Boundary value problems for degenerate ellipticparabolic equations of the fourth order
Author:
Robert G. Root
Journal:
Trans. Amer. Math. Soc. 324 (1991), 109134
MSC:
Primary 35M10; Secondary 35D05, 35J70
MathSciNet review:
986699
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Abstract: We consider boundary value problems for the fourthorder linear equation with smooth coefficients. The fourthorder part may degenerate on arbitrary subsets of i.e., for all matrices , with no restriction on where equality occurs. We assume the part of the operator is uniformly elliptic (of second order) on while is a parameter allowing us to increase modulus of ellipticity as much as needed. As in Fichera's secondorder ellipticparabolic equations [see, for example, Sulle equazioni differenziali lineari elliticoparaboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. (8) 5 (1956), 130], because of the degeneracy, there may be characteristic portions of the boundary; however, we restrict our attention to the noncharacteristic case. We define a weak solution to the Dirichlet problem and obtain existence and uniqueness results. The question of regularity is addressed; elliptic regularization is used to obtain a Sobolevtype global regularity result. The equation models an anisotropic, inhomogeneous plate under tension that can lose stiffness at any point and in any direction. The regularity result has the satisfying physical interpretation that sufficient tension results in a smooth solution.
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 [3]
 G. Duvaut and J.L. Lions, Inequalities of mechanics and physics, Springer, Berlin and New York, 1976. MR 0521262 (58:25191)
 [4]
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 , Degenerate ellipticparabolic equations of second order, Comm. Pure Appl. Math. 20 (1967), 797872. MR 0234118 (38:2437)
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 A. Kufner, O. John, and S. Fucik, Function spaces, Noordhoff, Leyden, 1977. MR 0482102 (58:2189)
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 O. A. Oleinik, A boundary value problem for linear ellipticparabolic equations, Lecture Series, 46, Univ. of Maryland Inst. Fluid Dynamics and Appl. Math., College Park, 1965.
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 , Alcuni resultati sulle equazioni lineari e quasi lineari elliticoparaboliche a derivate parziali del secondo ordine, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (8) 40 (1966), 775784. MR 0229976 (37:5542)
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 , A problem of Fichera, Soviet Math. Dokl. 5 (1964), 11291133.
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 , On the smoothness of solutions of degenerate elliptic and parabolic equations, Soviet Math. Dokl. 6 (1965), 972975.
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 O. A. Oleinik and E. V. Radkevic, Second order equations with nonnegative characteristic form, Plenum, London, 1973. MR 0457908 (56:16112)
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 R. S. Philips and L. Sarason, Singular symmetric positive first order differential operators, J. Math. Mech. 15 (1966), 235271. MR 0186902 (32:4357)
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 , Ellipticparabolic equations of the second order, J. Math. Mech. 17 (1968), 891917. MR 0219868 (36:2942)
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 , Selfadjoint boundary value problems for ellipticparabolic operators of the fourth order in regions with degeneracy at the boundary (in preparation).
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 S. Timoshenko and S. WoinowskyKrieger, Theory of plates and shells, McGrawHill, New York, 1959.
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 R. J. Weinacht, Asymptotic distribution of eigenvalues for a class of degenerate elliptic operators of the fourth order, Rend. Mat. (7) 6 (1986), no. 12, 159170 (1988). MR 973614 (89m:35169)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199109866999
PII:
S 00029947(1991)09866999
Keywords:
Ellipticparabolic,
degenerate elliptic,
4th order higher order,
elastic plate,
anisotropic,
inhomogeneous,
plate under tension
Article copyright:
© Copyright 1991 American Mathematical Society
