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Elmar Schrohe

Publications and source records attributed to Elmar Schrohe.

At least 37 records · Page 2Linked to original sources
arXiv2019

Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal $L^{q}$-regularity space for all times and is instantaneously smooth in space and time, where the maximal $L^{q}$-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

math.AP↗
arXiv2018

Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations

We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov's theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of Abelian groups.

math.OA↗
arXiv2017

Spectral Invariance of Pseudodifferential Boundary Value Problems on Manifolds with Conical Singularities

We prove the spectral invariance of the algebra of classical pseudodifferential boundary value problems on manifolds with conical singularities in the Lp-setting. As a consequence we also obtain the spectral invariance of the classical Boutet de Monvel algebra of zero order operators with parameters. In order to establish these results, we show the equivalence of Fredholm property and ellipticity for both cases.

math.AP↗
arXiv2017

Getzler rescaling via adiabatic deformation and a renormalized local index formula

We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion by deforming the Dirac operator into a polynomial coefficient operator over the rescaled bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold.

math.OA↗
arXiv2016

Elliptic Operators Associated with Groups of Quantized Canonical Transformations

Given a Lie group $G$ of quantized canonical transformations acting on the space $L^2(M)$ over a closed manifold $M$, we define an algebra of so-called $G$-operators on $L^2(M)$. We show that to $G$-operators we can associate symbols in appropriate crossed products with $G$, introduce a notion of ellipticity and prove the Fredholm property for elliptic elements. This framework encompasses many known elliptic theories, for instance, shift operators associated with group actions on $M$, transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier integral operators associated with coisotropic submanifolds.

math.OA↗
arXiv2015

Spectral flow of exterior Landau-Robin hamiltonians

We study the spectral flow of Landau-Robin hamiltonians in the exterior of a compact domain with smooth boundary. This provides a method to study the spectrum of the exterior Landau-Robin hamiltonian's dependence on the choice of Robin data, even explaining the heuristics of how the spectrum of the Robin problem asymptotically tends to the spectrum of the Dirichlet problem. The main technical result concerns the continuous dependence of Landau-Robin hamiltonians on the Robin data in the gap topology. The problem can be localized to the compact boundary where the asymptotic behavior of the spectral flow in some special cases can be described.

math.SP↗
arXiv2014

Fourier Integral Operators of Boutet de Monvel Type

Given two compact manifolds $X,Y,$ with boundary and a boundary preserving symplectomorphism $χ:T^*Y\setminus0\to T^*X\setminus0$, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with $χ$. We study their mapping properties between Sobolev spaces, develop a calculus and prove a Egorov type theorem. We also introduce a notion of ellipticity which implies the Fredholm property. Finally, we show how -- in the spirit of a classical construction by A. Weinstein -- a Fredholm operator of this type can be associated with $χ$ and a section of the Maslov bundle. If $\dim Y>2$ or the Maslov bundle is trivial, the index is independent of the section and thus an invariant of the symplectomorphism.

math.FA↗
arXiv2014

On a Class of Fourier Integral Operators on Manifolds with Boundary

We study a class of Fourier integral operators on compact manifolds with boundary, associated with a natural class of symplectomorphisms, namely, those which preserve the boundary. A calculus of Boutet de Monvel's type can be defined for such Fourier integral operators, and appropriate continuity properties established. One of the key features of this calculus is that the local representations of these operators are given by operator-valued symbols acting on Schwartz functions or temperate distributions. Here we focus on properties of the corresponding local phase functions, which allow to prove this result in a rather straightforward way.

math.OA↗
arXiv2014

Bounded imaginary powers of cone differential operators on higher order Mellin-Sobolev spaces and applications to the Cahn-Hilliard equation

