# zbMATH — the first resource for mathematics

Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions. (English) Zbl 1366.49004
The authors analyze a free boundary/shape optimization governed by the linear eigenvalue with indefinite weight $\Delta \varphi + \lambda m\varphi = 0\;\text{in}\,\Omega,\;\;\partial_n\varphi+\beta \varphi=0\;\text{on}\,\partial\Omega,$ where $$\Omega$$ is a bounded open and connected set in $$\mathbb{R}^N$$ with Lipschitz boundary $$\partial\Omega$$ and the weight $$m$$ is a bounded measurable function changing sign in $$\Omega$$ i.e. $$\Omega_m^+= \{x\in \Omega:\,m(x)>0\}$$ has a measure strictly between $$0$$ and $$|\Omega|$$ and fulfils $$-1\leq m(x)\leq \kappa\;\text{a.e.}\,x\in \Omega,\;\kappa>0.$$ The optimization problem has the form $\inf_{m\in \mathcal{M}}\lambda(m),\;\mathcal{M}=\left\{m\in L^{\infty}:-1\leq m\leq \kappa,\;|\Omega_m^+|>0,\;\int_\Omega m\geq -m_0|\Omega|\right\}.$ The biological motivation for considering such a problem is formulated.

##### MSC:
 49J15 Existence theories for optimal control problems involving ordinary differential equations 49K20 Optimality conditions for problems involving partial differential equations 49Q10 Optimization of shapes other than minimal surfaces 49R05 Variational methods for eigenvalues of operators 49M05 Numerical methods based on necessary conditions
Full Text:
##### References:
  Afrouzi, GA; Brown, KJ, On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions, Proc. Am. Math. Soc., 127, 125-130, (1999) · Zbl 0903.35045  Kolmogorov, AN; Petrovsky, IG; Piskunov, NS, Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Math. Bull., 1, 1-26, (1937) · Zbl 0018.32106  Bandle, C, Isoperimetric inequality for some eigenvalues of an inhomogeneous, free membrane, SIAM J. Appl. Math., 22, 142-147, (1972) · Zbl 0237.35069  Bandle, C.: Isoperimetric inequalities and applications, monographs and studies in mathematics, vol. 7. Pitman (Advanced Publishing Program), Boston (1980) · Zbl 0436.35063  Berestycki, H.: Personal communication (2012) · Zbl 0043.14401  Berestycki, H; Hamel, F; Roques, L, Analysis of the periodically fragmented environment model. I. species persistence, J. Math. Biol., 51, 75-113, (2005) · Zbl 1066.92047  Berestycki, H; Lachand-Robert, T, Some properties of monotone rearrangement with applications to elliptic equations in cylinders, Math. Nachr., 266, 3-19, (2004) · Zbl 1084.35003  Bôcher, M, The smallest characteristic numbers in a certain exceptional case, Bull. Am. Math. Soc., 21, 6-9, (1914) · JFM 45.0491.02  Cantrell, RS; Cosner, C, Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. R. Soc. Edinburgh Sect. A, 112, 293-318, (1989) · Zbl 0711.92020  Cantrell, RS; Cosner, C, Diffusive logistic equations with indefinite weights: population models in disrupted environments II, SIAM J. Math. Anal., 22, 1043-1064, (1991) · Zbl 0726.92024  Cantrell, RS; Cosner, C, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29, 315-338, (1991) · Zbl 0722.92018  Cantrell, R.S., Cosner, C.: Spatial ecology via reaction-diffusion equations. Wiley Series in Mathematical and Computational Biology. Wiley, Chichester (2003) · Zbl 1059.92051  Chanillo, S; Grieser, D; Imai, M; Kurata, K; Ohnishi, I, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys., 214, 315-337, (2000) · Zbl 0972.49030  Chanillo, S., Grieser, D., Kurata, K.: The free boundary problem in the optimization of composite membranes. In: Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999), vol 268. Contemp. Math., p 61-81. Am. Math. Soc., Providence, RI (2000) · Zbl 0988.35124  Chanillo, S; Kenig, CE, Weak uniqueness and partial regularity for the composite membrane problem, J. Eur. Math. Soc. (JEMS), 10, 705-737, (2008) · Zbl 1154.35096  Chanillo, S; Kenig, CE; To, T, Regularity of the minimizers in the composite membrane problem in $${\mathbb{R}}^2$$, J. Funct. Anal., 255, 2299-2320, (2008) · Zbl 1154.49026  Colbois, B., El Soufi, A.: Spectrum of the Laplacian with weights. Working paper or preprint (2016) · Zbl 1036.58026  Cox, S.J., McLaughlin, J. R.: Extremal eigenvalue problems for composite membranes. I, II. Appl. Math. Optim. 