Article
MSC:
35J40,
35S15,
35S30,
42A38,
42B37,
45N05 | MR 3238843 | Zbl 06362262 | DOI: 10.21136/MB.2014.143858
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Keywords:
wave factorization; pseudodifferential equation; boundary value problem; integral equation
wave factorization; pseudodifferential equation; boundary value problem; integral equation
Summary:
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.
References:
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[2] Gel'fand, I. M., Shilov, G. E.: Generalized Functions. Vol. I: Properties and Operations. Translated from the Russian. Academic Press, New York (1964). MR 0166596
[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). MR 1795504 | Zbl 0961.35193
[4] Vasilyev, V. B.: Elliptic equations and boundary value problems in non-smooth domains\. 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). MR 2867419
[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. MR 3337282
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