Sobolev Spaces in Mathematics III

Applications in Mathematical Physics

herausgegeben von: Prof. Victor Isakov

Verlag: Springer New York

Buchreihe : International Mathematical Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Geometrization of Rings as a Method for Solving Inverse Problems

In the boundary value inverse problems on manifolds, it is required to recover a Riemannian manifold ʊ from its boundary inverse data (the elliptic or hyperbolic Dirichlet-to-Neumann map, spectral data, etc). We show that for a class of elliptic and hyperbolic problems the required manifold is identical with the spectrum of a certain algebra determined by the inverse data and, consequently, to recover the manifold it suffices to represent the corresponding algebra in the relevant canonical form.

Mikhail Belishev

The Ginzburg-Landau Equations for Superconductivity with Random Fluctuations

Thermal fluctuations and material inhomogeneities have a large effect on superconducting phenomena, possibly inducing transitions to the non-superconducting state. To gain a better understanding of these effects, the Ginzburg—Landau model is studied in situations for which the described physical processes are subject to uncertainty. An adequate description of such processes is possible with the help of stochastic partial differential equations. The boundary value problem of Neumann type for the stochastic Ginzburg—Landau equations with additive and multiplicative white noise is investigated. We use white noise with minimal restriction on its independence property. The existence and uniqueness of weak and strong statistical solutions are proved. Our approach is based on using difference schemes for the Ginzburg— Landau equations.

Andrei Fursikov, Max Gunzburger, Janet Peterson

Carleman Estimates with Second Large Parameter for Second Order Operators

We prove Carleman type estimates with two large parameters for general linear partial differential operators of second order. Using the second large parameter, from results for scalar equations we derive Carleman estimates for dynamical Lamé system with residual stress. These estimates are used to prove the Hölder and Lipschitz stability for the continuation of solutions under pseudoconvexity assumptions. So, the first uniqueness and stability of the continuation results are established for an important anisotropic system of elasticity without the assumption that this anisotropic system is close to an isotropic system.

Victor Isakov, Nanhee Kim

Sharp Spectral Asymptotics for Dirac Energy

I derive sharp semiclassical asymptotics of \(\int {|e_h } \left( {x,y,0} \right)|^2 \omega \left( {x,y} \right)dxdy\) where e _h (x, y, τ) is the Schwartz kernel of the spectral projector and ω(x, y) is singular as x = y. I also consider asymptotics of more general expressions.

Victor Ivrii

Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control → Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates

Abstrect

Abstract Uniform stabilization with nonlinear boundary feedback is asserted for classes of hyperbolic and Petrowski type multidimensional partial differential equations with variable coefficients (in space), as a consequence of the continuity (boundedness) of the corresponding purely Boundary Control → Boundary observation open-loop map of dissipative character, of interest in its own right. The interior is assumed inaccessible. There are explicit hyper-bolic/Petrowski type dynamical PDE classes where such a property holds and classes where it fails. When available, it has a number of attractive and unexpected consequences. In particular, when accompanied by exact controllability of the corresponding open-loop linear model, it implies uniform stabilization with optimal decay rates—when a nonlinear function of the boundary observation closes up the loop, to generate the corresponding boundary feedback dissipative problem.

Irena Lasiecka, Roberto Triggiani

Uniform Asymptotics of Green's Kernels for Mixed and Neumann Problems in Domains with Small Holes and Inclusions

Abstrect

Abstract Uniform asymptotic approximations of Green's kernels for the harmonic mixed and Neumann boundary value problems in domains with singularly perturbed boundaries are obtained. We consider domains with small holes (in particular, cracks) or inclusions. Formal asymptotic algorithms are supplied with rigorous estimates of the remainder terms.

Vladimir Maz'ya, Alexander Movchan

Finsler Structures and Wave Propagation

Abstrect

We discuss connections between the study of wave propagation for general classes of hyperbolic PDEs (beyond the “standard wave equation”) and aspects of Finsler geometry. In particular, we investigate how understanding of the behavior of differential operators (and pseudodifferential operators) arising in such study can enhance one's understanding of Finsler geometry. We also discuss a problem in harmonic analysis motivated by a construction of Katok in Finsler geometry, which gives rise to an interesting variant of the Pinsky phenomenon, for pointwise Fourier inversion.

Michael Taylor

Backmatter

Titel: Sobolev Spaces in Mathematics III
herausgegeben von: Prof. Victor Isakov
Verlag: Springer New York
Electronic ISBN: 978-0-387-85652-0
Print ISBN: 978-0-387-85651-3
DOI: https://doi.org/10.1007/978-0-387-85652-0

Springer Professional

Inhaltsverzeichnis

Frontmatter

Geometrization of Rings as a Method for Solving Inverse Problems

The Ginzburg-Landau Equations for Superconductivity with Random Fluctuations

Carleman Estimates with Second Large Parameter for Second Order Operators

Sharp Spectral Asymptotics for Dirac Energy

Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control → Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with Optimal Decay Rates

Uniform Asymptotics of Green's Kernels for Mixed and Neumann Problems in Domains with Small Holes and Inclusions

Finsler Structures and Wave Propagation

Backmatter

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