On the Placement of an Obstacle So As to Optimize the Dirichlet Heat Trace

We prove that among all domains of $\mathbb{R}^n$ bounded by two spheres of given radii, $Z(t)$, the trace of the heat kernel with Dirichlet boundary conditions, achieves its minimum when the spheres are concentric (i.e., for the spherical shell). The supremum is attained when the interior sphere is in contact with the outer sphere. This is shown to be a special case of a more general theorem characterizing the optimal placement of a spherical obstacle inside a convex domain so as to maximize or minimize the trace of the Dirichlet heat kernel. In this case the minimizing position of the center of the obstacle belongs to the “heart” of the domain, while the maximizing situation occurs either in the interior of the heart or at a point where the obstacle is in contact with the outer boundary. Similar statements hold for the optimal positions of the obstacle for any spectral property that can be obtained as a positivity-preserving or positivity-reversing transform of $Z(t)$, including the spectral zeta functi...

[1]  L. Hermi,et al.  On Riesz Means of Eigenvalues , 2007, 0712.4088.

[2]  Evans M. Harrell,et al.  On the Placement of an Obstacle or a Well so as to Optimize the Fundamental Eigenvalue , 2001, SIAM J. Math. Anal..

[3]  Critical metrics of the trace of the heat kernel on a compact manifold , 2002 .

[4]  Shing-Tung Yau,et al.  On the Schrödinger equation and the eigenvalue problem , 1983 .

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

[6]  D. Vassilevich,et al.  Heat kernel expansion: user's manual , 2003, hep-th/0306138.

[7]  On two functionals connected to the Laplacian in a class of doubly connected domains , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[8]  Ahmad El Soufi,et al.  Extremal First Dirichlet Eigenvalue of Doubly Connected Plane Domains and Dihedral Symmetry , 2007, SIAM J. Math. Anal..

[9]  F. Pacard.  The role of minimal surfaces in the study of the Allen-Cahn equation. , 2012 .

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

[11]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[12]  Peter B. Gilkey,et al.  The asymptotics of the Laplacian on a manifold with boundary , 1990 .

[13]  L. Brasco,et al.  The Heart of a Convex Body , 2012, 1202.5223.

[14]  Generalized isoperimetric inequalities , 1973 .

[15]  Terence Tao,et al.  Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions , 2002 .

[16]  UNIFORM BOUNDS FOR EIGENFUNCTIONS OF THE LAPLACIAN ON MANIFOLDS WITH BOUNDARY* , 2001, math/0103080.

[17]  R. Laugesen,et al.  Extremals for Eigenvalues of Laplacians under Conformal Mapping , 1998 .

[18]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[19]  Daniel Daners,et al.  Dirichlet problems on varying domains , 2003 .

[20]  L. Brasco,et al.  The location of the hot spot in a grounded convex conductor , 2010, 1012.4742.

[21]  R. Kiwan,et al.  Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue , 2007, 0712.2033.

[22]  Paul Garabedian,et al.  Lectures on partial differential equations , 1964 .

[23]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[24]  I. Singer,et al.  R-Torsion and the Laplacian on Riemannian manifolds , 1971 .

[25]  Peter Sarnak,et al.  Extremals of determinants of Laplacians , 1988 .

[26]  J. Hersch.  THE METHOD OF INTERIOR PARALLELS APPLIED TO POLYGONAL OR MULTIPLY CONNECTED MEMBRANES , 1963 .

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

[28]  Ahmad El Soufi,et al.  Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold , 2007, 0705.1263.