346 research outputs found
On the placement of an obstacle so as to optimize the Dirichlet heat trace
We prove that among all doubly connected domains of bounded by two
spheres of given radii, , 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 obstaclefor any spectral property that can be obtained
as a positivity-preserving or positivity-reversing transform of
,including the spectral zeta function and, through it, the regularized
determinant.Comment: in SIAM Journal on Mathematical Analysis, Society for Industrial and
Applied Mathematics, 201
Spectral partitions for Sturm-Liouville problems
We look for best partitions of the unit interval that minimize certain
functionals defined in terms of the eigenvalues of Sturm-Liouville problems.
Via \Gamma-convergence theory, we study the asymptotic distribution of the
minimizers as the number of intervals of the partition tends to infinity. Then
we discuss several examples that fit in our framework, such as the sum of
(positive and negative) powers of the eigenvalues and an approximation of the
trace of the heat Sturm-Liouville operator
Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators
In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method.
In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle,
we obtain certain asymptotic estimates about the integral kernel of heat operators.
As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel.
Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means
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Low Eigenvalues of Laplace and Schrödinger Operators
This workshop brought together researchers interested in eigenvalue problems for Laplace and Schr¨dinger operators. The main topics o of discussions and investigations covered Dirichlet and Neumann eigenvalue problems, inequalities for the spectral gap, isoperimertic problems and sharp Lieb–Thirring type inequalities. The focus included not only the analytic and geometric sides of the problems, but also related probabilistic and computational aspects
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Advanced Computational Engineering
The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications
Proceedings of the FEniCS Conference 2017
Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg
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