Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Degenerate elliptic operators as regularizers


Author: R. N. Pederson
Journal: Trans. Amer. Math. Soc. 280 (1983), 533-553
MSC: Primary 35J70
DOI: https://doi.org/10.1090/S0002-9947-1983-0716836-8
MathSciNet review: 716836
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The spaces $ {\mathcal{K}_{mk}}$, introduced in the Nehari Volume of Journal d'Analyse Mathématique, for nonnegative integer values of $ m$ and arbitrary real values of $ k$ are extended to negative values of $ m$. The extension is consistent with the equivalence $ \parallel {\zeta ^j}u{\parallel_{m,k}}\sim\parallel u{\parallel_{m,k - j}}$, the inequality $ \parallel {D^\alpha }u{\parallel_{m,k}} \leqslant {\text{const}}\parallel u{\parallel_{m + \vert\alpha \vert,k + \vert\alpha \vert}}$, and the generalized Cauchy-Schwarz inequality $ \vert\langle {u,v} \rangle \vert \leqslant \parallel u\,{\parallel_{m,k}}\parallel v\parallel_{ - m, - k}$. (Here $ \langle u, \upsilon \rangle$ is the $ {L_2}$ scalar product.) There exists a second order degenerate elliptic operator which maps $ {\mathcal{K}_{m,k}}\,1 - 1$ onto $ {\mathcal{K}_{m - 2,k}}$. These facts are used to simplify proof of regularity theorems for elliptic and hyperbolic problems and to give new results concerning growth rates at the boundary for the coefficients of the operator and the forcing function. (See Notices Amer. Math. Soc. 28 (1981), 226.)


References [Enhancements On Off] (What's this?)

  • [1] F. E. Browder, On the regularity of solutions of elliptic differential equations, Comm. Pure Appl. Math. 9 (1956), 351-361. MR 0090740 (19:862a)
  • [2] G. Fichera, Linear elliptic differential systems and boundary value problems, Springer-Verlag, Berlin and New York, 1965. MR 0209639 (35:536)
  • [3] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333-418. MR 0100718 (20:7147)
  • [4] L. Hörmander, Linear partial differential operators, Springer-Verlag, Berlin, 1963.
  • [5] J. Kohn and L. Nirenberg, Degenerate elliptic parabolic equations of the second order, Comm. Pure Appl. Math. 20 (1967), 797-872. MR 0234118 (38:2437)
  • [6] P. D. Lax, On the Cauchy Problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955). MR 0078558 (17:1212c)
  • [7] S. Mizohata, Unicité du prolongment des solution des equations elliptiques du quatrieme ordre, Proc. Japan Acad. 34 (1958), 687-692. MR 0105553 (21:4292)
  • [8] L. Nirenberg, Remarks on strongly elliptic equations, Comm. Pure Appl. Math. 8 (1955), 648-674. MR 0075415 (17:742d)
  • [9] O. Oleinik and E. V. Radkevič, Second order elliptic equations with nonnegative characteristic form, Plenum Press, New York, 1973. MR 0457908 (56:16112)
  • [10] O. Oleinik, The Cauchy Problem for hyperbolic equations of second order degenerating in a region and on its boundary, Dokl. Akad. Nauk SSSR 169 (1966), 525-528. MR 0203267 (34:3120)
  • [11] R. N. Pederson, An equivalent norm for the Sobolev space $ {\mathcal{K}_m}$, J. Analyse Math. 36 (1979), 213-216. MR 581814 (81h:46035)
  • [12] -, On the Unique Continuation Theorem for certain second and fourth order elliptic equations, Comm. Pure Appl. Math. 11 (1958), 67-80. MR 0098900 (20:5350)
  • [13] L. Schwarz, Théorie des distributions. I, II, Hermann, Paris, 1950-1951.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J70

Retrieve articles in all journals with MSC: 35J70


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0716836-8
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society