Annales Academić Scientiarum Fennicć
Mathematica
Volumen 35, 2010, 421-438

SOLUTIONS OF THE DIVERGENCE AND KORN INEQUALITIES ON DOMAINS WITH AN EXTERNAL CUSP

Ricardo G. Durán and Fernando López García

Universidad de Buenos Aires, Departamento de Matemática
1428 Buenos Aires, Argentina; rduran 'at' dm.uba.ar

Universidad de Buenos Aires, Departamento de Matemática
1428 Buenos Aires, Argentina; flopezg 'at' dm.uba.ar

Abstract. This paper deals with solutions of the divergence for domains with external cusps. It is known that the classic results in standard Sobolev spaces, which are basic in the variational analysis of the Stokes equations, are not valid for this class of domains. For some bounded domains \Omega \subset Rⁿ presenting power type cusps of integer dimension m \le n - 2, we prove the existence of solutions of the equation div u = f in weighted Sobolev spaces, where the weights are powers of the distance to the cusp. The results obtained are optimal in the sense that the powers cannot be improved. As an application, we prove existence and uniqueness of solutions of the Stokes equations in appropriate spaces for cuspidal domains. Also, we obtain weighted Korn type inequalities for this class of domains.

2000 Mathematics Subject Classification: Primary 26D10, 35Q30; Secondary 76D03.

Key words: Divergence operator, weighted Sobolev spaces, Korn inequality.

Reference to this article: R.G. Durán and F. López García: Solutions of the divergence and Korn inequalities on domains with an external cusp. Ann. Acad. Sci. Fenn. Math. 35 (2010), 421-438.

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doi:10.5186/aasfm.2010.3527