (3.1)Introduction(17) (3.2)First order elliptic system(19) (3.3)Local behavior of V(22) (3.4)Carleman estimates with more singular weights(26) (3.5)Proof of (SUCP)(29) (4)Strong unique continuation for an elasticity system with residual stress(32) (4.1)Introduction(32) (4.2)Transformation of the original system into new system(36) (4.3)The asymptotic behavior of U near 0(37) (4.4)Carleman estimates(50) (4.5)Proof of Theorem 4.1.2(55) (4.6)Another proof of SUCP with fourth order reduced system(56) (5)Uniqueness in inverse problems for an elasticity system with residual stress by single measurement(59) (5.1)Introduction(59) (5.2)Derivation of a new system(63) (5.3)Carleman estimates(65) (5.4)Main results(67) Appendix(72)

    G. Alessandrini, A. Morassi, Strong unique continuation for the Lame
    system of elasticity, Comm. in P.D.E., 26, 1787-1810, 2001.
    S. Alinhac, Non-unicite pour des
    op$acute{e}$rateurs differentiels a
    caracteristiques complexes simples, Ann. Sci. Ec. Norm.
    Sup., 13, 385-393, 1980.
    S. Alinhac, M. S. Baouendi, Uniqueness for the
    characteristic Cauchy problem and strong unique continuation for
    higher order partial differential inequalities, Amer. J. Math., 102, 179-217, 1980.
    S. Alinhac, M. S. Baouendi, A counterexample to strong
    uniqueness for partial differential equations of
    Schrodinger's type, Comm. in P.D.E., 19,
    1723-1733, 1994.
    D.D. Ang, M. Ikehata, D.D. Trong and M. Yamaoto, Unique continuation for a
    stationary isotropic Lame system with varaiable coefficients, Comm. in PDE.,
    23, 371-385, 1998.
    N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations
    or inequalities of second order, J. Math. Pures Appl., 36, 235-249, 1957.
    N. Aronszajn, A. Krzywicki, J. Szarski, A unique continuation theorem for exterior differential forms
    on Riemannian manifolds, Ark. Mat., 4, 417-453, 1962.
    P. Le Borgne, Strong uniqueness for fourth order elliptic differential operators,
    Indiana Univ. Math. J., 50 No. 1, 353-381, 2001.
    A.L. Bukhgeim, Introduction to the Theory of Inverse
    Problems, Nauka, Novosibrisk, 1988.
    A.L. Bukhgeim and M.V. Klibanov, Global uniqueness of a
    class of multidimensional inverse problems, English translation,
    Soviet Math. Dokl., 24, 244-247. Nauka, Novosibrisk, 1988.
    L. De Carli and T. Okaji, Strong unique continuation
    property for the Dirac equation, Publ. RIMS, Kyoto Univ., 35, 825-846, 1999.
    J. Cheng, Y.C. Hon M. Yamamoto, Stability in line unique
    continuation of harmonic functions : general dimensions, Inverse
    and Ill-Posed Problems., 6, 319-326, 1998.
    J. Cheng, M. Yamamoto, Unique continuation on a line for
    harmonic functions , Inverse Problems., 14, 869-882, 1998.
    J. Cheng, M. Yamamoto, Q. Zhou, Unique continuation on a
    hyperplane for wave equation, Chin. Ann. of Math.,
    (20B)4, 385-392, 1999.
    J. Cheng, C.L. Lin, G. Nakamura, Unique continuation
    along curves and hypersurfaces for second order anisotropic
    hyperbolic systems with real analytic coefficients, Preprint.
    F. Colombini and C. Grammatico, Some remarks on strong
    unique continuation for the Laplace operator and its power, Comm.
    in P.D.E., 24, 1079-1094, 1999.
    H.O. Cordes, Uber die Bestimmtheit der
    Losungen elliptischer Differentialgleichungen durch
    Anfangsvorgaben, Nachr. Akad. Wiss. Gottingen Math.
    Phys., 11, 239-258, 1956.
    B. Dehman and L. Robbiano, La propriete du
    prolongement unique pour un systeme elliptique: le systeme
    Lame, J. Math. Pures Appl., 72, 475-492, 1993.
