BMO and Lipschitz approximation by solutions of elliptic equations

Mateu, Joan; Netrusov, Yuri; Orobitg, Joan; Verdera, Joan

Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1057-1081.

On considère le problème de l’approximation qualitative par des solutions d’une équation elliptique, homogène, à coefficients constants, dans les normes de Lipschitz et BMO. Notre méthode est bien connue : on trouve une condition suffisante pour l’approximation en se réduisant à un problème de synthèse spectrale dans un certain espace de Lizorkin-Triebel doté de sa topologie faible $*$. Deux exemples, dont l’origine est dans une construction de Hedberg, montrent que nos conditions sont fines.

We consider the problem of qualitative approximation by solutions of a constant coefficients homogeneous elliptic equation in the Lipschitz and BMO norms. Our method of proof is well-known: we find a sufficient condition for the approximation reducing matters to a weak $*$ spectral synthesis problem in an appropriate Lizorkin-Triebel space. A couple of examples, evolving from one due to Hedberg, show that our conditions are sharp.

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