Annales Academiæ Scientiarum Fennicæ

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 **R**^{n} 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.

Full document as PDF file

doi:10.5186/aasfm.2010.3527

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