    M. Eller and V. Isakov, Carleman estimates with two large
    parameters and applications, Proceedings of AMS summer research
    conference in Boulder, Colorado, July 1999, Contemporary Math.,
    268, 117-137, 2000.
    M. Eller, V. Isalov, G.Nakamura and D.Tataru, Uniqueness
    and stability in the Cauchy problem for Maxwell' and elastivity
    systems, Stud. Math. Appl., 31, 329-349, 2002.
    N. Garofalo and F.H. Lin, Monotonicity properties of
    variational integrals, Ap weights and unique continuation,
    Indiana Univ. Math. J., 35, 245-268, 1986.
    N. Garofalo and F.H. Lin Unique continuation for elliptic
    operators: a geometric-variational approach, Comm. Pure Appl.
    Math., 40, 347-366, 1987.
    C. Grammatico, A Result on Strong Unique Continuation for
    the Laplace Operator, Birkhauser Verlag, 1993.
    M. Giaquinta, Introduction to regularity theory for
    nonlinear elliptic systems, Comm. in P.D.E., 22, 1475-1491,
    1997.
    A. Hoger, On the determination of residual stress in an
    elastic body, J. Elasticity, 16, 303-324, 1986.
    L. Hormander, Uniqueness theorems for second order
    elliptic differential equations, Comm. in P.D.E., 8, No. 1,
    21-64, 1983.
    L. Hormander, ``The Analysis of Linear Partial
    Differential Operators I-IV', Springer-Verlag, Berlin, 1985.
    M. Ikehata, G. Nakamura, M. Yamamoto Uniqueness in
    inverse problems for the isotropic Lame system, J. Math. Sci.
    Univ. Tokyo, 5, 627-692, 1998.
    V. Isakov, A non-hyperpolic Cauchy problem for b
    c and its application to elasticity theory, Comm. Pure.
    Appl. Math., 39, 747-767, 1986.
    V. Isakov, Uniqueness of the continuation across a
    time-like hyperplane and related inverse problems for hyperbolic
    equations, Comm. P.D.E., 14, 465-478, 1989.
    V. Isakov, 'Inverse Source Problems', Mathematical
    Surveys and Monographs No. 34, Ams, Providence, 1990.
    V. Isakov, G. Nakamura, J.N. Wang Uniqueness and stability
    in the Cauchy problem for the elasticity system with residual
    stress, Preprint.
    D. Jerison, Carleman inequalities for the Dirac and
    Laplace operator and unique continuation, Adv. Math., 62,
    118-134, 1986.
    D. Jerison, C. Kenig Unique continuation and absence of
    positive eigenvalues for Schrodinger operators, Ann. Math.
    121, 463-488, 1985.
    C. Kenig, A. Ruiz, C. Sogge, Sobolev inequalives and
    unique continuation for second order constant coefficient elliptic
    operators, Duke. Math., 55, 329-347, 1987.
    A. Khaidarov, Carleman estimates and inverse problems
    for second order hyperbolic equations, Math. USSR Sbornik,
    58, 267-277, 1987.
    Y.M. Kim, MIT thesis, 1989.
    M. Klibanov, Inverse problems and Carleman estimates,
    Inverse Problems, 8, 575-596, 1992.
    A. Krzywcki, J. Szarski A unique contiunation theorem for
    exterior differential forms on Riemannian manifolds, Ark.
    for Mat., 4, 417-453, 1962.
    M. Kubo, Identification of the potential term of the wave
    equation, Preceedings of the Japan Acad., 71, 174-176,
    1995.
    V.D. Kupradze, Three-dimensional problems of the
    mathematical theory of elasticity and thermoelasticity, Vol. 25, North-Holland, 1979.
    N. Lerner, Uniqueness for an ill-posed problem,
    J. Diff. Eq., 71, 255-260, 1988.
    F. H. Lin, Unique continuation for elliptic operators: a
    geometric-variational approach, Comm. Pure. Appl. Math.,
    XL, 347-366, 1987.
    C.S. Man, Hartig's law and linear elasticity with initial
    stress, Inverse Prob., 14, 313-319, 1998.
