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Continuity of solutions for the Δϕ-Laplacian operator

Part of: Spectral theory and eigenvalue problems Linear function spaces and their duals Miscellaneous topics - Partial differential equations Existence theories

Published online by Cambridge University Press:  11 September 2020

Natalí A. Cantizano ,
Ariel M. Salort  and
Juan F. Spedaletti
Natalí A. Cantizano
Affiliation:
Departamento de Matemática, FCFMyN, Universidad Nacional de San Luis, Instituto de Matemática Aplicada San Luis, IMASL, CONICET, Italia avenue 1556, San Luis 5700, San Luis, Argentina ([email protected])
Ariel M. Salort
Affiliation:
Departamento de Matemática FCEyN, Universidad de Buenos Aires and IMAS – CONICET, Ciudad Universitaria, Pabellón I, Buenos Aires, Argentina ([email protected])
Juan F. Spedaletti
Affiliation:
Departamento de Matemática, FCFMyN, Universidad Nacional de San Luis, Instituto de Matemática Aplicada San Luis, IMASL, CONICET, Italia avenue 1556, office 155, San Luis 5700, San Luis, Argentina ([email protected])
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Abstract

In this paper we give sufficient conditions to obtain continuity results of solutions for the so called ϕ-Laplacian Δϕ with respect to domain perturbations. We point out that this kind of results can be extended to a more general class of operators including, for instance, nonlocal nonstandard growth type operators.

