Comptes Rendus
Singular sets of Sobolev functions
[Ensembles singuliers des fonctions de Sobolev]
Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 539-544.

Nous sommes intéressés à trouver des fonctions de Sobolev dont l'ensemble des singularités est « grand ». Étant donné N,k, 1<p<∞, kp<N, pour chaque sous-ensemble A compact de N , dont la « box-dimension » supérieure est plus petite que Nkp, nous construisons une fonction de Sobolev uW k,p ( N ) qui est singulière précisément sur A. Nous introduisons les notions de dimensions singulières inférieure et supérieure de l'espace de Sobolev, et montrons que ses valeurs sont Nkp.

We are interested in finding Sobolev functions with “large” singular sets. Given N,k, 1<p<∞, kp<N, for any compact subset A of N , such that its upper box dimension is less than Nkp, we construct a Sobolev function uW k,p ( N ) which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to Nkp.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02316-6

Darko Žubrinić 1

1 Department of Applied Mathematics, Faculty of Electrical Engineering, Unska 3, 10000 Zagreb, Croatia
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Darko Žubrinić. Singular sets of Sobolev functions. Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 539-544. doi : 10.1016/S1631-073X(02)02316-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02316-6/

[1] D.R. Adams; L.I. Hedberg Function Spaces and Potential Theory, Springer-Verlag, 1996

[2] N. Aronszajn; K.T. Smith Theory of Bessel potentials I, Ann. Inst. Fourier (Grenoble), Volume 13 (1956), pp. 125-185

[3] T. Bagby; W. Ziemer Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc., Volume 194 (1974), pp. 129-148

[4] A.P. Calderón Lebesgue spaces of differentiable functions and distributions, Partial Differential Equations, Proc. Sympos. Pure Math., 4, American Mathematical Society, Providence, RI, 1961, pp. 33-49

[5] J. Deny Les potentiels d'energie finie, Acta Math., Volume 82 (1950), pp. 107-183

[6] J. Deny; J.L. Lions Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble), Volume 5 (1953–1954), pp. 305-370

[7] K.J. Falconer Fractal Geometry, Wiley, New York, 1990

[8] H. Federer Geometric Measure Theory, Springer-Verlag, 1969

[9] B. Fuglede Extremal length and functional completion, Acta Math., Volume 98 (1957), pp. 171-219

[10] B. Fuglede On generalized potentials of functions in the Lebesgue classes, Math. Scand., Volume 8 (1960), pp. 287-304

[11] M. Grillot Prescribed singular submanifolds of some quasilinear elliptic equations, Nonlinear Anal., Volume 34 (1998), pp. 839-856

[12] V.P. Havin; V.G. Mazya Use of (p,l)-capacity in problems of the theory of exceptional sets, Mat. Sb., Volume 90 (1973) no. 132, pp. 558-591 Math. USSR-Sb. 19 (1973) 547–580

[13] J. Heinonen; T. Kilpeläinen; O. Martio Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993

[14] S. Jaffard; Y. Meyer Wavelet methods for pointwise regularity and local oscillations of functions, Mem. Amer. Math. Soc., Volume 123 (1996) no. 587

[15] J. Keesling Hausdorff dimension, Topology Proc., Volume 11 (1986) no. 2, pp. 349-383

[16] T. Kilpeläinen Singular solutions to p-Laplacian type equations, Ark. Mat., Volume 37 (1999), pp. 275-289

[17] L. Korkut; M. Pašić; D. Žubrinić Some qualitative properties of solutions of quasilinear elliptic equations and applications, J. Differential Equations, Volume 170 (2001), pp. 247-280

[18] J. Malý; W.P. Ziemer Fine Regularity of Solutions of Elliptic Partial Differential Equations, American Mathematical Society, 1997

[19] N.G. Meyers Continuity properties of potentials, Duke Math. J., Volume 42 (1975), pp. 157-166

[20] L. Mou Removability of singular sets of harmonic maps, Arch. Rational Mech. Anal., Volume 127 (1994), pp. 199-217

[21] Yu.G. Reshetnyak On the concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Zh., Volume X (1969) no. 5, pp. 1108-1138 (in Russian); Siberian Math. J. 13 (1969) 818–842

[22] J. Serrin Isolated singularities of solutions of quasi-linear equations, Acta Math., Volume 113 (1965), pp. 219-240

[23] E.M. Stein Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970

[24] L. Veron Singularities of Solutions of Second Order Quasilinear Equations, Addison-Wesley–Longman, 1996

[25] W.P. Ziemer Weakly Differentiable Functions; Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Math., Springer-Verlag, 1989

[26] D. Žubrinić Generating singularities of solutions of quasilinear elliptic equations, J. Math. Anal. Appl., Volume 244 (2000), pp. 10-16

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