Singular sets of Sobolev functions

Darko Žubrinić

doi:10.1016/S1631-073X(02)02316-6

Singular sets of Sobolev functions
[Ensembles singuliers des fonctions de Sobolev]

Darko Žubrinić ¹

¹ Department of Applied Mathematics, Faculty of Electrical Engineering, Unska 3, 10000 Zagreb, Croatia

Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 539-544.

Résumé
Abstract

Nous sommes intéressés à trouver des fonctions de Sobolev dont l'ensemble des singularités est « grand ». Étant donné $N, k \in ℕ$ , 1<p<∞, kp<N, pour chaque sous-ensemble A compact de $ℝ^{N}$ , dont la « box-dimension » supérieure est plus petite que N−kp, nous construisons une fonction de Sobolev $u \in W^{k, p} (ℝ^{N})$ 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 N−kp.

We are interested in finding Sobolev functions with “large” singular sets. Given $N, k \in ℕ$ , 1<p<∞, kp<N, for any compact subset A of $ℝ^{N}$ , such that its upper box dimension is less than N−kp, we construct a Sobolev function $u \in W^{k, p} (ℝ^{N})$ 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 N−kp.

Reçu le : 2001-12-21
Accepté le : 2002-02-04
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02316-6

Affiliations des auteurs :

Darko Žubrinić ¹

¹ Department of Applied Mathematics, Faculty of Electrical Engineering, Unska 3, 10000 Zagreb, Croatia

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     title = {Singular sets of {Sobolev} functions},
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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/

Bibliographie
Cité par

[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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