On the Dirichlet and Neumann problems in multi-dimensional cone

Vladimir Vasilyev

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 333-340
  • ISSN: 0862-7959

Abstract

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We consider an elliptic pseudodifferential equation in a multi-dimensional cone, and using the wave factorization concept for an elliptic symbol we describe a general solution of such equation in Sobolev-Slobodetskii spaces. This general solution depends on some arbitrary functions, their quantity being determined by an index of the wave factorization. For identifying these arbitrary functions one needs some additional conditions, for example, boundary conditions. Simple boundary value problems, related to Dirichlet and Neumann boundary conditions, are considered. A certain integral representation for this case is given.

How to cite

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Vasilyev, Vladimir. "On the Dirichlet and Neumann problems in multi-dimensional cone." Mathematica Bohemica 139.2 (2014): 333-340. <http://eudml.org/doc/261945>.

@article{Vasilyev2014,
abstract = {We consider an elliptic pseudodifferential equation in a multi-dimensional cone, and using the wave factorization concept for an elliptic symbol we describe a general solution of such equation in Sobolev-Slobodetskii spaces. This general solution depends on some arbitrary functions, their quantity being determined by an index of the wave factorization. For identifying these arbitrary functions one needs some additional conditions, for example, boundary conditions. Simple boundary value problems, related to Dirichlet and Neumann boundary conditions, are considered. A certain integral representation for this case is given.},
author = {Vasilyev, Vladimir},
journal = {Mathematica Bohemica},
keywords = {wave factorization; pseudodifferential equation; boundary value problem; integral equation; wave factorization; pseudodifferential equation; boundary value problem; integral equation},
language = {eng},
number = {2},
pages = {333-340},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Dirichlet and Neumann problems in multi-dimensional cone},
url = {http://eudml.org/doc/261945},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Vasilyev, Vladimir
TI - On the Dirichlet and Neumann problems in multi-dimensional cone
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 333
EP - 340
AB - We consider an elliptic pseudodifferential equation in a multi-dimensional cone, and using the wave factorization concept for an elliptic symbol we describe a general solution of such equation in Sobolev-Slobodetskii spaces. This general solution depends on some arbitrary functions, their quantity being determined by an index of the wave factorization. For identifying these arbitrary functions one needs some additional conditions, for example, boundary conditions. Simple boundary value problems, related to Dirichlet and Neumann boundary conditions, are considered. A certain integral representation for this case is given.
LA - eng
KW - wave factorization; pseudodifferential equation; boundary value problem; integral equation; wave factorization; pseudodifferential equation; boundary value problem; integral equation
UR - http://eudml.org/doc/261945
ER -

References

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  1. Eskin, G. I., Boundary Value Problems for Elliptic Pseudodifferential Equations. Translated from the Russian, Translations of Mathematical Monographs 52 AMS, Providence (1981). (1981) MR0623608
  2. Gel'fand, I. M., Shilov, G. E., Generalized Functions. Vol. I: Properties and Operations. Translated from the Russian, Academic Press, New York (1964). (1964) MR0166596
  3. Vasil'ev, V. B., Wave Factorization of Elliptic Symbols: Theory and Applications. Introduction to the theory of boundary value problems in non-smooth domains, Kluwer Academic Publishers, Dordrecht (2000). (2000) Zbl0961.35193MR1795504
  4. Vasilyev, V. B., [unknown], Pseudo-Differential Operators: Analysis, Applications and Computations L. Rodino, M. W. Wong, H. Zhu Operator Theory: Advances and Applications 213 Birk-häuser, Basel 105-121 (2011). (2011) MR2867419
  5. Vasilyev, V. B., General boundary value problems for pseudo-differential equations and related difference equations, Advances in Difference Equations (2013), Article ID 289, 7 pages. (2013) MR3337282

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