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1996 | 33 | 1 | 79-83
Tytuł artykułu

A counterexample to the $L^{p}$-Hodge decomposition

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We construct a bounded domain $Ω ⊂ ℝ^2$ with the cone property and a harmonic function on Ω which belongs to $W_0^{1,p}(Ω)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no $L^p$-Hodge decomposition in $L^{p}(Ω,ℝ^2)$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in $W^{1,p}(Ω)$ for all p > 4.
Rocznik
Tom
33
Numer
1
Strony
79-83
Opis fizyczny
Daty
wydano
1996
Twórcy
  • Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
Bibliografia
  • [1] F. Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), 417-443.
  • [2] M.-E. Bogovskiĭ, Solutions of some vector analysis problems connected with the operators div and grad, Trudy Sem. Sobolev. Akad. Nauk SSSR, Sibirsk. Otdel. Mat., 1 (1980), 5-40 (in Russian).
  • [3] J.-E. Brennan, The integrability of the derivative in conformal mapping, J. London Math. Soc. 18 (1978), 261-272.
  • [4] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. I, Springer, 1990.
  • [5] L. Greco and T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 17-35.
  • [6] L. Greco, T. Iwaniec and G. Moscariello, Limits of the improved integrability of the volume forms, Indiana Univ. Math. J. 44 (1995), 305-339.
  • [7] T. Iwaniec, p-harmonic tensors and quasiregular mappings, Ann. of Math. 136 (1992), 589-624.
  • [8] T. Iwaniec, $L^p$-theory of quasiregular mappings, in: Lecture Notes in Math. 1508, Springer, 1992, 39-64.
  • [9] T. Iwaniec and A. Lutoborski, Integral estimates for null-Lagrangians, Arch. Rational Mech. Anal. 125 (1993), 25-79.
  • [10] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29-81.
  • [11] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129-143.
  • [12] T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143-161.
  • [13] H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains, Indiana Univ. Math. J. 40 (1991), 1-27.
  • [14] V. G. Maz'ya and A. A. Solov'ev, On an integral equation for the Dirichlet problem in a plane domain with a cusp on the boundary, Mat. Sb. 180 (1989), 1211-1233 (in Russian); English transl.: Math. USSR-Sb. 68 (1991), 61-83.
  • [15] Ch. Pommerenke, On the integral means of the derivative of a univalent function, J. London Math. Soc. 32 (1985), 254-258.
  • [16] C. Scott, $L^p$-theory of differential forms on manifolds, preprint.
  • [17] C.-G. Simader, On Dirichlet's Boundary Value Problem, Lecture Notes in Math. 268, Springer, 1972.
  • [18] B. Stroffolini, On weakly A-harmonic tensors, Studia Math. 114 (1995), 289-301.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-bcpv33z1p79bwm
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