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Green formulae for elliptic cone differential operators are established. This is achieved by an accurate description of the maximal domain of an elliptic cone differential operator and its formal adjoint; thereby utilizing the concept of a discrete asymptotic type. From this description, the singular coefficients replacing the boundary traces in classical Green formulas are deduced.
We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator-valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator As in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of As as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices Ps of As, interpreted as Mellin-edge representations of P.
Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge Sobolev spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. Our methods from the calculus of boundary value problems on a manifold with edges will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator.
We investigate crack problems, where the crack boundary has conical singularities. Elliptic operators with two-sided elliptc boundary conditions on the plus and minus sides of the crack will be interpreted as elements of a corner algebra of boundary value problems. The corresponding operators will be completed by extra edge conditions on the crack boundary to Fredholm operators in corner Sobolev spaces with double weights, and there are parametrices within the calculus.
Contents: Chapter 3: Eta Invariant and the Spectral Flow 3.1. Introduction 3.2. The Classical Spectral Flow 3.2.1. Definition and main properties 3.2.2. The spectral flow formula for periodic families 3.3. The Atiyah–Patodi–Singer Eta Invariant 3.3.1. Definition of the eta invariant 3.3.2. Variation under deformations of the operator 3.3.3. Homotopy invariance. Examples 3.4. The Eta Invariant of Families with Parameter (Melrose’s Theory) 3.4.1. A trace on the algebra of parameter-dependent operators 3.4.2. Definition of the Melrose eta invariant 3.4.3. Relationship with the Atiyah–Patodi–Singer eta invariant 3.4.4. Locality of the derivative of the eta invariant. Examples 3.5. The Spectral Flow of Families of Parameter-Dependent Operators 3.5.1. Meromorphic operator functions. Multiplicities of singular points 3.5.2. Definition of the spectral flow 3.6. Higher Spectral Flows 3.6.1. Spectral sections 3.6.2. Spectral flow of homotopies of families of self-adjoint operators 3.6.3. Spectral flow of homotopies of families of parameter-dependent operators 3.7. Bibliographical Remarks
Contents: Chapter 4: Pseudodifferential Operators 4.1. Preliminary Remarks 4.1.1. Why are pseudodifferential operators needed? 4.1.2. What is a pseudodifferential operator? 4.1.3. What properties should the pseudodifferential calculus possess? 4.2. Classical Pseudodifferential Operators on Smooth Manifolds 4.2.1. Definition of pseudodifferential operators on a manifold 4.2.2. Hörmander’s definition of pseudodifferential operators 4.2.3. Basic properties of pseudodifferential operators 4.3. Pseudodifferential Operators in Sections of Hilbert Bundles 4.3.1. Hilbert bundles 4.3.2. Operator-valued symbols. Specific features of the infinite-dimensional case 4.3.3. Symbols of compact fiber variation 4.3.4. Definition of pseudodifferential operators 4.3.5. The composition theorem 4.3.6. Ellipticity 4.3.7. The finiteness theorem 4.4. The Index Theorem 4.4.1. The Atiyah–Singer index theorem 4.4.2. The index theorem for pseudodifferential operators in sections of Hilbert bundles 4.4.3. Proof of the index theorem 4.5. Bibliographical Remarks
In this paper, by a new constructive method, the authors reprove the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate fields. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C¹ functions. In particular, the authors give the optimal control time of the system. Finally, a new application is also given.
Contents: Chapter 1: Localization (Surgery) in Elliptic Theory 1.1. The Index Locality Principle 1.1.1. What is locality? 1.1.2. A pilot example 1.1.3. Collar spaces 1.1.4. Elliptic operators 1.1.5. Surgery and the relative index theorem 1.2. Surgery in Index Theory on Smooth Manifolds 1.2.1. The Booß–Wojciechowski theorem 1.2.2. The Gromov–Lawson theorem 1.3. Surgery for Boundary Value Problems 1.3.1. Notation 1.3.2. General boundary value problems 1.3.3. A model boundary value problem on a cylinder 1.3.4. The Agranovich–Dynin theorem 1.3.5. The Agranovich theorem 1.3.6. Bojarski’s theorem and its generalizations 1.4. (Micro)localization in Lefschetz theory 1.4.1. The Lefschetz number 1.4.2. Localization and the contributions of singular points 1.4.3. The semiclassical method and microlocalization 1.4.4. The classical Atiyah–Bott–Lefschetz theorem
Boundary value problems on manifolds with conical singularities or edges contain potential operators as well as trace and Green operators which play a similar role as the corresponding operators in (pseudo-differential) boundary value problems on a smooth manifold. There is then a specific asymptotic behaviour of these operators close to the singularities. We characterise potential operators in terms of actions of cone or edge pseudo-differential operators (in the neighbouring space) on densities supported by sbmanifolds which also have conical or edge singularities. As a byproduct we show the continuity of such potentials as continuous perators between cone or edge Sobolev spaces and subspaces with asymptotics.
