On the rigidity theorem for elliptic genera

We give a detailed proof of the rigidity theorem for elliptic genera. Using the Lefschetz fixed point formula we carefully analyze the relation between the characteristic power series defining the elliptic genera and the equivariant elliptic genera. We show that equivariant elliptic genera converge to Jacobi functions which are holomorphic. This implies the rigidity of elliptic genera. Our approach can be easily modified to give a proof of the rigidity theorem for the elliptic genera of level .

     

Nonstrictly hyperbolic systems of conservation laws of the conjugate type

We consider an inverse boundary problem for a general second order self-adjoint elliptic differential operator on a compact differential manifold with boundary. The inverse problem is that of the reconstruction of the manifold and operator via all but finite number of eigenvalues and traces on the boundary of the corresponding eigenfunctions of the operator. We prove that the data determine the manifold and the operator to within the group of the generalized gauge transformations. The proof is based upon a procedure of the reconstruction of a canonical object in the orbit of the group. This object, the canonical Schrödinger operator, is uniquely determined via its incomplete boundary spectral data.     

A Proof of the Uniformization Theorem on S2

We present a simple proof for the uniformization theorem on 2-sphere by methods of elliptic partial differential equations.     

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

We study the well-posedness of the bidomain model that is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model as a system of coupled parabolic and elliptic PDEs for two potentials and ODEs representing the ionic activity. We first reformulate the parabolic and elliptic PDEs into a single parabolic PDE by the introduction of a bidomain operator. We properly define and analyze this operator, basically a non-differential and non-local operator. We then present a proof of existence, uniqueness and regularity of a local solution in time through a semigroup approach, but that applies to fairly general ionic models. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution in time is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh–Nagumo, Aliev–Panfilov and MacCulloch models.     

Cobordism invariance of the index of Callias-type operators

We introduce a notion of cobordism of Callias-type operators overcomplete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application, we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two noncompact but topologically simpler manifolds. As another application, we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.     

Singularly perturbed Dirichlet boundary-value problem for a stationary system in the linear elasticity theory

We consider a singularly perturbed Dirichlet boundary-value problem for an elliptic operator of the linear elasticity theory in a bounded domain with a small cavity. The main result is the proof of the theorem about the convergence of eigenelements of the perturbed boundary-value problem to eigenelements of the corresponding limiting boundary-value problem, when the parameter ? which defines the diameter of the small cavity tends to zero.     

de Rham－Hodge－Signature算子的局部指标定理

整体的Ａｔｉｙａｈ－Ｓｉｎｇｅｒ指标定理对于一般的椭圆微分算子都成立．但对于局部指标定理来说，人们只能对具体的算子给予证明．并且各种情况需个别处理．由［６］知本文中证得的ｄｅＲｈａｍ－Ｈｏｄｇｅ－Ｓｉｇｎａｔｕｒｅ算子的局部指标既不是α型也不是β型的．这和经典的椭圆算子情形不同．     

A nonstandard proof of the spectral theorem for unbounded self-adjoint operators

We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint operator due to Raab.     

Solvability and properties of solutions of nonlinear elliptic equations

The paper contains an exposition of variational and topological methods of investigating general nonlinear operator equations in Banach spaces. Application is given of these methods to the proof of solvability of boundary-value problems for nonlinear elliptic equations of arbitrary order, to the problem of eigenfunctions, and to bifurcation of solutions of differential equations. Results are presented of investigations of the properties of generalized solutions of quasilinear elliptic equations of higher order.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 9, pp. 131–254, 1976.     

ζ-determinant and the Quillen determinant on the Grassmannian of elliptic self-adjoint boundary conditions

We announce a proof of the equality of the ζ-determinant of a Dirac operator over an odd-dimensional manifold with boundary with the Quillen determinant section computed in a canonical trivialization over the Grassmannian of elliptic self-adjoint boundary conditions.     

