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

show all references

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

[1]

Krzysztof Fujarewicz, Krzysztof Łakomiec. Adjoint sensitivity analysis of a tumor growth model and its application to spatiotemporal radiotherapy optimization. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1131-1142. doi: 10.3934/mbe.2016034

[2]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2907-2946. doi: 10.3934/dcds.2020391

[3]

Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381

[4]

Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial & Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1

[5]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022

[6]

Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial & Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611

[7]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks & Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189

[8]

Tomasz Adamowicz, Przemysław Górka. The Liouville theorems for elliptic equations with nonstandard growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2377-2392. doi: 10.3934/cpaa.2015.14.2377

[9]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019

[10]

Barbara Kaltenbacher, Gunther Peichl. The shape derivative for an optimization problem in lithotripsy. Evolution Equations & Control Theory, 2016, 5 (3) : 399-430. doi: 10.3934/eect.2016011

[11]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623

[12]

Caglar S. Aksezer. On the sensitivity of desirability functions for multiresponse optimization. Journal of Industrial & Management Optimization, 2008, 4 (4) : 685-696. doi: 10.3934/jimo.2008.4.685

[13]

Patrick Winkert, Rico Zacher. A priori bounds for weak solutions to elliptic equations with nonstandard growth. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 865-878. doi: 10.3934/dcdss.2012.5.865

[14]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006

[15]

Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, Jürgen Sprekels. An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 353-368. doi: 10.3934/dcdss.2013.6.353

[16]

Markus Muhr, Vanja Nikolić, Barbara Wohlmuth, Linus Wunderlich. Isogeometric shape optimization for nonlinear ultrasound focusing. Evolution Equations & Control Theory, 2019, 8 (1) : 163-202. doi: 10.3934/eect.2019010

[17]

Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055

[18]

Günter Leugering, Jan Sokołowski, Antoni Żochowski. Control of crack propagation by shape-topological optimization. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2625-2657. doi: 10.3934/dcds.2015.35.2625

[19]

Stanislav Antontsev, Michel Chipot, Sergey Shmarev. Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1527-1546. doi: 10.3934/cpaa.2013.12.1527

[20]

Elena Celledoni, Markus Eslitzbichler, Alexander Schmeding. Shape analysis on Lie groups with applications in computer animation. Journal of Geometric Mechanics, 2016, 8 (3) : 273-304. doi: 10.3934/jgm.2016008

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]