    K. Miller, Non-unique continuation for certain ODE's in
    Hilbert space and for uniformly parabolic and elliptic equations
    in self-adjoint divergence form, Springer Lecture Notes in
    Mathematics., vol. 316, 85-101, 1973.
    G. Nakamura and G. Uhlmann, Global uniqueness for an
    inverse boundary problem arising in elasticity, 118,
    457-474, 1994.
    G. Nakamura and J.N. Wang, Unique continuation for the
    elasticity system with residual stress and its application, Siam J. Math. Anal.
    35, no. 2, 304--317, 2003.
    G. Nakamura and J.N. Wang, Unique continuation for the two-dimensional
    anisotropic elasticity system and its application to inverse problems, submitted.
    M. Ohtsu, S. Yuyama, T. Imanaka, Theoretical treatment of
    acoustic emission sources in micorfracturing due to disbonding,
    J. Acoustical Soc. America, 82, 506-51, 1987.
    T. Okaji, Strong unique continuation property for time
    harmonic Maxwell equations, Bull. Acad. Pol. Sci., 11,
    95-100, 1963.
    T. Okaji, Strong unique continuation property for
    elliptic systems of normal type in two independent variables,
    Tohoku Math. J. 54(2002), 309--318.
    T. Okaji, Strong unique continuation property for time
    harmonic Maxwell equations, J. Math. Soc. Japan 54(2002), no. 1, 89--122.
    Y. Pan, Unique continuation for Schrodinger
    operators with singular potentials, Comm. in PDE., 17,
    953-965, 1992.
    A. Plis, On non-uniqueness in Cauchy problem
    for an elliptic second order differentialequation, J. Math. Soc.
    Japan, 54, No. 1, 89-122, 2002.
    L. Rachele, An inverse problem in elastodynamics:
    uniqueness of the wave speeds in the interior, J. Diff. Equ., 162, 300-325, 2000.
    R. Regbaoui, Strong uniqueness for second order
    differential operators, J. Diff. Eq., 141, 201--217, 1997.
    R. Regbaoui, Unique continuation for differential
    equations of Schrodinger's type, Comm. Anal. Geo., 7, 303-323, 1999.
    L. Robbiano, Theoreme d`unicite adapte au
    controle des solutions des problemes hyperboliques,
    Commnications in P.D.E.,16, 789-800, 1991.
    R. Robertson, Boundary identifiability of residual stress
    via the Dirichlet to Neumann map, Inverse Problems, 13,
    1107-1119, 1997.
    C. Sogge, Strong uniqueness theorems for second order
    elliptic differential equations, Amer. J. Math., 112,
    943-984, 1990.
    V. Vogeslang, Absence of embedded eigenvalues of the Dirac
    equation for long range potentials, Analysis, 7, 259-274,
    1987.
    V. Vogeslang, On the strong continuation principle for
    inequalities of Maxwell type, Math. Ann., 289, 285-295,
    1991.
    T.H. Wolff, "Fourier analysis and partial differential
    equations" , Miraflores de la Sierra, 99-128, 1992.
    T.H. Wolff, A counterexample in a Unique Continuation
    problem , Comm. Anal. Geom., 1, 79-102, 1994.
    T.H. Wolff, Note on counterexamples in strong unique
    continuation problems , Proc. Amer. Math. Soc., 114,
    351-356, 1992.
    T.H. Wolff, Unique continuation for |u|< V|u|
    and related problems, Revista. Math. Iberoamericana., 6,
    155-200, 1990.
    M. Yamamoto, Uniqueness and stability in multidimensional
    hyperbolic inverse problems, J. Math. Pures Appl., 78,
    65-98, 1999.
    M. Yamamoto, On an inverse problem of determining source
    terms in Maxwell's equations with a single measurement, Inverse
    problems, tomography, and image processing (Newark, DE, 1997),
    241--256, Plenum, New York, 1998.
    N. Weck, Unique continuation for systems with Lame
    principal part, Math. Methods Appl. Sci., 24, 595--605,
    2001.

    下載圖示 校內:立即公開
    校外:2004-04-23公開
    QR CODE