Oscillation criteria for elliptic systems

Authors:
W. Allegretto and C. A. Swanson

Journal:
Proc. Amer. Math. Soc. **27** (1971), 325-330

MSC:
Primary 35.11

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276592-4

MathSciNet review:
0276592

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Oscillation criteria are established for quasilinear elliptic partial differential systems of second order in unbounded domains of Euclidean space. The main departures from earlier investigations are: (1) systems of partial differential equations are considered; (2) the equations are nonlinear; (3) the matrices involved are not required to be positive definite; and (4) a direct method of the Calculus of Variations is used rather than the earlier majorization method.

**[1]**W. Allegretto,*Eigenvalue comparison and oscillation criteria for elliptic operators*, J. London Math. Soc. (to appear). MR**0284687 (44:1911)****[2]**-,*A comparison theorem for nonlinear operators*, Ann. Scuola Norm. Sup. Pisa (to appear).**[3]**-,*Comparison and oscillation theorems for elliptic equations*, Ph.D. Thesis, The University of British Columbia, April, 1969.**[4]**-,*Oscillation criteria for*4*th order elliptic equations*, Boll. Un. Mat. Ital.**3**(1970), 357-361. MR**0265793 (42:702)****[5]**I. M. Glazman,*On the negative part of the spectrum of one-dimensional and multi-dimensional differential operators on vector-functions*, Dokl. Akad. Nauk SSSR**119**(1958), 421-424. (Russian) MR**20**#4674. MR**0098212 (20:4674)****[6]**V. B. Headley,*Elliptic equations of order*, J. Math. Anal. Appl.**25**(1969), 558-568. MR**38**#3565. MR**0235255 (38:3565)****[7]**V. B. Headley and C. A. Swanson,*Oscillation criteria for elliptic equations*, Pacific J. Math.**27**(1968), 501-506. MR**38**#4797. MR**0236502 (38:4797)****[8]**Kurt Kreith,*Oscillation theorems for elliptic equations*, Proc. Amer. Math. Soc.**15**(1964), 341-344. MR**33**#6098. MR**0197939 (33:6098)****[9]**-,*A Sturm theorem for strongly elliptic systems and applications*, Bull. Amer. Math. Soc.**75**(1969), 1025-1027. MR**0249824 (40:3065)****[10]**L. M. Kuks,*Sturm's theorem and the oscillation of solutions of strongly elliptic systems*, Dokl. Akad. Nauk SSSR**142**(1962), 32-35 = Soviet Math. Dokl.**3**(1962), 24-27. MR**26**#441. MR**0142874 (26:441)****[11]**M. Morse,*A generalization of the Sturm separation and comparison theorems in -space*, Math. Ann.**103**(1930), 52-69. MR**1512617****[12]**C. A. Swanson,*An identity for elliptic equations with applications*, Trans. Amer. Math. Soc.**134**(1968), 325-333. MR**38**#400. MR**0232074 (38:400)****[13]**-,*Comparison and oscillation theory of linear differential equations*, Mathematics in Science and Engineering, vol. 48, Academic Press, New York, 1968. MR**0463570 (57:3515)****[14]**-,*Oscillation criteria for nonlinear matrix differential inequalities*, Proc. Amer. Math. Soc.**24**(1970), 824-827.**[15]**-,*Comparison theorems for elliptic differential systems*, Pacific J. Math.**33**(1970), 445-450. MR**0262650 (41:7255)****[16]**E. C. Tomastik,*Oscillation of nonlinear matrix differential equations of second order*, Proc. Amer. Math. Soc.**19**(1968), 1427-1431. MR**38**#372. MR**0232046 (38:372)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
35.11

Retrieve articles in all journals with MSC: 35.11

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276592-4

Keywords:
Quasilinear elliptic system,
oscillation criterion,
prepared matrix

Article copyright:
© Copyright 1971
American Mathematical Society