The quantum cosmological wavefunction for a quadratic gravity theory derived from the heterotic string effective action is obtained near the inflationary epoch and during the initial Planck era. Neglecting derivatives with respect to the scalar field, the wavefunction would satisfy a third-order differential equation near the inflationary epoch which has a solution that is singular in the scale factor limit a(t) → 0. When scalar field derivatives are included, a sixth-order differential equation is obtained for the wavefunction and the solution by Mellin transform is regular in the a → 0 limit. It follows that inclusion of the scalar field in the quadratic gravity action is necessary for consistency of the quantum cosmology of the theory at very early times.
Toeplitz operators, and ellipticity of boundary value problems with global projection conditions
(2003)
Ellipticity of (pseudo-) differential operators A on a compact manifold X with boundary (or with edges) Y is connected with boundary (or edge) conditions of trace and potential type, formulated in terms of global projections on Y together with an additional symbolic structure. This gives rise to operator block matrices A with A in the upper left corner. We study an algebra of such operators, where ellipticity is equivalent to the Fredhom property in suitable scales of spaces: Sobolev spaces on X plus closed subspaces of Sobolev spaces on Y which are the range of corresponding pseudo-differential projections. Moreover, we express parametrices of elliptic elements within our algebra and discuss spectral boundary value problems for differential operators.
For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.
We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernal of this domain. The Lefschetz number is proved to be the sum of usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points.
On the existence of a non-zero lower bound for the number of Goldbach partitions of an even integer
(2002)
The Goldbach partitions of an even number greater than 2, given by the sums of two prime addends, form the non-empty set for all integers 2n with 2 ≤ n ≤ 2 × 1014. It will be shown how to determine by the method of induction the existence of a non-zero lower bound for the number of Goldbach partitions of all even integers greater than or equal to 4. The proof depends on contour arguments for complex functions in the unit disk.
The derivation of the standard model from a higher-dimensional action suggests a further study of the fibre bundle formulation of gauge theories to determine the variations in the choice of structure group that are allowed in this geometrical setting. The action of transformations on the projection of fibres to their submanifolds are characteristic of theories with fewer gauge vector bosons, and specific examples are given, which may have phenomenological relevance. The spinor space for the three generations of fermions in the standard model is described algebraically.
The genus-dependence of multi-loop superstring ams is estimated at large orders in perturbation theory using the super-Schottky group parameterization of supermoduli space. Restriction of the integration region to a subset of supermoduli space and a single fundamental domain of the super-modular group suggests an exponential dependence on the genus. Upper bounds for these estimates are obtained for arbitrary N-point superstring scattering amplitudes and are shown to be consistent with exact results obtained for special type II string amplitudes for orbifold or Calabi-Yau compactifications. The genus-dependence is then obtained by considering the effect of the remaining contribution to the superstring amplitudes after the coefficients of the formally divergent parts of the integrals vanish as a result of a sum over spin structures. The introduction of supersymmetry therefore leads to the elimination of large-order divergences in string pertubation theory, a result which is based only on the supersymmetric generalization of the polyakov measure and not the gauge group of the string model.
We study pseudodifferential operators on a cylinder IR x B with cross section B that conical singularities. Configurations of that kind are the local model of cornere singularities with base spaces B. Operators A in our calculus are assumed to have symbols α which are meromorphic in the complex covariable with values in the space of all cone operators on B. In case α is dependent of the axial variable t ∈ IR, we show an explicit formula for solutions of the homogeneous equation. Each non-bjectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula.
Elliptic equations on configurations W = W1 ∪ ... ∪ Wn with edge Y and components Wj of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here, edges. Starting from edge-degenerate operators on Wj, j = 1, ..., N, we construct an algebra with extra "transmission" conditions on Y that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator-valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on W\Y. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics.
