Hadamard’s Formula Inside and Out

Our goal is to explore boundary variations of spectral problems from the calculus of moving surfaces point of view. Hadamard’s famous formula for simple Laplace eigenvalues under Dirichlet boundary conditions is generalized in a number of significant ways, including Neumann and mixed boundary conditions, multiple eigenvalues, and second order variations. Some of these formulas appear for the first time here. Furthermore, we present an analytical framework for deriving general formulas of the Hadamard type.The presented analysis finds direct applications in shape optimization and other variational problems. As a specific application, we discuss equilibrium and stable shapes of electron bubbles.

[1]  Paul Garabedian,et al.  Partial Differential Equations , 1964 .

[2]  Vladimir Kozlov,et al.  On the Hadamard formula for nonsmooth domains , 2006 .

[3]  L. Rayleigh,et al.  The theory of sound , 1894 .

[4]  J. Wisdom,et al.  A way to compute the gravitational potential for near-spherical geometries , 2006 .

[5]  E. B. Christoffel.  Untersuchungen über die mit dem Fortbestehen linearer partieller Differentialgleichungen verträglichen Unstetigkeiten , 1877 .

[6]  J. Dalton.  LXVI. Experimental inquiry into the proportion of the several gases or elastic fluids constituting the atmosphere , 1806 .

[7]  Tullio Levi-Civita,et al.  The absolute differential calculus (calculus of tensors) , 1927 .

[8]  M. Schiffer.  Variation of domain functionals , 1954 .

[9]  M. Grinfel'd.  Continuum methods in the theory of phase transitions in solids , 1988 .

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

[11]  M. Bareket.  On an Isoperimetric Inequality for the First Eigenvalue of a Boundary Value Problem , 1977 .

[12]  M. Lighthill.  Dynamics of a Dissociating Gas , 1959, The Journal of the Royal Aeronautical Society.

[13]  C. Truesdell,et al.  The Classical Field Theories , 1960 .

[14]  Instability of the 2S electron bubbles. , 2003, Physical review letters.

[15]  Menahem Schiffer,et al.  Convexity of domain functionals , 1952 .

[16]  G. Allaire,et al.  A level-set method for shape optimization , 2002 .

[17]  G. G. Stokes.  On a Difficulty in the Theory of Sound , 1848 .

[18]  P. Grinfeld.  Hamiltonian Dynamic Equations for Fluid Films , 2010 .

[19]  J. Hadamard,et al.  Leçons sur la propagation des ondes et les equations de l'hydrodynamique , 2015 .

[20]  R. M. Bowen,et al.  On displacement derivatives , 1971 .

[21]  Shape Optimization and Electron Bubbles , 2009 .

[22]  Lehel Banjai,et al.  Eigenfrequencies of fractal drums , 2007 .

[23]  Richard Bellman,et al.  Plastic Flow and Fracture in Solids , 1962 .

[24]  H. Maris,et al.  Negative Pressures and Cavitation in Liquid Helium , 2000 .

[25]  C. Truesdell.  On Curved Shocks in Steady Plane Flow of an Ideal Fluid , 1952 .

[26]  Konstantin A. Lurie,et al.  Applied Optimal Control Theory of Distributed Systems , 1993 .

[27]  M. J. Grinfel'd,et al.  Thermodynamic methods in the theory of heterogeneous systems , 1991 .

[28]  W. Hayes.  The vorticity jump across a gasdynamic discontinuity , 1957, Journal of Fluid Mechanics.

[29]  S. Osher,et al.  Level Set Methods for Optimization Problems Involving Geometry and Constraints I. Frequencies of a T , 2001 .

[30]  M. J. Lighthill,et al.  Dynamics of a dissociating gas Part I Equilibrium flow , 1957, Journal of Fluid Mechanics.

[31]  Gilbert Strang,et al.  The Laplacian eigenvalues of a polygon , 2004 .

[32]  Menahem Schiffer,et al.  Hadamard's Formula and Variation of Domain-Functions , 1946 .

[33]  T. Thomas.  Extended Compatibility Conditions for the Study of Surfaces of Discontinuity in Continuum Mechanics , 1957 .

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