A super-twisted Dirac operator and Novikov inequalities

A super-twisted Dirac operator is constructed and deformed suitably. Following Shubin’s approach to Novikov inequalities associated to the deformed de Rham-Hodge operator, we give a for mula for the index of the super-twisted Dirac operator, and Novikov type inequalities for the deformed operator. In particular, we obtain a purely analytic proof of the Hopf index theorem for general vector bundles.     

Stokeswaves with vorticity

The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep body of water under the force of gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite cylinder with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. The proof combines a generalized degree theory, global bifurcation theory, and Whyburn’s lemma in topology with the Schauder theory for elliptic problems and the maximum principle.     

On Carvalho's -theoretic formulation of the cobordism invariance of the index

We give an analytic proof of the fact that the index of an elliptic operator on the boundary of a compact manifold vanishes when the principal symbol comes from the restriction of a -theory class from the interior. The proof uses non-commutative residues inside the calculus of cusp pseudodifferential operators of Melrose.

     

On the asymptotic behavior of solutions of quasilinear elliptic equations

The asymptotic behavior of solutions of second-order quasilinear elliptic and nonhyperbolic partial differential equations defined on unbounded domains inR ⁿ contained in\(\{ x_1 ,...,x_n :\left| {x_n } \right|< \lambda \sqrt {x_1^2 + ...x_{n - 1}^2 } \) for certain sublinear functions λ is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data has appropriate asymptotic behavior at infinity. We prove Phragmèn-Lindelöf theorems for large classes of nonhyperbolic operators, without «lower order terms”, including uniformly elliptic operators and operators with well-definedgenre, using special barrier functions which are constructed by considering an operator associated to our original operator. We also estimate the rate at which a solution converges to its limiting function at infinity in terms of properties of the top order coefficienta _nn of the operator and the rate at which the boundary values converge to their limiting function; these results are proven using appropriate barrier functions which depend on the behavior of the coefficients of the operator and the rate of convergence of boundary values.     

Analyticity of solutions of semi-linear equations with double characteristics

We prove the analyticity and Gevrey regularity of solutions of elliptic degenerate semi-linear differential equations principle part of which is a linear operator with double characteristics considered first by Gilioli and Treves. A new elementary proof for hypoellipticity in the weak sense is given.     

Multi-valued boundary value problems involving Leray-Lions operators and discontinuous nonlinearities

We prove an existence result for a class of Dirichlet boundary value problems with discontinuous nonlinearity and involving a Leray-Lions operator. The proof combines monotonicity methods for elliptic problems, variational inequality techniques and basic tools related to monotone operators. Our work generalizes a result obtained in Carl [4].     

Analyticity for Some Operator Functions from Statistical Quantum Mechanics

For rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schr?dinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain boundedly maps the space of square integrable functions to the space of essentially bounded functions. Dedicated to Günter Albinus Submitted: November 21, 2008. Accepted: March 31, 2009.     

Fundamental solution and Fourier series in eigenfunctions of degenerate elliptic operator

We study the degenerate elliptic differential operator of the second order in the divergence form. The operator is assumed to be symmetric. The weight function which is describing the degeneration of the coefficients (or singularity) assumed to be in the Muckenhoupt class. We prove the uniform estimates for the fundamental solution of this operator and obtain the conditions which guarantee the absolute and uniform convergence of Fourier series in eigenfunctions. These results might be applied to the ground of Fourier method.     

Eigenvalue approximation from below using non-conforming finite elements

This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators,with emphasis on obtaining lower bounds. In addition,this article also contains some new materials for eigenvalue approximations of the Laplace operator,which include:1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below;2) the proof of the fact that the non-conforming EQ rot1 element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements;3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q rot1 element approximate the true eigenvalues from below for the L-shaped domain. Finally,we list several unsolved problems.     

Global L theory and regularity for the 3D nonlinear Wigner-Poisson-Fokker-Planck system

A global existence, uniqueness and regularity theorem is proved for the simplest Markovian Wigner-Poisson-Fokker-Planck model incorporating friction and dissipation mechanisms. The proof relies on Green function and energy estimates under mild formulation, making essential use of the Husimi function and the elliptic regularization of the Fokker-Planck operator.