As is known from Kondratyev's work, boundary value problems for elliptic operators on a manifold with conical singularities and boundary are controlled by a principal symbolic hierarchy, where the conormal symbols belong to the typical new components, compared with the smooth case, with interior and boundary symbols. A similar picture may be expected on manifolds with corners when the base of the cone itself is a manifold with conical or edge singularities. This is a natural situation in a number of applications, though with essential new difficulties. We investigate here corresponding conormal symbols in terms of a calculus of holomorphic parameter-dependent edge boundary value problems on the base. We show that a certain kernel cut-off procedure generates all such holomorphic families, modulo smoothing elements, and we establish conormal symbols as an algebra as is necessary for a parametrix constructions in the elliptic case.
Relative elliptic theory
(2002)
This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms.
When studyind elliptic operators on manifolds with nonisolated singularities one naturally encounters families of conormal symbols (i.e. operators elliptic with parameter p ∈ IR in the sense of Agranovich-Vishik) parametrized by the set of singular points. For homotopies of such families we define the notion of spectral flow, which in this case is an element of the K-group of the parameter space. We prove that the spectral flow is equal to the index of some family of operators on the infinite cone.
We study an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted Lp-space. Under suitable conditions we show that the resolvent (λ - A )-¹ exists in a sector of the complex plane and decays like 1/|λ| as |λ| -> ∞. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with striaght conical degeneracy and establish maximal regularity for the Cauchy problem u - Δu = f, u(0) = 0.
Operators on manifolds with corners that have base configurations with geometric singularities can be analysed in the frame of a conormal symbolic structure which is in spirit similar to the one for conical singularities of Kondrat'ev's work. Solvability of elliptic equations and asymptotics of solutions are determined by meromorphic conormal symbols. We study the case when the base has edge singularities which is a natural assumption in a number of applications. There are new phenomena, caused by a specific kind of higher degeneracy of the underlying symbols. We introduce an algebra of meromorphic edge operators that depend on complex parameters and investigate meromorphic inverses in the parameter-dependent elliptic case. Among the examples are resolvents of elliptic differential operators on manifolds with edges.
The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f : M -> M in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschtz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah-Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschtz formula on a compact complex manifold with boundary
Local asymptotic types
(2002)
We construct algebras of Volterra pseudodifferential operators that contain, in particular, the inverses of the most natural classical systems of parabolic boundary value problems of general form. Parabolicity is determined by the invertibility of the principal symbols, and as a result is equivalent to the invertibility of the operators within the calculus. Existence, uniqueness, regularity, and asymptotics of solutions as t → ∞ are consquences of the mapping properties of the operators in exponentially weighted Sobolev spaces and subspaces with asymptotics. An important aspect of this work is that the microlocal and global kernel structure of the inverse operator (solution operator) of a parabolic boundary value problem for large times is clarified. Moreover, our approach naturally yields qualitative pertubation results for the solvability theory of parabolic boundary value problems. To achieve these results, we assign t = ∞ the meaning of a conical point and treat the operators as totally characteristic pseudodifferential boundary value problems.
We prove a theorem describing the behaviour of the relative index of families of Fredholm operators under surgery performed on spaces where the operators act. In connection with additional conditions (like symmetry conditions) this theorem results in index formulas for given operator families. By way of an example, we give an application to index theory of families of boundary value problems.
Contents: 1 Introduction. Denfitions and Discussions 2 Solvability of the Cauchy Type Functional Equations 2.1 The Case of a P-configuration 2.2 The Case of a Z-configuration 2.3 Multiplicative Cauchy type functional equations 3 Problems in Analysis Reducing to Cauchy Type Functional Equations 3.1 Some problems in Integral Geometry and Cauchy Functional Equations 3.2 First Boundary Problem for Hyperbolic Differential Equations and Cauchy Type Functional Equations 4 Functional Equations Determining Polynomials
Anisotropic edge problems
(2002)
We investigate elliptic pseudodifferential operators which degenerate in an anisotropic way on a submanifold of arbitrary codimension. To find Fredholm problems for such operators we adjoint to them boundary and coboundary conditions on the submanifold.The algebra obtained this way is a far reaching generalisation of Boutet de Monvel's algebra of boundary value problems with transmission property. We construct left and right regularisers and prove theorems on hypoellipticity and local solvability.
This note is devoted to the study on the global existence of a shock wave for the supersonic flow past a curved wedge. When the curved wedge is a small perturbation of a straight wedge and the angle of the wedge is less than some critical value, wwe show that a shock attached at the wedge will exist globally.
