Verchota, Gregory Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. (English) Zbl 0589.31005 J. Funct. Anal. 59, 572-611 (1984). The author considers the classical layer potentials for harmonic functions on the boundary \(\partial G\) of a bounded Lipschitz domain G in \({\mathbb{R}}^ n\) for use in Dirichlet and Neumann problems. It is shown that these potentials are invertible operators on \(L^ 2(\partial G)\) and some subspaces. In the case \(n=2\) the layer potentials are shown to be invertible on every \(L^ p(\partial G)\), \(1<p<\infty\). Reviewer: G.Dziuk Cited in 7 ReviewsCited in 382 Documents MSC: 31B25 Boundary behavior of harmonic functions in higher dimensions 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) Keywords:Lipschitz boundary; regularity; layer potentials; harmonic functions; Dirichlet and Neumann problems PDFBibTeX XMLCite \textit{G. Verchota}, J. Funct. Anal. 59, 572--611 (1984; Zbl 0589.31005) Full Text: DOI References: [1] Calderón, A. P., Algebra of Singular Integral Operators, (Proc. Sympos. Pure Math., Vol. X (1967), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · Zbl 0222.44007 [2] Calderón, A. P., Cauchy integrals on Lipschitz curves and related operators, (Proc. Nat. Acad. Sci USA, 74 (1977)), 1324-1327 · Zbl 0373.44003 [3] Courant, R.; Hilbert, D., (Methods of Mathematical Physics, Vol. II (1962), Interscience: Interscience New York), 258-261 [4] Coifman, R. R.; Meyer, Y., Au delà des opérateurs pseudo-différentiels, (Astérisque 57 (1978), Société Mathématique de France) · Zbl 0483.35082 [5] Coifman, R. R.; McIntosh, A.; Meyer, Y., L’intégrale de Cauchy définit un opérateur Borné sur \(L^2\) pour les Courbes Lipschitiennes, Ann. of Math., 116, 361-388 (1982) · Zbl 0497.42012 [6] Dahlberg, B. E.J, Estimates of harmonic measure, Arch. Rational Mech. Anal., 65, 275-288 (1977) · Zbl 0406.28009 [7] Dahlberg, B. E.J, On the Poisson integral for Lipschitz and \(C^1\) domains, Studia Math., 66, 7-24 (1979) · Zbl 0422.31008 [8] Dahlberg, B. E.J, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math., 67, 279-314 (1980) · Zbl 0449.31002 [9] Fabes, E. B.; Jodeit, M.; Riviére, N. M., Potential techniques for boundary value problems on \(C^1\) Domains, Acta Math., 141, 165-186 (1978) · Zbl 0402.31009 [10] Fabes, E. B.; Jodeit, M.; Lewis, J. E., Double layer potentials for domains with corners and edges, Indiana Univ. Math. J., 26, 95-114 (1977) · Zbl 0363.35010 [11] Hunt, R.; Wheeden, R. L., On the boundary values of harmonic functions, Trans. Amer. Math. Soc., 132, 307-322 (1968) · Zbl 0159.40501 [12] Hunt, R.; Wheeden, R. L., Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc., 147, 507-527 (1970) · Zbl 0193.39601 [13] Jerison, D. S.; Kenig, C. E., The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc., 4, 203-207 (1981) · Zbl 0471.35026 [15] Jerison, D. S.; Kenig, C. E., Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math., 46, 80-147 (1982) · Zbl 0514.31003 [16] Kenig, C. E., Weighted \(H^p\) spaces on Lipschitz domains, Amer. J. Math., 102, 129-163 (1980) · Zbl 0434.42024 [17] Morrey, C. B., Multiple Integrals in the Calculus of Variations (1966), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0142.38701 [18] Necas, J., Sur les domaines du type \(N\), Czechoslovak. Math. J., 12, 274-287 (1962) · Zbl 0106.27001 [19] Necas, J., Les méthodes directes en théorie des équations élliptiques (1967), Academia: Academia Prague · Zbl 1225.35003 [20] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J · Zbl 0207.13501 [21] Verchota, G., Layer Potentials and Boundary Value Problems for Laplace’s Equation on Lipschitz Domains, (Thesis (1982), University of Minnesota) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.