Keywords

Orlicz–Sobolevnonstandard growth

MSC classification

Primary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 35R99: None of the above, but in this section 35R11: Fractional partial differential equations 35P99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 4 , August 2021 , pp. 1355 - 1382
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Kufner, O., John, A. and Fučík, S.. Function spaces, (Leyden: Noordhoff International Publishing; Prague: Academia, 1977). MR 0482102.Google Scholar
Allaire, G.. Shape optimization by the homogenization method, Applied Mathematical Sciences, vol. 146, (New York: Springer-Verlag, 2002), MR1859696.10.1007/978-1-4684-9286-6CrossRefGoogle Scholar
Baroncini, C. and Bonder, J. F.. An extension of a theorem of V. Šverák to variable exponent spaces. Commun. Pure Appl. Anal. 14 (2015), 1987–2007. MR3359555.CrossRefGoogle Scholar
Baroncini, C., Bonder, J. F. and Spedaletti, J. F.. Continuity results with respect to domain perturbation for the fractional p-Laplacian. Appl. Math. Lett. 75 (2018), 59–67. MR3692161.CrossRefGoogle Scholar
Baruah, D., Harjulehto, P. and Hästö, P. Capacities in generalized Orlicz spaces. J. Funct. Spaces 2018 (2018), Art. ID 8459874, 10. MR3864628.Google Scholar
Bonder, J. F., Pérez-Llanos, M. and Salort, A. M.. A hölder infinity Laplacian obtained as limit of Orlicz fractional Laplacians, preprint, arXiv:1807.01669, (2018).Google Scholar
Bonder, J. F. and Salort, A. M.. Fractional order Orlicz–Sobolev spaces. J. Funct. Anal. 277 (2019), 333–367. MR3952156.CrossRefGoogle Scholar
Bucur, D. and Buttazzo, G.. Variational methods in shape optimization problems, Progress in Nonlinear Differential Equations and their Applications, vol. 65, (Boston, MA: Birkhöauser Boston, Inc., 2005). MR2150214.CrossRefGoogle Scholar
Bucur, D. and Trebeschi, P.. Shape optimisation problems governed by nonlinear state equations. Proc. R. Soc. Edinburgh Sect. A 128 (1998), 945–963. MR1642112.CrossRefGoogle Scholar
Bucur, D. and Zolésio, J-P.. N-dimensional shape optimization under capacitary constraint. J. Differ. Equ. 123 (1995), 504–522. MR1362884.CrossRefGoogle Scholar
Cioranescu, D. and Murat, F. A strange term coming from nowhere [MR0652509 (84e:35039a); MR0670272 (84e:35039b)], Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., vol. 31, (Boston, MA: Birkhäuser Boston, 1997), pp. 45–93. MR1493040.Google Scholar
da Silva, J. V., Salort, A. M., Silva, A. and Spedaletti, J. F.. A constrained shape optimization problem in Orlicz–Sobolev spaces. J. Diff. Equ. 267 (2019), 5493–5520. MR3991565.CrossRefGoogle Scholar
Diening, L., Harjulehto, P., Hästö, P. and Ružička, M.. Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, (Heidelberg: Springer, 2011). MR2790542.10.1007/978-3-642-18363-8CrossRefGoogle Scholar
Ekeland, I. and Témam, R. Convex analysis and variational problems, English ed., Classics in Applied Mathematics, vol. 28, (Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1999), Translated from the French. MR1727362.CrossRefGoogle Scholar
Harjulehto, P., Hästö, P., Koskenoja, M. and Varonen, S.. Sobolev capacity on the space W 1,p(·)(ℝn). J. Funct. Spaces Appl. 1 (2003), 17–33. MR2011498.10.1155/2003/895261CrossRefGoogle Scholar
Heinonen, J., Kilpeläinen, T. and Martio, O. Nonlinear potential theory of degenerate elliptic equations (New York: Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, 1993), Oxford Science Publications. MR1207810.Google Scholar
Henrot, A. Extremum problems for eigenvalues of elliptic operators (Basel: Frontiers in Mathematics, Birkhäuser Verlag, 2006). MR2251558.CrossRefGoogle Scholar
Henrot, A. and Pierre, M. Variation et optimisation de formes, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 48, (Berlin: Springer, 2005), Une analyse géométrique. [A geometric analysis]. MR2512810.10.1007/3-540-37689-5CrossRefGoogle Scholar
Krasnoselskii, M. A. and Rutickii, J. B. Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. (Groningen: Noordhoff Ltd., 1961). MR0126722.Google Scholar
Montenegro, M. Strong maximum principles for supersolutions of quasilinear elliptic equations. Nonlinear Anal. 37 (1999), Ser. A: Theory Methods, 431–448. MR1691019.10.1016/S0362-546X(98)00057-1CrossRefGoogle Scholar
Ohno, T. and Shimomura, T. Musielak–Orlicz–Sobolev spaces with zero boundary values on metric measure spaces. Czechoslovak Math. J. 66 2016), 371–394. MR3519608.CrossRefGoogle Scholar
Orlicz, W. and Birnbaum, Z. W.. Uber die verallgemeinerung des begriffes der zueinander konjugierten potenzen. Studia Math. 3 (1931), 1–67.Google Scholar
Pironneau, O. Optimal shape design for elliptic systems, Springer Series in Computational Physics (New York: Springer-Verlag, 1984). MR725856.CrossRefGoogle Scholar
Rao, M. M. and Ren, Z. D. Applications of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 250 (New York: Marcel Dekker, Inc., 2002). MR1890178.CrossRefGoogle Scholar
Salort, A. M.. Eigenvalues and minimizers for a non-standard growth non-local operator. J. Diff. Equ. 268 (2020), 5413–5439. MR4066053.CrossRefGoogle Scholar
Sokolowski, J. and Zolésio, J-P. Introduction to shape optimization, Springer Series in Computational Mathematics, vol. 16 (Berlin: Springer-Verlag, 1992), Shape sensitivity analysis. MR1215733.CrossRefGoogle Scholar
Šverák, V.. On optimal shape design. J. Math. Pures Appl. 72 (1993), 537–551. MR1249408.Google Scholar
Tartar, L. The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana vol. 7, (Berlin; UMI, Bologna: Springer-Verlag, 2009), A personalized introduction. MR2582099.Google Scholar