In this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfies certain nondegenerate condition, then from the results in [20], we know that there exists a unique blowup point at the blowup time such that the first order derivates of smooth solution blow up meanwhile the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of shock curve, moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in the neighbourhood of the blowup point.
Contents: Introduction 1 Operators with the transmission property 1.1 Operators on a manifold with boundary 1.2 Conditions with pseudodifferential projections 1.3 Projections and Fredholm families 2 Boundary value problems not requiring the transmission property 2.1 Interior operators 2.2 Edge amplitude functions 2.3 Boundary value problems 3 Operators with global projection conditions 3.1 Construction for boundary symbols 3.2 Ellipticity of boundary value problems with projection data 3.3 Operators of order zero
Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We consider smooth symbols and ellipticity without additional boundary conditions which is the relevant case on a manifold with boundary. Starting from a class of symbols that has been investigated before for integer orders in boundary value problems with the transmission property we study operators of arbitrary real orders that play a similar role for operators without the transmission property. Moreover, we show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We finally investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary.
In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].
We introduce the Volterra calculus of pseudodifferential operators with an anisotropic analytic parameter based on "twisted" operator-valued Volterra symbols. We establish the properties of the symbolic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side. In particular, we investigate the kernel cut-off operator via direct oscillatory integral techniques purely on symbolic level. We discuss the notion of parabolic for the calculus of Volterra operators, and construct Volterra parametrices for parabolic operators within the calculus.
Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This paper is aimed at describing refined asymptotics in the Dirichlet problem in a ball. The beauty of explicit formulas highlights the structure of asymptotic expansions in the calculi on singular varieties.
We introduce the calculus of Mellin pseudodifferential operators parameters based on "twisted" operator-valued Volterra symbols as well aas the abstract Mellin calclus with holomorphic symbols. We establish the properties of the symblic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side, e. g., for the Leibniz-product, kernel cut-off, and Mellin quantization. Moreover, we introduce the notion of parabolicity for the calculi of Volterra Mellin operators, and construct Volterra parametrices for parabolic operators within the calculi.
This is a survey of recent results concerning the general index locality principle, associated surgery, and their applications to elliptic operators on smooth manifolds and manifolds with singularities as well as boundary value problems. The full version of the paper is submitted for publication in Russian Mathematical Surveys.
We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the K-groups of A are topologically determined. In case the manifold, its boundary, and the cotangent space of its interior have torsion free K-theory, we get Ki(A,k) congruent Ki(C(X))⊕Ksub(1-i)(Csub(0)(T*X)),i = 0,1, with k denoting the compact ideal, and T*X denoting the cotangent bundle of the interior. Using Boutet de Monvel's index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis. For the case of orientable, two-dimensional X, Ksub(0)(A) congruent Z up(2g+m) and Ksub(1)(A) congruent Z up(2g+m-1), where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ k ⊂ G ⊂ A, with A/G commutative and G/k isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L²(R+).
The paper deals with elliptic theory on manifolds with boundary represented as a covering space. We compute the index for a class of nonlocal boundary value problems. For a nontrivial covering, the index defect of the Atiyah-Patodi-Singer boundary value problem is computed. We obtain the Poincaré duality in the K-theory of the corresponding manifolds with singularities.
This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These operators are bounded in these (Lipshitz and Djrbashian) spaces. As an application, we show a theorem about the division by good-inner functions in the mentioned classes is proved.
Adiabatic vacuum states are a well-known class of physical states for linear quantum fields n Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetme manifolds (factoriality, quasiequivalence, local definteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.
Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write Ssub(A)(D) for the space of solutions to thesystem Au = 0 in a domain D ⊂ X. Using reproducing kernels related to various Hilbert structures on subspaces of Ssub(A)(D) we show explicit identifications of the dual spaces. To prove the "regularity" of reproducing kernels up to the boundary of D we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the ∂-Neumann problem. The duality itself takes place only for those domains D which possess certain convexity properties with respect to A.
