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Hyperjacobians, determinantal ideals and weak solutions to variational problems. (English) Zbl 0562.49005

The paper deals with the solution of the classification problem of homogeneous null Lagrangians satisfying nth order divergence identity. Results are in reference to earlier research by J. M. Ball and J. C. Currie [J. Funct. Anal. 41, 135-174 (1981; Zbl 0459.35020)] where it is noted that certain null Lagrangians can be written as higher order divergences. In this earlier paper, the problem of affecting similar classification of homogeneous null Lagrangians (nth order divergences) is raised. Thus the main result of the current paper is that any homogeneous null Lagrangian is necessarily an affine combination of nth order hyperjacobians which are higher order Jacobian matrices. Earlier formulas for hyperjacobias are limited to polynomial functions. Also, the fact that the differential polynomials are higher order divergences is missing from earlier work. The characterization of nth order divergences as hyperjacobians depends on the fact that symbolic powers of the same determinantal ideal are the same as its actual power. The transform theory is used to demonstrate the applicability of the above result to the classification.
It is noted that nth order hyperjacobians of degree k can be defined with m fewer derivatives in Sobolev space where \(m=n mod k\). Then the compact embeddings of Sobolev space (over bounded domains) can be used to obtain sequential weak continuity results for hyperjacobians. Hence, existence result for minimizers of certain special types of quasi-convex variational problems with weakened growth conditions on integrands are derived.
Reviewer: N.Warsi

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs

Citations:

Zbl 0459.35020
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References:

[1] DOI: 10.1017/S0305004100058394 · Zbl 0475.49011 · doi:10.1017/S0305004100058394
[2] DOI: 10.1016/0021-8693(77)90211-3 · Zbl 0358.15033 · doi:10.1016/0021-8693(77)90211-3
[3] DOI: 10.1070/RM1975v030n05ABEH001522 · Zbl 0334.58007 · doi:10.1070/RM1975v030n05ABEH001522
[4] Escherich, Denkschr. Kais. Akad. Wiss. Wien 43 pp 1– (1882)
[5] DOI: 10.1007/BF01392548 · Zbl 0435.14015 · doi:10.1007/BF01392548
[6] DOI: 10.1016/0022-1236(81)90085-9 · Zbl 0459.35020 · doi:10.1016/0022-1236(81)90085-9
[7] DOI: 10.1007/BF00279992 · Zbl 0368.73040 · doi:10.1007/BF00279992
[8] DOI: 10.2307/2374195 · Zbl 0454.58021 · doi:10.2307/2374195
[9] DOI: 10.1063/1.524104 · Zbl 0416.58028 · doi:10.1063/1.524104
[10] Adams, Sobolev Spaces (1975)
[11] DOI: 10.1017/S0305004100055572 · Zbl 0387.35050 · doi:10.1017/S0305004100055572
[12] DOI: 10.2307/2304053 · Zbl 0063.08426 · doi:10.2307/2304053
[13] DOI: 10.1093/qmath/14.1.193 · Zbl 0116.02504 · doi:10.1093/qmath/14.1.193
[14] DOI: 10.1112/jlms/s1-42.1.595 · Zbl 0171.00401 · doi:10.1112/jlms/s1-42.1.595
[15] DOI: 10.1016/0001-8708(78)90037-3 · Zbl 0394.14022 · doi:10.1016/0001-8708(78)90037-3
[16] Hodge, Methods of Algebraic Geometry II (1954) · Zbl 0055.38705
[17] DOI: 10.1007/BF01226242 · Zbl 0251.13012 · doi:10.1007/BF01226242
[18] Gurevich, Foundations of the Theory of Algebraic Invariants (1964)
[19] Grace, The Algebra of Invariants (1903) · JFM 34.0114.01
[20] Gordan, Vorlesung Über Invariantentheorie 2 (1887)
[21] DOI: 10.1016/0021-8693(79)90167-4 · Zbl 0432.13002 · doi:10.1016/0021-8693(79)90167-4
[22] Gegenbauer, Denkschr. Kais. Akad. Wiss. Wien 43 pp 17– (1882)
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