A nonhyperbolic Cauchy problem for \(\square _ b\square _ c\) and its applications to elasticity theory. (English) Zbl 0649.35015

Isakov, Victor

Commun. Pure Appl. Math. 39, 747-767 (1986).

The author proves the uniqueness and stability of the ill-posed problems for a function u satisfying the differential equation \[ (\square_ b\square_ c+A_ 3)u=f\quad \quad on(-T,T)\times G,\quad G\subset R^ n, \] and the boundary condition \((\partial /\partial N)^ ju=g_ j\), \(j\leq 3\), on \((-T,T)\times \Gamma '\), where \(\Gamma '\) is a part of the boundary of G, N is a normal to it, \(A_ 3\) is a differential operator of the third order, \(\square_ b=b\frac{\partial^ 2}{\partial t^ 2}- \Delta\), \(b\neq c\). He proves Carleman-type estimates for \(\square_ b\square_ c\) and other inequalities for the Lamé equations of the classical elasticity theory.

Reviewer: S.M.Zverev

MSC:

35G10	Initial value problems for linear higher-order PDEs
35R25	Ill-posed problems for PDEs
35L30	Initial value problems for higher-order hyperbolic equations
74G30	Uniqueness of solutions of equilibrium problems in solid mechanics
74H25	Uniqueness of solutions of dynamical problems in solid mechanics
35B35	Stability in context of PDEs

Keywords:

uniqueness; stability; ill-posed; third order; Carleman-type estimates; Lamé equations; elasticity theory

References:

[1]	Bell, SIAM J. Math. Anal. 12 pp 759– (1981)
[2]	Bukhgeim, Soviet Math. Doklady Vol. 24 pp 244– (1981)
[3]	Russian Dokl. Akad. Nauk, SSSR 260 pp 269– (1981)
[4]	Hile, Comm. Part. Diff. Equat. 1 pp 437– (1976)
[5]	Hörmander, Math. Scand. 6 pp 213– (1958) · Zbl 0088.30201 · doi:10.7146/math.scand.a-10546
[6]	Math. Scand. 7 pp 177– (1959) · Zbl 0090.08001 · doi:10.7146/math.scand.a-10571
[7]	Linear Partial Differential Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1976.
[8]	Isakov, Soviet Math. Doklady 22 pp 639– (1980)
[9]	Russian Dokl. Akad. Nauk SSSR 255 pp 18– (1980)
[10]	John, Comm. Pure Appl. Math. 13 pp 551– (1960)
[11]	and , Uniquenss and continuous dependence of a null solution of the Cauchy problem for a nonlinear elliptic system, Math. Research, editor, Vol. 1, Akademie-Verlag, Berlin, 1979, pp. 151–160.
[12]	Kohn, Comm. Pure Appl. Math. 18 pp 269– (1965)
[13]	Nirenberg, Comm. Pure Appl. Math. 10 pp 89– (1957)
[14]	Lectures on Partial Linear Differential Equations, Conf. Board on the Math. Sc. Regional Conference, Series in Math., AMS, No. 19, 1973, Providence, RI, 1973.
[15]	Payne, Contemp. Math. 4 pp 1– (1981) · Zbl 0476.73015 · doi:10.1090/conm/004/641222
[16]	Plis, Comm. Pure Appl. Math. 14 pp 599– (1961)
[17]	Sakamoto, J. Math. Kyoto Univ. 10 pp 349– (1970)
[18]	Uniqueness and Nonuniqueness in the Cauchy Problem, Progress in Mathematics, Vol. 3, Birkhauser, Boston, Basel, Stuttgart, 1983.