Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.

Pseudo holomorphic curves in symplectic manifolds

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ∂ Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ*γ which will produce the same voltage and current measurements on ∂ Ω.



We prove the converse in two dimensions (i.e., if γ1 and γ2 produce the same boundary measurements, then γ1, = Ψ*γ2 for an appropriate Ψ) for conductivities which are near a constant.

/pdf/an-anisotropic-inverse-boundary-value-problem-14js24t5mz.pdf

An anisotropic inverse boundary value problem

Generalized Analytic Functions. By I. N. Vekua. English Translation Editor I. N. Sneddon. Pergamon Press. 1962. Pp. 668. 105s.

Tutorial introduction Algorithms of CM, TM, and CTM Coupling techniques Boundary element methods Other kinds of boundary methods Comparisons Part I: Collocation Trefftz method 1 - Basic algorithms and theory Notations and preliminaries Approximation problems Error estimates Debye-Huckel equation Stability analysis 2 - Motz's problem and its variants Introduction Basic algorithms of CTM Error bounds and integration approximation Variants of Motz's problem Concluding remarks 3 - Coupling techniques Introduction Description of generalized TMs Penalty TMs Simplified hybrid TMs Penalty plus hybrid TMs Lagrange multiplier TM Effective condition number Numerical experiments 4 - Biharmonic equations with singularities Introduction The Green formulas of A2u The collocation Trefftz methods Error bounds Part II: Collocation methods 5 - Collocation methods Introduction Description of collocation methods Error analysis Robin boundary conditions Inverse inequalities Final remarks 6 - Combinations of collocation and finite element methods Introduction Combinations of FEMs Linear algebraic equations of combination of FEM and CM Uniformly Vh0-elliptic inequality Uniformly Vh0-elliptic inequality involving integration approximation Final remarks 7 - Radial basis function collocation methods Introduction Radial basis functions Description of radial basis function collocation methods Inverse estimates for radial basis functions Numerical experiments Comparisons and conclusions Part III: Advanced topics 8 - Combinations with high-order FEMs Introduction Combinations of TM and Lagrange FEMs Global superconvergence Adini's elements 9 - Eigenvalue problems Introduction New numerical algorithms for eigenvalue problems Error bounds of eigenvalues Error bounds of eigenfunctions Computational models and numerical experiments Eigenvalues for the singularity problem Summaries and discussions 10 - The Helmholtz equation Introduction The Trefftz method Error analysis Summaries and discussions 11 - Explicit harmonic solutions of Laplace's equation Introduction Harmonic functions Harmonic solutions involving Neumann conditions Extensions and analysis on singularity New models of singularities for Laplace's equation Concluding remarks Appendix - Historic review of boundary methods Potential theory Existence and uniqueness Reduction in dimensions and Green's formula Integral equations Extended Green's formula Pre-electronic computer era Electronic computer era Boundary integral equation and boundary element methods

/pdf/trefftz-and-collocation-methods-1x7y3q2kmy.pdf

Trefftz and Collocation Methods

This is an introductory text for beginners who have a basic knowledge in complex analysis, functional analysis and partial differential equations. The Riemann and Riemann-Hilbert boundary value problems are discussed for analytic functions, for generalized Cauchy-Riemann systems and for generalized Beltrami systems. Related problems, such as the Poincare problem, pseudoparabolic systems, half Dirichlet problems for the Dirac operator in Clifford analysis and elliptic second order equations are also considered. Estimates for solutions to linear equations existence and uniqueness results are thus available for related nonlinear problems; the method is explained by constructing entire solutions to nonlinear Beltrami equations.

Complex Analytic Methods for Partial Differential Equations: An Introductory Text

Complex analytic methods for partial differential equations

Boundary value problems for elliptic equations and systems

1. Systems of Elliptic Equations 2. Singularities of Solutions to Elliptic Equations 3. Evolutionary Equations 4. Clifford Analysis

Transformations, transmutations, and kernel functions

/pdf/a-hierarchy-of-integral-operators-m4lzr4i760.pdf

Heinrich Begehr

Papers

Complex Analytic Methods for Partial Differential Equations: An Introductory Text

Complex analytic methods for partial differential equations

Boundary value problems for elliptic equations and systems

Transformations, transmutations, and kernel functions

A Hierarchy of Integral Operators