Stability in shape optimization with second variation

We are interested in the question of stability in the field of shape optimization, with focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the hessian of the considered shape function, so that critical stable domains (i.e. such that the first order derivative vanishes and the second order one is positive) are local minima for smooth perturbations; as we are in an infinite dimensional framework, and that in most applications there is a norm-discrepancy phenomenon, this type of result require a lot of work. We show that these hypotheses are satisfied by classical functionals, involving the perimeter, the Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how we can easily deal with constraints and/or invariance of the functionals. As an application, we retrieve or improve previous results from the existing literature, and provide new local stability results. We finally test the sharpness of our results by showing that the local minimality is in general not valid for non-smooth perturbations.

[1]  N. Fusco,et al.  Communications in Mathematical Physics Minimality via Second Variation for a Nonlocal Isoperimetric Problem , 2013 .

[2]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[3]  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 .

[4]  M. Abramowitz,et al.  Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables , 1966 .

[5]  STABLE CONSTANT MEAN CURVATURE HYPERSURFACES ARE AREA MINIMIZING IN SMALL L 1 NEIGHBORHOODS , 2008, 0811.3126.

[6]  B. Fuglede.  Stability in the isoperimetric problem for convex or nearly spherical domains in ⁿ , 1989 .

[7]  G. P. Leonardi,et al.  A Selection Principle for the Sharp Quantitative Isoperimetric Inequality , 2010, Archive for Rational Mechanics and Analysis.

[8]  F. Caubet,et al.  Stability of critical shapes for the drag minimization problem in Stokes flow , 2013 .

[9]  F. Maggi,et al.  Sharp stability inequalities for the Plateau problem , 2014 .

[10]  C. Nitsch.  An isoperimetric result for the fundamental frequency via domain derivative , 2012, 1201.5328.

[11]  G. Philippis,et al.  Faber–Krahn inequalities in sharp quantitative form , 2013, 1306.0392.

[12]  M. Novaga,et al.  Existence and Stability for a Non-Local Isoperimetric Model of Charged Liquid Drops , 2013, Archive for Rational Mechanics and Analysis.

[13]  A sharp quantitative isoperimetric inequality in higher codimension , 2015 .

[14]  Fabien Caubet,et al.  DETECTING AN OBSTACLE IMMERSED IN A FLUID BY SHAPE OPTIMIZATION METHODS , 2011 .

[15]  Karsten Große-Brauckmann,et al.  Stable constant mean curvature surfaces minimize area , 1996 .

[16]  N. Fusco,et al.  A quantitative form of Faber–Krahn inequality , 2017 .

[17]  Robin Neumayer,et al.  A Strong Form of the Quantitative Wulff Inequality , 2015, SIAM J. Math. Anal..

[18]  A. Ioffe.  Necessary and Sufficient Conditions for a Local Minimum. 3: Second Order Conditions and Augmented Duality , 1979 .

[19]  H. Weinberger,et al.  Some isoperimetric inequalities for membrane frequencies and torsional rigidity , 1961 .

[20]  G. Pólya,et al.  Isoperimetric inequalities in mathematical physics , 1951 .

[21]  Marc Dambrine,et al.  On variations of the shape Hessian and sufficient conditions for the stability of critical shapes. , 2002 .

[22]  J. Simon,et al.  Second variations for domain optimization problems , 1989 .

[23]  Michel Dambrine,et al.  On Second Order Shape Optimization Methods for Electrical Impedance Tomography , 2007, SIAM J. Control. Optim..

[24]  A. Figalli,et al.  A mass transportation approach to quantitative isoperimetric inequalities , 2010 .

[25]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[26]  M. C. Delfour,et al.  Shapes and Geometries - Metrics, Analysis, Differential Calculus, and Optimization, Second Edition , 2011, Advances in design and control.

[27]  Alessio Figalli,et al.  Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies , 2014, Communications in Mathematical Physics.

[28]  N. Fusco,et al.  The sharp quantitative isoperimetric inequality , 2008 .

[29]  Matthew MacDonald,et al.  Shapes and Geometries , 1987 .

[30]  N. Fusco,et al.  Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities , 2009 .

[31]  M. Dambrine,et al.  ON THE ERSATZ MATERIAL APPROXIMATION IN LEVEL-SET METHODS , 2010 .

[32]  Peter Sternberg,et al.  On the first and second variations of a nonlocal isoperimetric problem , 2007 .

[33]  Reinhold Böhme,et al.  Zur Struktur der Lösungsmenge des Plateauproblems , 1973 .

[34]  Arian Novruzi,et al.  Structure of shape derivatives , 2002 .

[35]  L. W.,et al.  The Theory of Sound , 1898, Nature.

[36]  N. Fusco,et al.  A strong form of the quantitative isoperimetric inequality , 2011, 1111.4866.

[37]  Marc Dambrine,et al.  About stability of equilibrium shapes , 2000 .

[38]  Jimmy Lamboley,et al.  Structure of shape derivatives around irregular domains and applications , 2006 .

[39]  E. Casas,et al.  Second Order Optimality Conditions and Their Role in PDE Control , 2015 .

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

[41]  B. White.  A strong minimax property of nondegenerate minimal submanifolds. , 1994 .

[42]  Winfried Sickel,et al.  Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations , 1996, de Gruyter series in nonlinear analysis and applications.

[43]  Jack K. Hale,et al.  Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations , 2005 .