Spectral optimization problems

In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered.

[1]  Dorin Bucur,et al.  N-Dimensional Shape Optimization under Capacitary Constraint , 1995 .

[2]  F. Brock.  Continuous Steiner‐Symmetrization , 1995 .

[3]  W. Ziemer.  Weakly differentiable functions , 1989 .

[4]  G. Szegő,et al.  Inequalities for Certain Eigenvalues of a Membrane of Given Area , 1954 .

[5]  Dorin Bucur,et al.  An alternative approach to the Faber–Krahn inequality for Robin problems , 2009 .

[6]  Tanguy Briançon.  Regularity of optimal shapes for the Dirichlet's energy with volume constraint , 2004 .

[7]  V. Komkov.  Optimal shape design for elliptic systems , 1986 .

[8]  George Polya,et al.  On the characteristic frequencies of a symmetric membrane , 1955 .

[9]  M. Kohler-Jobin.  Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique II. Seconde partie: cas inhomogène: une inégalité isopérimétrique entre la fréquence fondamentale d'une membrane et l'énergie d'équilibre d'un problème de Poisson , 1978 .

[10]  G. Buttazzo,et al.  Shape optimization for Dirichlet problems: Relaxed solutions and optimality conditions , 1990 .

[11]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[12]  D. Bucur,et al.  Shape optimisation problems governed by nonlinear state equations , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  G. Buttazzo,et al.  On the characterization of the compact embedding of Sobolev spaces , 2009 .

[14]  L. A. Cafferelli,et al.  An Optimal Partition Problem for Eigenvalues , 2007, J. Sci. Comput..

[15]  Marie-Hélène Bossel.  Membranes élastiquement liées: extension du théorème de Rayleigh-Faber-Krahn et de l'inégalité de Cheeger , 1986 .

[16]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[17]  R. Benguria,et al.  A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions , 1992 .

[18]  Michel Pierre,et al.  Lipschitz continuity of state functions in some optimal shaping , 2005 .

[19]  F. Almgren.  Review: Enrico Giusti, Minimal surfaces and functions of bounded variation , 1987 .

[20]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

[21]  D. Bucur,et al.  A Variational Approach to the Isoperimetric Inequality for the Robin Eigenvalue Problem , 2010 .

[22]  José M. Arrieta,et al.  Neumann Eigenvalue Problems on Exterior Perturbations of the Domain , 1995 .

[23]  H. Attouch.  Variational convergence for functions and operators , 1984 .

[24]  Giuseppe Buttazzo,et al.  An existence result for a class of shape optimization problems , 1993 .

[25]  Isabel N. Figueiredo,et al.  On the attainable eigenvalues of the Laplace operator , 1999 .

[26]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[27]  Qatu,et al.  Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, Vol 146 , 2003 .

[28]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[29]  Daniel Daners,et al.  A Faber-Krahn inequality for Robin problems in any space dimension , 2006 .

[30]  Dorin Bucur,et al.  Variational Methods in Shape Optimization Problems , 2005, Progress in Nonlinear Differential Equations and Their Applications.

[31]  Giuseppe Buttazzo,et al.  Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization (Mps-Siam Series on Optimization 6) , 2005 .

[32]  On the motion of rigid bodies in a viscous incompressible fluid , 2003 .

[33]  J. Keller,et al.  Range of the first two eigenvalues of the laplacian , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[34]  Antoine Henrot,et al.  Le stade ne minimise pas λ2 parmi les ouverts convexes du plan , 2001 .

[35]  Gianni Dal Maso.  $\Gamma $-convergence and $\mu $-capacities , 1987 .

[36]  Edouard Oudet,et al.  Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions , 2003 .

[37]  G. Buttazzo,et al.  On some rescaled shape optimization problems , 2009, 0911.4561.

[38]  Gianni Dal Maso,et al.  Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators , 1997 .

[39]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[40]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[41]  Dorin Bucur,et al.  Shape optimization problems for eigenvalues of elliptic operators , 2006 .

[42]  Dorin Bucur,et al.  Minimization of the third eigenvalue of the Dirichlet Laplacian , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[43]  Marie-Hélène Bossel.  Membranes élastiquement liées inhomogènes ou sur une surface: Une nouvelle extension du théorème isopérimétrique de Rayleigh-Faber-Krahn , 1988 .

