Approximation of harmonic functions
Annales de l'Institut Fourier, Volume 30 (1980) no. 2, p. 97-107

Let u be harmonic in a bounded domain D with smooth boundary. We prove that if the boundary values of u belong to L p (σ), where p2 and σ denotes the surface measure of D, then it is possible to approximate u uniformly by function of bounded variation. An example is given that shows that this result does not extend to p<2.

Soit D un domaine borné à frontière régulière, et u une fonction harmonique dans D. On montre que si les valeurs de u à la frontière appartiennent à L p (σ) avec p2 (σ étant la mesure de surface à la frontière), u est approchable uniformément par des fonctions à variation bornée, et on montre que le résultat ne s’étend pas au cas p<2.

@article{AIF_1980__30_2_97_0,
     author = {Dahlberg, Bj\"orn E. J.},
     title = {Approximation of harmonic functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {30},
     number = {2},
     year = {1980},
     pages = {97-107},
     doi = {10.5802/aif.787},
     zbl = {0417.31005},
     mrnumber = {82i:31010},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1980__30_2_97_0}
}
Dahlberg, Björn E. J. Approximation of harmonic functions. Annales de l'Institut Fourier, Volume 30 (1980) no. 2, pp. 97-107. doi : 10.5802/aif.787. http://www.numdam.org/item/AIF_1980__30_2_97_0/

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