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ELLIPTIC EXTENSIONS IN THE DISK WITH OPERATORS IN DIVERGENCE FORM

Published online by Cambridge University Press:  20 August 2012

ORAZIO ARENA
Affiliation:
Dipartimento di Costruzioni e Restauro, Università di Firenze, Piazza Brunelleschi 6, I-50121 Firenze, Italy (email: [email protected])
CRISTINA GIANNOTTI*
Affiliation:
Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri, I- 62032 Camerino (Macerata), Italy (email: [email protected])
*
For correspondence; e-mail: [email protected]
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Abstract

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Let $\varphi _0$ and $\varphi _1$ be regular functions on the boundary $\partial D$ of the unit disk $D$ in $\mathbb {R}^2$, such that $\int _{0}^{2\pi }\varphi _1\,d\theta =0$ and $\int _{0}^{2\pi }\sin \theta (\varphi _1-\varphi _0)\,d\theta =0$. It is proved that there exist a linear second-order uniformly elliptic operator $L$ in divergence form with bounded measurable coefficients and a function $u$ in $W^{1,p}(D)$, $1 \lt p \lt 2$, such that $Lu=0$ in $D$ and with $u|_{\partial D}= \varphi _0$ and the conormal derivative $\partial u/\partial N|_{\partial D}=\varphi _1$.

MSC classification

Type
Research Article
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

References

[1]Giannotti, C. & Manselli, P., ‘On elliptic extensions in the disk’, Potential Anal. 33 (2010), 249262.CrossRefGoogle Scholar
[2]Ladyzhenskaya, O. A. & Ural’tseva, N. N., Linear and Quasilinear Elliptic Equations (Academic Press, New York, 1968).Google Scholar
[3]Wolff, T. H., ‘Some constructions with solutions of variable coefficient elliptic equations’, J. Geom. Anal. 3 (1993), 423511.CrossRefGoogle Scholar