The Analysis of Solutions of Elliptic Equations

1. Front Matter

   Pages i-xx

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2. List of Main Notations

   • Nikolai N. Tarkhanov

   Pages 1-3

3. Removable Singularities

   • Nikolai N. Tarkhanov

   Pages 5-66

4. Laurent Series

   • Nikolai N. Tarkhanov

   Pages 67-114

5. Representation of Solutions with Non-Discrete Singularities

   • Nikolai N. Tarkhanov

   Pages 115-189

6. Uniform Approximation

   • Nikolai N. Tarkhanov

   Pages 191-270

7. Mean Approximation

   • Nikolai N. Tarkhanov

   Pages 271-318

8. BMO Approximation

   • Nikolai N. Tarkhanov

   Pages 319-344

9. Conditional Stability

   • Nikolai N. Tarkhanov

   Pages 345-371

10. The Cauchy Problem

    • Nikolai N. Tarkhanov

    Pages 373-409

11. Quasiconformality

    • Nikolai N. Tarkhanov

    Pages 411-449

12. Back Matter

    Pages 451-484

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This book is intended as a continuation of my book "Parametrix Method in the Theory of Differential Complexes" (see [291]). There, we considered complexes of differential operators between sections of vector bundles and we strived more than for details. Although there are many applications to for maximal generality overdetermined systems, such an approach left me with a certain feeling of dissat- faction, especially since a large number of interesting consequences can be obtained without a great effort. The present book is conceived as an attempt to shed some light on these new applications. We consider, as a rule, differential operators having a simple structure on open subsets of Rn. Currently, this area is not being investigated very actively, possibly because it is already very highly developed actively (cf. for example the book of Palamodov [213]). However, even in this (well studied) situation the general ideas from [291] allow us to obtain new results in the qualitative theory of differential equations and frequently in definitive form. The greater part of the material presented is related to applications of the L- rent series for a solution of a system of differential equations, which is a convenient way of writing the Green formula. The culminating application is an analog of the theorem of Vitushkin [303] for uniform and mean approximation by solutions of an elliptic system. Somewhat afield are several questions on ill-posedness, but the parametrix method enables us to obtain here a series of hitherto unknown facts.

• Cauchy problem
• Potential theory
• differential equation
• differential operator
• functional analysis
• partial differential equation
• partial differential equations

`Summarising this is a well-written book. It provides the complete survey of the history and the recent state of the subject topic. The book will be of use to postgraduate students and researchers in PDE's, in theory of approximations and expansions, in potential theory or in functional analysis. It is warmly recommended as a text for seminars and courses, as well as for independent study for everyone with a basic knowledge on distributions and pseudodifferential operators.'
Acta Scientiarum Mathematicarum

• Institute of Physics, Krasnoyarsk, Russia

  Nikolai N. Tarkhanov

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