Contents: 1 Introduction 2 Statement of the problem and definitions 3 The main results 4 Proof of theorem 2 4.1 Reduction of problem (2) to functional - integral equations 4.2 The uniqueness of a solution of equation (2) 4.3 The existence of a solution of equation (2) 5 Proof of theorem 1 6 Proof of theorem 3 7 First boundary problem for hyperbolic differential equations 7.1 Statement of the problem 7.2 The formulation of the result and a sketch of the proof
Asymptotic algebras
(2001)
Contents: 1 Introduction 2 Main result 3 Construction of the asymptotic solutions 3.1 Derivation of the equations for the profiles 3.2 Exsistence of the principal profile 3.3 Determination of Usub(2) and the remaining profiles 4 Stability of the samll global solutions. Justification of One Phase Nonlinear Geometric Optics for the Kirchhoff-type equations 4.1 Stability of the global solutions to the Kirchhoff-type symmetric hyperbolic systems 4.2 The nonlinear system of ordinary differential equations with the parameter 4.3 Some energies estimates 4.4 The dependence of the solution W(t, ξ) on the function s(t) 4.5 The oscillatory integrals of the bilinear forms of the solutions 4.6 Estimates for the basic bilinear form Γsub(s)(t) 4.7 Contraction mapping 4.8 Stability of the global solution 4.9 Justification of One Phase Nonlinear Geometric Optics for the Kirchhoff-type equations
Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.
In this paper, we study the existence of positive solutions of a one-parameter family of logistic equations on R+ or on R. These equations are stationary versions of the Fisher equations and the KPP equations. We also study the blow up region of a sequence of the solutions when the parameter approachs a critical value and the nonexistence of positive solutions beyond the critical value. We use the direct method and the sub and super solution method.
We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus.
We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus.
We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus.
The inhomogeneous ∂-equations is an inexhaustible source of locally unsolvable equations, subelliptic estimates and other phenomena in partial differential equations. Loosely speaking, for the anaysis on complex manifolds with boundary nonelliptic problems are typical rather than elliptic ones. Using explicit integral representations we assign a Fredholm complex to the Dolbeault complex over an arbitrary bounded domain in C up(n).
We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B,1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A up(z), z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the pure imaginary powers. Examples concern unique solvability and maximal regularity of the solution of the Cauchy problem u' - Δu = f, u(0) = 0, for the Laplacian on conical manifolds.
Boundary value problems for pseudodifferential operators (with or without the transmission property) are characterised as a substructure of the edge pseudodifferential calculus with constant discrete asymptotics. The boundary in this case is the edge and the inner normal the model cone of local wedges. Elliptic boundary value problems for non-integer powers of the Laplace symbol belong to the examples as well as problems for the identity in the interior with a prescribed number of trace and potential conditions. Transmission operators are characterised as smoothing Mellin and Green operators with meromorphic symbols.
Contents: Introduction 1 Edge calculus with parameters 1.1 Cone asymptotics and Green symbols 1.2 Mellin edge symbols 1.3 The edge symbol algebra 1.4 Operators on a manifold with edges 2 Corner symbols and iterated asymptotics 2.1 Holomorphic corner symbols 2.2 Meromorphic corner symbols and ellipicity 2.3 Weighted corner Sobolev spaces 2.4 Iterated asymptotics 3 The edge corner algebra with trace and potential conditions 3.1 Green corner operators 3.2 Smoothing Mellin corner operators 3.3 The edge corner algebra 3.4 Ellipicity and regularity with asymptotics 3.5 Examples and remarks
A function has vanishing mean oscillation (VMO) on R up(n) if its mean oscillation - the local average of its pointwise deviation from its mean value - both is uniformly bounded over all cubes within R up(n) and converges to zero with the volume of the cube. The more restrictive class of functions with vanishing lower oscillation (VLO) arises when the mean value is replaced by the minimum value in this definition. It is shown here that each VMO function is the difference of two functions in VLO.
It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.
We continue the investigation of the calculus of Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour (SG symbols). Here we analyse what happens when one restricts the choice of amplitude and phase functions to the subclass of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a "classical" counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1); for such operators we refine the known results about the analogous problem for general SG hyperbolic operators. The material contained here will be used in a forthcoming paper to obtain a Weyl formula for a class of operators defined on manifolds with cylindrical ends, improving the results obtained in [9].
We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.
For a domain D subset of IRn with singular points on the boundary and a weight function ω infinitely differentiable away from the singularpoints in D, we consider a C*-algebra G (D; ω) of operators acting in the weighted space L² (D, ω). It is generated by the operators XD F-¹ σ F XD where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for G (D; ω). When combined with the local principle, this leads to describing the Fredholm operators in G (D; ω).
The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature o singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied.