Extending earlier results on the existence of bounded imaginary powers for cone differential operators on weighted $L^p$-spaces $\mathcal{H}^{0,γ}_p(\mathbb{B})$ over a manifold with conical singularities, we show how the same assumptions also yield the existence of bounded imaginary powers on higher order Mellin-Sobolev spaces $\mathcal{H}^{s,γ}_p(\mathbb{B})$, $s\geq0$. As an application we then consider the Cahn-Hilliard equation on a manifold with (possibly warped) conical singularities. Relying on our work for the case of straight cones, we first establish $R$-sectoriality (and thus maximal regularity) for the linearized equation and then deduce the existence of a short time solution with the help of a theorem by Clément and Li. We also obtain the short time asymptotics of the solution near the conical point.

math.AP↗
arXiv2013

Determinants of Classical SG-Pseudodifferential Operators

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG-pseudodifferential operators on R^n and suitable manifolds, using a finite-part integral regularization technique. This allows us to define a zeta-regularized determinant det A for classical parameter-elliptic SG-operators A of order (μ,m), with μ>0, m\ge0. For m=0, the asymptotics of TR exp(-tA) as t\to 0 and of TR (λ-A)^{-k}$ as |λ|\to\infty are derived. For suitable pairs (A,A_0) we show that det A/det A_0 coincides with the so-called relative determinant det(A,A_0).

math.AP↗
arXiv2013

The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

We consider the Cahn-Hilliard equation on a manifold with conical singularities and show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clement and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.

math.AP↗
arXiv2013

Families index for Boutet de Monvel operators

We define the analytical and the topological indices for continuous families of operators in the C*-closure of the Boutet de Monvel algebra. Using techniques of C*-algebra K-theory and the Atiyah-Singer theorem for families of elliptic operators on a closed manifold, we prove that these two indices coincide.

math.KT↗
arXiv2012

C*-Algebra approach to the index theory of boundary value problems

Boutet de Monvel's calculus provides a pseudodifferential framework which encompasses the classical differential boundary value problems. In an extension of the concept of Lopatinski and Shapiro, it associates to each operator two symbols: a pseudodifferential principal symbol, which is a bundle homomorphism, and an operator-valued boundary symbol. Ellipticity requires the invertibility of both. If the underlying manifold is compact, elliptic elements define Fredholm operators. Boutet de Monvel showed how then the index can be computed in topological terms. The crucial observation is that elliptic operators can be mapped to compactly supported $K$-theory classes on the cotangent bundle over the interior of the manifold. The Atiyah-Singer topological index map, applied to this class, then furnishes the index of the operator. Based on this result, Fedosov, Rempel-Schulze and Grubb have given index formulas in terms of the symbols. In this paper we survey previous work how C*-algebra K-theory can be used to give a proof of Boutet de Monvel's index theorem for boundary value problems, and how the same techniques yield an index theorem for families of Boutet de Monvel operators. The key ingredient of our approach is a precise description of the K-theory of the kernel and of the image of the boundary symbol.

math.KT↗
arXiv2011

Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.

math.AP↗
arXiv2011

Spectral triples and the geometry of fractals

For each K-homolgy element of the Sierpinski gasket we construct a spectral triple which will generate that element. We show that there must be certain limits on the choice of the K-homology element if the geometric properties of the gasket shall be recoverable from that spectral triple. For a big subgroup of the K-homology group we show that our spectral triples will recover the metric, the dimension and the Hausdorff measure on the gaket.

math.OA↗
arXiv2010

FE-BE coupling for a transmission problem involving microstructure

We analyze a finite element/boundary element procedure to solve a non-convex contact problem for the double-well potential. After relaxing the associated functional, the degenerate minimization problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which may then be solved numerically. The convergence of the Galerkin approximations to certain macroscopic quantities and a corresponding a posteriori estimate for the approximation error are discussed.

math.NA↗
arXiv2009

Adaptive FE-BE Coupling for Strongly Nonlinear Transmission Problems with Coulomb Friction

We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the p-Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of L^p- and L^2-Sobolev spaces.

math.NA↗