×

zbMATH — the first resource for mathematics

Remarks on the regularity of the minimizers of certain degenerate functionals. (English) Zbl 0607.49003
The authors study the partial or global regularity of the minimizers for integrals functionals of the type \(\int_{\Omega}f(x,u,Du)dx\) where \(\Omega\) is a bounded open set in \({\mathbb{R}}^ n\), \(u: \Omega\to {\mathbb{R}}^ N\) and f is subjected to a growth condition in the last variable, which is polynomial like \(| p|^ m\), \(m\geq 2\). The treatment is very careful and articulate and includes also many hard cases as, for example, that of degenerate integrands. The theorems they obtain extend and improve various well-known results on the subject (see the references).
Reviewer: A.Salvadori

MSC:
49J10 Existence theories for free problems in two or more independent variables
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35D10 Regularity of generalized solutions of PDE (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] M. CHIPOT, L.C. EVANS- Linearization at infinity and Lipschitz estimates for certain problems in the Calculus of variations, pre-print
[2] N. FUSCO, J. HUTCHINSON- Partial regularity for minimizers of certain functionals having non quadratic growth. Manuscripta math.54 (1985) 121,11 43 · Zbl 0587.49005 · doi:10.1007/BF01171703
[3] M. GIAQUINTA- Multiple integrals in the Calculus of Variations and Nonlinear elliptic systems. Princeton Univ. Press, Princeton 1983 · Zbl 0516.49003
[4] M. GIAQUINTA, E. GIUSTI- On the regularity of minima of variational integrals. Acta Math.148 (1982) 31-46 · Zbl 0494.49031 · doi:10.1007/BF02392725
[5] M. GIAQUINTA, E. GIUSTI- Differentiability of minima of non-differentiable functionals. Inventiones Math.72 (1983) 285-298 · Zbl 0513.49003 · doi:10.1007/BF01389324
[6] M. GIAQUINTA, E. GIUSTI- Quasi-minima. Ann. Inst. H. Poincaré, Analyse non linéaire1 (1984) 79-107
[7] M. GIAQUINTA, E. GIUSTI- Sharp estimates for the derivatives of local minima of variat?nal integrals. Boll. UMI (6)3-A (1984) 239-248
[8] M. GIAQUINTA, P.A. IVERT- Partial regularity for minima of variational integrals. Ank. für Math · Zbl 0637.49005
[9] M. GIAQUINTA, G. MODICA- Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. H. Poincaré, Analyse nonlinéaire3 (1986) · Zbl 0594.49004
[10] M. GIAQUINTA, J. SOUCEK- Harmonic maps into a hemisphere. Ann. Sc. Norm. Sup. Pisa12 (1985) 81-90 · Zbl 0599.58017
[11] D. GILBARG, N.S. TRUDINGER- Elliptic partial differential equations of second order. Springer Verlag, Heidelberg 1977 · Zbl 0361.35003
[12] W. JÄGER, H. KAUL- Relationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems. J. reine angew. Math.343 (1983) 146-161
[13] O.A. LADYZHENSKAYA, N.N. URAL’TSEVA- Linear and quasi-linear elliptic equations. Acad. Press, New York, 1968
[14] C.B. MORREY- Multiple integrals in the Calculus of Variations. Springer Verlag Hildelberg 1966
[15] J. MOSER- On Harnack’s inequality for elliptic differential equations. Comm. Pure Appl. Math.14 (1961) 577-591 · Zbl 0111.09302 · doi:10.1002/cpa.3160140329
[16] R. SCOEN, K. UHLENBECK- Regularity of minimizing harmonic maps into the sphere. Inventiones Math.78 (1984) 89-100 · Zbl 0555.58011 · doi:10.1007/BF01388715
[17] K. UHLENBECK- Regularity for a class of nonlinear elliptic systems. Acta Math.138 (1977) 219-240 · Zbl 0372.35030 · doi:10.1007/BF02392316
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.