Abstract
In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied. Given a density function in a rearrangement class generated by a step function, find the density such that the principal eigenvalue is as small as possible. Considering a membrane of known fixed mass and with fixed boundary of prescribed shape consisting of two different materials, our results determine the way to distribute these materials such that the basic frequency of the membrane is minimal. We obtain some qualitative aspects of the optimizer and then we determine nearly optimal sets which are approximations of the minimizer for specific ranges of parameters values. A numerical algorithm is proposed to derive the optimal shape and it is proved that the numerical procedure converges to a local minimizer. Numerical illustrations are provided for different domains to show the efficiency and practical suitability of our approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anane, A.: Simplicité et isolation de la première valeur propre du \(p\)-Laplacien avec poids. C. R. Acad. Sci. Paris Stér. I Math. 305, 725–728 (1987)
Anedda, C., Cuccu, F.: Steiner symmetry in the minimization of the first eigenvalue in problems involving the \(p\)-Laplacian. Proc. Am. Math. Soc. 144, 3431–3440 (2016)
Antunes, P.R.S., Mohammadi, S.A., Voss, H.: A nonlinear eigenvalue optimization problem: optimal potential functions. Nonlinear Anal. RWA 40, 307–327 (2018)
Burton, G.R.: Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. H. Poincare Anal. Non Linaire 6(4), 295–319 (1989)
Bozorgnia, F.: Convergence of inverse power method for first eigenvalue of \(p\)-Laplace Operator. Numer. Func. Anal. Opt. 37, 1378–1384 (2016)
Chanillo, S., Grieser, D., Kurata, K.: The free boundary problem in the optimization of composite membranes. Contemp. Math. 268, 61–81 (2000)
Chanillo, S., Grieser, D., Imai, M., Kurata, K., Ohnishi, I.: Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun. Math. Phys. 214, 315–337 (2000)
Chen, W., Chou, C.-S., Kao, C.-Y.: Minimizing eigenvalues for inhomogeneous rods and plates. J. Sci. Comput. 69, 983–1013 (2016)
Cox, S.J., McLaughlin, J.R.: Extremal eigenvalue problems for composite membranes, I and II. Appl. Math. Optim. 22(153–167), 169–187 (1990)
Cuccu, F., Emamizadeh, B., Porru, G.: Optimization of the first eigenvalue in problems involving the p-Laplacian. Proc. Am. Math. Soc. 137, 1677–1687 (2009)
Del Pezzo, L.M., Fernández Bonder, J.: An optimization problem for the first weighted eigenvalue problem plus a potential. Proc. Am. Math. Soc. 138, 3551–3567 (2010)
Derlet, A., Gossez, J.-P., Takáč, P.: Minimization of eigenvalues for a quasilinear elliptic Neumann problem with indefinite wieght. J. Math. Anal. Appl. 371, 69–79 (2010)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, New York (1998)
Juutinen, P., Lindqvist, P., Manfredi, J.J.: The \(\infty \)-eigenvalue problem. Arch. Ration. Mech. Anal. 148, 89–105 (1999)
Kang, D., Kao, C.-Y.: Minimization of inhomogeneous biharmonic eigenvalue problems. Appl. Math. Model. 51, 587–604 (2017)
Kao, C.-Y., Su, S.: Efficient rearrangement algorithm for shape optimization on elliptic eigenvalue problems. J. Sci. Comput. 54, 492–512 (2013)
Kawohl, B., Lucia, M., Prashanth, S.: Simplicity of the principal eigenvalue for indefinite quasilinear problems. Adv. Differ. Equ. 12, 407–434 (2007)
Lê, A.: Eigenvalue problems for the \(p\)-Laplacian. Nonlinear Anal. 64, 1057–1099 (2006)
Lieb, E., Loss, M.: Analysis, 2nd edn. American Mathematical Society, Providence (2001)
Lindqvist, P.: On the equation \(\text{ div }(| \nabla u|^{p-2} \nabla u) + \lambda |u|^{p-2} u = 0\). Proc. Am. Math. Soc. 109, 157–164 (1990)
Lindqvist, P.: A nonlinear eigenvalue problem. Topics in mathematical analysis, Ser. Anal. Appl. Comput. 3, 175–203 (2008)
Lindqvist, P., Manfredi, J.J.: Note on \(\infty \)-harmonic functions. Revista matemática de la universidad complutense de Madrid 10, 1–9 (1997)
Mohammadi, A., Bahrami, F.: A nonlinear eigenvalue problem arising in a nanostructured quantum dot. Commun. Nonlinear Sci. Numer. Simul. 19, 3053–3062 (2014)
Mohammadi, S.A.: Extremal energies of Laplacian operator: different configurations for steady vortices. J. Math. Anal. Appl. 448, 140–155 (2017)
Mohammadi, S.A., Voss, H.: A minimization problem for an elliptic eigenvalue problem with nonlinear dependence on the eigenparameter. Nonlinear. Anal. RWA 31, 119–131 (2016)
Mohammadi, A., Bahrami, F., Mohammadpour, H.: Shape dependent energy optimization in quantum dots. Appl. Math. Lett. 25, 1240–1244 (2012)
Mohammadi, A., Bahrami, F.: Extremal principal eigenvalue of the bi-Laplacian operator. Appl. Math. Model. 40, 2291–2300 (2016)
Pielichowski, W.: The optimization of eigenvalue problems involving the \(p\)-Laplacian. Univ. Iagel. Acta Math. 42, 109–122 (2004)
Struwe, M.: Variational Methods. Springer, Berlin (1990)
Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)
Vàzquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)
Yu, Y.: Some properties of the ground states of the infinity Laplacian. Indiana Univ. Math. J. 56, 947–964 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mohammadi, S.A., Bozorgnia, F. & Voss, H. Optimal Shape Design for the p-Laplacian Eigenvalue Problem. J Sci Comput 78, 1231–1249 (2019). https://doi.org/10.1007/s10915-018-0806-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-018-0806-7