22(2):153-167, 169-187 (1990) · Zbl 0709.73044  Derlet, A; Gossez, J-P; Takáč, P, Minimization of eigenvalues for a quasilinear elliptic Neumann problem with indefinite weight, J. Math. Anal. Appl., 371, 69-79, (2010) · Zbl 1197.35105  Fisher, RA, The advance of advantageous genes, Ann. Eugen., 7, 335-369, (1937) · JFM 63.1111.04  Fleming, WH, A selection-migration model in population genetics, J. Math. Biol., 2, 219-233, (1975) · Zbl 0325.92009  Girouard, A; Polterovich, I, Shape optimization for low Neumann and Steklov eigenvalues, Math. Methods Appl. Sci., 33, 501-516, (2010) · Zbl 1186.35121  Harrell II, E.M., Kröger, P., Kurata, K.: On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue. SIAM J. Math. Anal. 33(1), 240-259 (2001) (electronic) · Zbl 0994.47015  Henrot, A.: Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006) · Zbl 1109.35081  Henrot, A; Oudet, E, Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions, Arch. Ration. Mech. Anal., 169, 73-87, (2003) · Zbl 1055.35080  Henrot, A; Privat, Y, What is the optimal shape of a pipe?, Arch. Ration. Mech. Anal., 196, 281-302, (2010) · Zbl 1304.76022  Hess, P; Kato, T, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differ. Equ., 5, 999-1030, (1980) · Zbl 0477.35075  Hintermüller, M; Kao, C-Y; Laurain, A, Principal eigenvalue minimization for an elliptic problem with indefinite weight and Robin boundary conditions, Appl. Math. Optim., 65, 111-146, (2012) · Zbl 1242.49094  Jha, K; Porru, G, Minimization of the principal eigenvalue under Neumann boundary conditions, Numer. Funct. Anal. Optim., 32, 1146-1165, (2011) · Zbl 1239.47061  Kao, C-Y; Lou, Y; Yanagida, E, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Math. Biosci. Eng., 5, 315-335, (2008) · Zbl 1167.35426  Kawohl, B, On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems, Arch. Rational Mech. Anal., 94, 227-243, (1986) · Zbl 0603.49030  Kohn, RV; Strang, G, Optimal design and relaxation of variational problems, I, Comm. Pure Appl. Math., 39, 113-137, (1986) · Zbl 0609.49008  Kohn, RV; Strang, G, Optimal design and relaxation of variational problems, II, Comm. Pure Appl. Math., 39, 139-182, (1986) · Zbl 0621.49008  Kohn, RV; Strang, G, Optimal design and relaxation of variational problems, III, Comm. Pure Appl. Math., 39, 353-377, (1986) · Zbl 0694.49004  Krein, MG, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, Am. Math. Soc. Transl., 2, 163-187, (1955) · Zbl 0066.33404  Laugesen, RS, Eigenvalues of the Laplacian on inhomogeneous membranes, Am. J. Math., 120, 305-344, (1998) · Zbl 0908.35086  Lou, Y; Yanagida, E, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Jpn. J. Indust. Appl. Math., 23, 275-292, (2006) · Zbl 1185.35059  Nelson, E, Analytic vectors, Ann. Math., 2, 572-615, (1959) · Zbl 0091.10704  Privat, Y; Trélat, E; Zuazua, E, Complexity and regularity of maximal energy domains for the wave equation with fixed initial data, Discrete Contin. Dyn. Syst. Ser. A, 35, 6133-6153, (2015) · Zbl 1332.93069  Rakotoson, J. -M.: Réarrangement relatif: Un instrument d’estimations dans les problèmes aux limites. [An estimation tool for limit problems], vol 64. Math. Appl. (Berlin). Springer, Berlin (2008) · Zbl 1332.93069  Roques, L; Hamel, F, Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., 210, 34-59, (2007) · Zbl 1131.92068  Skellam, JG, Random dispersal in theoretical populations, Biometrika, 38, 196-218, (1951) · Zbl 0043.14401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.