Quasiharmonic fields and Beltrami operators
Capone, Claudia
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002), p. 363-377 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

A quasiharmonic field is a pair $\Cal{F} = [B,E]$ of vector fields satisfying $\operatorname{div} B=0$, $\operatorname{curl} E=0$, and coupled by a distorsion inequality. For a given $\Cal F$, we construct a matrix field $\Cal A=\Cal A[B,E]$ such that ${\Cal A} E=B$. This remark in particular shows that the theory of quasiharmonic fields is equivalent (at least locally) to that of elliptic PDEs. Here we stress some properties of our operator $\Cal A[B,E]$ and find their applications to the study of regularity of solutions to elliptic PDEs, and to some questions of G-convergence.

Publié le : 2002-01-01
Classification:  30C65,  35B40,  35B45,  35D10,  35J20,  35J60,  47B99,  47F05 
@article{119326,
     author = {Claudia Capone},
     title = {Quasiharmonic fields and Beltrami operators},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {43},
     year = {2002},
     pages = {363-377},
     zbl = {1069.35024},
     mrnumber = {1922134},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119326}
}

Capone, Claudia. Quasiharmonic fields and Beltrami operators. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 363-377. http://gdmltest.u-ga.fr/item/119326/

Astala K. Recent Connections and Applications of Planar Quasiconformal Mappings, Progr. Math. 168 36-51 (1998). (1998) | MR 1645796 | Zbl 0908.30019

De Giorgi E. Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. U.M.I. 4 (1968), 135-137. (1968) | MR 0227827

De Giorgi E.; Spagnolo S. Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. U.M.I. 8 (1973), 391-411. (1973) | MR 0348255 | Zbl 0274.35002

Formica M.R. On the $\Gamma$-convergence of Laplace-Beltrami Operators in the plane, Ann. Acad. Sci. Fenn. Math. 25 (2000), 423-438. (2000) | MR 1762427 | Zbl 0955.30016

Francfort G.A.; Murat F. Optimal bounds for conduction in two-dimensional, two-phase, Anisotropic media, Non-classical continuum mechanics, R.J. Knops and A.A. Lacey, Eds., London Mathematical Society Lecture Note Series 122, Cambridge, 1987, pp.197-212. | MR 0926503 | Zbl 0668.73018

Keller J.B. A theorem on the conductivity of a composite medium, J. Math. Phys. 5 (1964), 548-549. (1964) | MR 0161559 | Zbl 0129.44001

Koshelev A.I. Regularity of solutions of quasilinear elliptic systems, Engl.trans.: Russian Math. Survey 33 (1978), 1-52. (1978) | MR 0510669

Iwaniec T.; Sbordone C. Quasiharmonic fields, Ann. Inst. H. Poincaré-AN 18.5 (2001), 519-572. (2001) | MR 1849688 | Zbl 1068.30011

John O.; Malý J.; Stará J. Nowhere continuous solutions to elliptic systems, Comment. Math. Univ. Carolinae 30.1 (1989), 33-43. (1989) | MR 0995699

Leonetti F.; Nesi V. Quasiconformal solutions and a conjecture of Milton, J. Math. Pures Appl. 76 (1997), 109-124. (1997) | MR 1432370

Liusterinik, Sobolev Elements of Functional Analysis, (1965), Frederick Ungar Publishing Company London. (1965)

Marino A.; Spagnolo S. Un tipo di approssimazione dell'operatore $\sum D_i{a_ij}D_j$ con operatori $D_j(b D_j)$, Ann. Scu. Norm. Sup. Pisa 23 (1969), 657-673. (1969) | MR 0278128

Meyers N. An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scu. Norm. Pisa 17 (1963), 189-206. (1963) | MR 0159110

Milton G.W. Modelling the properties of composites by laminates, in Homogenization and effective moduli of materials and media, Ericksen, Kinderleher, Kohn, Lions, Eds., IMA volumes in mathematics and its applications 1, Springer-Verlag, New York, 1986, pp.150-174. | MR 0859415 | Zbl 0631.73011

Morrey On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166. (1938) | MR 1501936 | Zbl 0018.40501

Murat F. Compacite par Compensation, Ann. Scu. Norm. Pisa 5 (1978), 489-507. (1978) | MR 0506997 | Zbl 0399.46022

Murat F. H-convergence, Seminaire d'Analyse Fonctionelle et Numerique, University of Alger, 1977/78. | Zbl 0920.35019

Souček J. Singular solution to linear elliptic systems, Comment. Math. Univ. Carolinae 25 (1984), 273-281. (1984) | MR 0768815

Spagnolo S. Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scu. Norm. Sup. Pisa 22 (1968), 571-597. (1968) | MR 0240443

Spagnolo S. Some convergence problems, Sympos. Math. 18 (1976), 391-397. (1976) | MR 0509184 | Zbl 0332.46020

Tartar L. Homogeneisation et compacite par compensation, Cours Peccot, College de France, 1977. | Zbl 0406.35055

Tartar L. Compensated Compactness and Applications to Partial Differential Equations, ed. by R.J. Knops, Pitman, London, Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium IV (1979), no. 39, 136-212. | MR 0584398 | Zbl 0437.35004

Tartar L. Convergence d'operateurs differentiels, Analisi Convessa Appl., Roma (1974), 101-104. (1974)