[44]  Mark S. Ashbaugh,et al.  Open Problems on Eigenvalues of the Laplacian , 1999 .

[45]  Edouard Oudet,et al.  Numerical minimization of eigenmodes of a membrane with respect to the domain , 2004 .

[46]  Dorin Bucur,et al.  Optimal Partitions for Eigenvalues , 2009, SIAM J. Sci. Comput..

[47]  A. Chambolle,et al.  Uniqueness of the Cheeger set of a convex body , 2007, Pacific Journal of Mathematics.

[48]  G. Pólya,et al.  ON THE RATIO OF CONSECUTIVE EIGENVALUES , 1956 .

[49]  Giuseppe Buttazzo,et al.  Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations , 1989 .

[50]  L. Caffarelli,et al.  An area‐Dirichlet integral minimization problem , 2001 .

[51]  G. Buttazzo,et al.  Minimization of $\lambda_2(\Omega)$ with a perimeter constraint , 2009, 0904.2193.

[52]  Susanna Terracini,et al.  An optimal partition problem related to nonlinear eigenvalues , 2003 .

[53]  A. Henrot.  Minimization problems for eigenvalues of the Laplacian , 2003 .

[54]  Andrea Braides.  Γ-convergence for beginners , 2002 .

[55]  Antoine Henrot,et al.  Variation et optimisation de formes : une analyse géométrique , 2005 .

[56]  G. Buttazzo,et al.  An optimal design problem with perimeter penalization , 1993 .

[57]  V. Sverák,et al.  On optimal shape design , 1992 .

[58]  E. Krahn,et al.  Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises , 1925 .

[59]  Lucia De Luca,et al.  Metodi diretti nel Calcolo delle Variazioni , 2010 .

[60]  Par Marie-Hélèe Bossel.  Elastically supported membranes inhomogeneous or on a surface: a new extension of the isoperimetric Rayleigh-Faber-Krahn , 1988 .

[61]  Antoine Henrot,et al.  Variation et optimisation de formes , 2005 .

[62]  J. Zolésio,et al.  Introduction to shape optimization : shape sensitivity analysis , 1992 .

[63]  Elliptical membranes with smallest second eigenvalue , 1973 .

[64]  Marie-Thérèse Kohler-Jobin.  Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique I. Première partie: une démonstration de la conjecture isopérimétrique Pλ2 ≥ πj04/2 de Pólya et Szegö , 1978 .

[65]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[67]  Antoine Henrot,et al.  Extremum Problems for Eigenvalues of Elliptic Operators , 2006 .

[68]  P. Garabedian,et al.  Variational Problems in the Theory of Elliptic Partial Differential Equations , 1953 .

[69]  J. Heinonen,et al.  Nonlinear Potential Theory of Degenerate Elliptic Equations , 1993 .

[70]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[71]  L. Hedberg,et al.  Function Spaces and Potential Theory , 1995 .

[72]  Giuseppe Buttazzo,et al.  Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions , 1991 .

[73]  H. Weinberger,et al.  An optimal Poincaré inequality for convex domains , 1960 .

[74]  Hans F. Weinberger,et al.  An Isoperimetric Inequality for the N-Dimensional Free Membrane Problem , 1956 .

[75]  M. Brelot.  Classical potential theory and its probabilistic counterpart , 1986 .

[76]  Marie-Thérèse Kohler-Jobin.  Méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique , 1977 .

[77]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[78]  Gianni Dal Maso,et al.  Wiener's criterion and Γ-convergence , 1987 .

[79]  B. Fuglede.  Finely Harmonic Functions , 1972 .

[80]  Steven J. Cox,et al.  Where Best to Hold a Drum Fast , 2003, SIAM Rev..

[81]  Ronald F. Gariepy.  FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs) , 2001 .

[82]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[83]  Jimmy Lamboley,et al.  Regularity of the optimal shape for the first eigenvalue of the laplacian with volume and inclusion constraints , 2008, 0807.2196.

[84]  G. Buttazzo,et al.  Shape flows for spectral optimization problems , 2011, 1109.5243.

[85]  G. Buttazzo,et al.  Quasistatic Evolution in Debonding Problems via Capacitary Methods , 2008 .

[86]  Jan Sokolowski,et al.  Introduction to shape optimization , 1992 .

[87]  F. Brock.  Continuous rearrangement and symmetry of solutions of elliptic problems , 2000 .