On uniqueness, stability and pointwise estimates in the Cauchy problem for coupled elliptic equations
Author:
Philip W. Schaefer
Journal:
Quart. Appl. Math. 31 (1973), 321-328
MSC:
Primary 35J45
DOI:
https://doi.org/10.1090/qam/430523
MathSciNet review:
430523
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Abstract: Results of the kind cited in the title are obtained for the improperly posed Cauchy problem for a system of two coupled elliptic partial differential equations. We assume one stabilizing condition is imposed on one of the dependent variables. The results follow from an a priori inequality which is derived as a consequence of the logarithmic convexity of a suitable functional.
- James Conlan and G. N. Trytten, Pointwise bounds in the Cauchy problem for elliptic systems of partial differential equations, Arch. Rational Mech. Anal. 22 (1966), 143–152. MR 190508, DOI https://doi.org/10.1007/BF00276514
J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Yale Univ. Press, 1923
- Fritz John, A note on “improper” problems in partial differential equations, Comm. Pure Appl. Math. 8 (1955), 591–594. MR 75428, DOI https://doi.org/10.1002/cpa.3160080409
M. M. Lavrentiev, Some improperly posed problems of mathematical physics, Springer–Verlag, New York, 1967
- L. E. Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Rational Mech. Anal. 5 (1960), 35–45 (1960). MR 110875, DOI https://doi.org/10.1007/BF00252897
- L. E. Payne, Pointwise bounds in the Cauchy problem of elastic plates, J. Res. Nat. Bur. Standards Sect. B 65B (1961), 157–163. MR 142904
- L. E. Payne, On a priori bounds in the Cauchy problem for elliptic equations, SIAM J. Math. Anal. 1 (1970), 82–89. MR 262654, DOI https://doi.org/10.1137/0501008
C. Pucci, Some topics in parabolic and elliptic equations, Lecture Ser. No. 36, Inst. Fl. Dyn. Appl. Math., College Park, Md., 1958
- Carlo Pucci, Sui problemi di Cauchy non “ben posti”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 18 (1955), 473–477 (Italian). MR 87873
- Philip W. Schaefer, On the Cauchy problem for an elliptic system, Arch. Rational Mech. Anal. 20 (1965), 391–412. MR 192176, DOI https://doi.org/10.1007/BF00282360
- Philip W. Schaefer, Pointwise bounds in the Cauchy problem for an elliptic system, SIAM J. Appl. Math. 15 (1967), 665–677. MR 214906, DOI https://doi.org/10.1137/0115058
---, On the Cauchy problem for the nonlinear biharmonic equation, J. Math. Anal. Appl. 36 (1971), 660–673
- Philip W. Schaefer, Improvable estimates in some non-well-posed problems for a system of elliptic equations, SIAM J. Math. Anal. 4 (1973), 447–455. MR 326161, DOI https://doi.org/10.1137/0504039
J. Conlan and G. N. Trytten, Pointwise bounds in the Cauchy problem for elliptic systems of partial differential equations, Arch. Rat. Mech. Anal. 22 (1966), 143–152
J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Yale Univ. Press, 1923
F. John, A note on “improper” problems in partial differential equations, Comm. Pure Appl. Math. 8 (1955), 591–594
M. M. Lavrentiev, Some improperly posed problems of mathematical physics, Springer–Verlag, New York, 1967
L. E. Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Rat. Mech. Anal. 5 (1960), 35–45
---, Pointwise bounds in the Cauchy problem for elastic plates, J. Res. N.B.S. 65 (1961) 157–163
---, On a priori bounds in the Cauchy problem for elliptic equations, SIAM J. Math. Anal. 1 (1970), 82–89
C. Pucci, Some topics in parabolic and elliptic equations, Lecture Ser. No. 36, Inst. Fl. Dyn. Appl. Math., College Park, Md., 1958
---, Sui problemi di Cauchy non “ben posti", Rend. Acad. Naz. Lincei. 18 (1955), 473–477
P. W. Schaefer, On the Cauchy problem for an elliptic system, Arch. Rat. Mech. Anal. 20 (1965), 391–412
---, Pointwise bounds in the Cauchy problem for an elliptic system, SIAM J. Appl. Math. 15 (1967), 665–677
---, On the Cauchy problem for the nonlinear biharmonic equation, J. Math. Anal. Appl. 36 (1971), 660–673
---, Improvable estimates in some non-well-posed problems for a system of elliptic equations, SIAM J. Math. Anal. 4 (1973)
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Article copyright:
© Copyright 1973
American Mathematical Society