Content: Introduction 1 Anisotropic operators in a cylinder with a conical base 1.1 Manifolds with conical singularities and opertors of Fuchs type 1.2 Typical operators and symbol structures 2 Weighted wedge Sobolev spaces and edge asymptotics 2.1 Discrete edge asymptotics 2.2 Continuos edge asymptotics with discrete limit at infinity 2.3 Calculus with operator valued symbols 3 Corner asymptotics at infinity 3.1 The structure of singular functions 3.2 Operators with trace and potential conditions 3.3 Asymptotics and (anisotropic) elliptic regularity
We prove the existence of a limit in Hm(D) of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S ⊂ ∂D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchy problem in D with data on S, for an elliptic operator A of order m ≥ 1, whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of the double layer potential. A similar regularisation is constructed also for a mixed problem in D.
Content: 0.1 Preliminary Remarks Chapter 1: Noncommutative Analysis: Main Ideas, Definitions, and Theorems 1.1 Functions of One Operator (Functional Calculi) 1.2 Functions of Several Operators 1.3 Main Formulas of Operator Calculus 1.4 Main Tools of Noncommutative Analysis 1.5 Composition Laws and Ordered Representations
Crack problems are regarded as elements in a pseudo-differential algbra, where the two sdes int S± of the crack S are treated as interior boundaries and the boundary Y of the crack as an edge singularity. We employ the pseudo-differential calculus of boundary value problems with the transmission property near int S± and the edge pseudo-differential calculus (in a variant with Douglis-Nirenberg orders) to construct parametrices od elliptic crack problems (with extra trace and potential conditions along Y) and to characterise asymptotics of solutions near Y (expressed in the framework of continuous asymptotics). Our operator algebra with boundary and edge symbols contains new weight and order conventions that are necessary also for the more general calculus on manifolds with boundary and edges.
The topological significance of the spectral Atiyah-Patodi-Singer η-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.
We construct a new calculus of boundary value problems with the transmission property on a non-compact smooth manifold with boundary and conical exits to infinity. The symbols are classical both in covariables and variables. The operators are determined by principal symbol tuples modulo operators of lower orders and weights (such remainders are compact in weighted Sobolev spaces). We develop the concept of ellipticity, construct parametrices within the algebra and obtain the Fredholm property. For the existence of Shapiro-Lopatinskij elliptic boundary conditions to a given elliptic operator we prove an analogue of the Atiyah-Bott condition.
We prove the existence of Hp(D)-limit of iterations of double layer potentials constructed with the use of Hodge parametrix on a smooth compact manifold X, D being an open connected subset of X. This limit gives us an orthogonal projection from Sobolev space Hp(D) to a closed subspace of Hp(D)-solutions of an elliptic operator P of order p ≥ 1. Using this result we obtain formulae for Sobolev solutions to the equation Pu = f in D whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of double layer potentials. Similar regularization is constructed also for a P-Neumann problem in D.
We study the approach to the theory of hypergeometric functions in several variables via a generalization of the Horn system of differential equations. A formula for the dimension of its solution space is given. Using this formula we construct an explicit basis in the space of holomorphic solutions to the generalized Horn system under some assumptions on its parameters. These results are applied to the problem of describing the complement of the amoeba of a rational function, which was posed in [12].
Using the Riemannian connection on a compact manifold X, we show that the algebra of classical pseudo-differential operators on X generates a canonical deformation quantization on the cotangent manifold T*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization.
Problems for elliptic partial differential equations on manifolds M with singularities M' (here with piece-wise smooth geometry)are studied in terms of pseudo-differential algebras with hierarchies of symbols that consist of scalar and operator-valued components. Classical boundary value problems (with or without the transmission property) belong to the examples. They are a model for operator algebras on manifolds M with higher "polyhedral" singularities. The operators are block matrices that have upper left corners containing the pseudo-differential operators on the regular M\M' (plus certain Mellin and Green summands) and are degenerate (in streched coordinates) in a typical way near M'. By definition M' is again a manifold with singularities. The same is true of M'', and so on. The block matrices consist of trace, potential and Mellin and Green operators, acting between weighted Sobolev spaces on M(j) and M(k), with 0 ≤ j, k ≤ ord M; here M(0) denotes M, M(1) denotes M', etc. We generate these algebras, including their symbol hierarchies, by iterating so-called "edgifications" and "conifications" os algebras that have already been constructed, and we study ellipicity, parametrics and Fredholm property within these algebras.
We prove a theorem on analytic representation of integrable CR functions on hypersurfaces with singular points. Moreover, the behaviour of representing analytic functions near singular points is investigated. We are aimed at explaining the new effect caused by the presence of a singularity rather than at treating the problem in full generality.