| 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.
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.