# American Institute of Mathematical Sciences

September  2015, 14(5): 1987-2007. doi: 10.3934/cpaa.2015.14.1987

## An extension of a Theorem of V. Šverák to variable exponent spaces

 1 IMAS-CONICET and Departamento de Matemática, FCEN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 1 (1428), Buenos Aires , Argentina 2 Departamento de Matematica, FCEyN, UBA, 1428 Buenos Aires, Argentina

Received  November 2014 Revised  April 2015 Published  June 2015

In 1993, V. Šverák proved that if a sequence of uniformly bounded domains $\Omega_n\subset R^2$ such that $\Omega_n\to \Omega$ in the sense of the Hausdorff complementary topology, verify that the number of connected components of its complements are bounded, then the solutions of the Dirichlet problem for the Laplacian with source $f\in L^2(R^2)$ converges to the solution of the limit domain with same source. In this paper, we extend Šverák result to variable exponent spaces.
Citation: Carla Baroncini, Julián Fernández Bonder. An extension of a Theorem of V. Šverák to variable exponent spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1987-2007. doi: 10.3934/cpaa.2015.14.1987
##### References:
 [1] G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar [2] I. Babuška and R. Výborný, Continuous dependence of eigenvalues on the domain, Czechoslovak Math. J., 15 (1965), 169-178.  Google Scholar [3] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983, Théorie et applications. [Theory and applications].  Google Scholar [4] D. Bucur and P. Trebeschi, Shape optimisation problems governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006.  Google Scholar [5] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.  Google Scholar [6] D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topics in the Mathematical Modelling of Composite Materials, vol. 31 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1997, 45-93.  Google Scholar [7] L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer, Heidelberg, 2011. Google Scholar [8] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland Publishing Co., Amsterdam, 1976, Translated from the French, Studies in Mathematics and its Applications, Vol. 1.  Google Scholar [9] X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.  Google Scholar [10] R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal., 67 (1977), 25-39.  Google Scholar [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar [12] A. Henrot and M. Pierre, Variation et optimisation de formes, vol. 48 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer, Berlin, 2005, Une analyse géométrique. [A geometric analysis]. doi: 10.1007/3-540-37689-5.  Google Scholar [13] I. Hong, On an eigenvalue and eigenfunction problem of the equation $\Delta u+\lambda u=0$, Kōdai Math. Sem. Rep., 9 (1957), 179-190.  Google Scholar [14] I. Hong, A supplement to "On an eigenvalue and eigenfunction problem of the equation $\Delta u+\lambda u=0$'', Kōdai Math. Sem. Rep., 10 (1958), 27-37.  Google Scholar [15] I. Hong, On the equation $\Delta u+\lambda f(x,\,y)=0$ under the fixed boundary condition, Kōdai Math. Sem. Rep., 11 (1959), 95-108.  Google Scholar [16] T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. Google Scholar [17] T. Lukkari, Boundary continuity of solutions to elliptic equations with nonstandard growth, Manuscripta Math., 132 (2010), 463-482. doi: 10.1007/s00229-010-0355-3.  Google Scholar [18] J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, vol. 51 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/051.  Google Scholar [19] V. G. Mazja, The continuity at a boundary point of the solutions of quasi-linear elliptic equations, Vestnik Leningrad. Univ., 25 (1970), 42-55.  Google Scholar [20] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. doi: 10.1007/978-3-642-87722-3.  Google Scholar [21] M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar [22] J. Simon, Régularité de la solution d'une équation non linéaire dans $R^N$, in Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), vol. 665 of Lecture Notes in Math., Springer, Berlin, 1978, 205-227.  Google Scholar [23] V. Šverák, On optimal shape design, J. Math. Pures Appl., 72 (1993), 537-551.  Google Scholar [24] L. Tartar, The General Theory of Homogenization, vol. 7 of Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, Berlin, 2009. A personalized introduction. doi: 10.1007/978-3-642-05195-1.  Google Scholar

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##### References:
 [1] G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar [2] I. Babuška and R. Výborný, Continuous dependence of eigenvalues on the domain, Czechoslovak Math. J., 15 (1965), 169-178.  Google Scholar [3] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983, Théorie et applications. [Theory and applications].  Google Scholar [4] D. Bucur and P. Trebeschi, Shape optimisation problems governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006.  Google Scholar [5] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.  Google Scholar [6] D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topics in the Mathematical Modelling of Composite Materials, vol. 31 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1997, 45-93.  Google Scholar [7] L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer, Heidelberg, 2011. Google Scholar [8] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland Publishing Co., Amsterdam, 1976, Translated from the French, Studies in Mathematics and its Applications, Vol. 1.  Google Scholar [9] X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.  Google Scholar [10] R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal., 67 (1977), 25-39.  Google Scholar [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar [12] A. Henrot and M. Pierre, Variation et optimisation de formes, vol. 48 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer, Berlin, 2005, Une analyse géométrique. [A geometric analysis]. doi: 10.1007/3-540-37689-5.  Google Scholar [13] I. Hong, On an eigenvalue and eigenfunction problem of the equation $\Delta u+\lambda u=0$, Kōdai Math. Sem. Rep., 9 (1957), 179-190.  Google Scholar [14] I. Hong, A supplement to "On an eigenvalue and eigenfunction problem of the equation $\Delta u+\lambda u=0$'', Kōdai Math. Sem. Rep., 10 (1958), 27-37.  Google Scholar [15] I. Hong, On the equation $\Delta u+\lambda f(x,\,y)=0$ under the fixed boundary condition, Kōdai Math. Sem. Rep., 11 (1959), 95-108.  Google Scholar [16] T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. Google Scholar [17] T. Lukkari, Boundary continuity of solutions to elliptic equations with nonstandard growth, Manuscripta Math., 132 (2010), 463-482. doi: 10.1007/s00229-010-0355-3.  Google Scholar [18] J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, vol. 51 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/051.  Google Scholar [19] V. G. Mazja, The continuity at a boundary point of the solutions of quasi-linear elliptic equations, Vestnik Leningrad. Univ., 25 (1970), 42-55.  Google Scholar [20] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. doi: 10.1007/978-3-642-87722-3.  Google Scholar [21] M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar [22] J. Simon, Régularité de la solution d'une équation non linéaire dans $R^N$, in Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), vol. 665 of Lecture Notes in Math., Springer, Berlin, 1978, 205-227.  Google Scholar [23] V. Šverák, On optimal shape design, J. Math. Pures Appl., 72 (1993), 537-551.  Google Scholar [24] L. Tartar, The General Theory of Homogenization, vol. 7 of Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, Berlin, 2009. A personalized introduction. doi: 10.1007/978-3-642-05195-1.  Google Scholar
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