Abstract
The Fredholm radius of the integral operators of the theory of n-dimensional harmonic potentials on surfaces with edges, acting in certain weighted Hölder type spaces, is found. Results for integral operators are derived from theorems on operators in certain auxiliary contact problems.
Similar content being viewed by others
References
V. G. Maz'ya, “On potential theory for the Lamé system in a domain with a piecewise smooth boundary,” in: Partial Differential Equations and Their Applications (Proc. All-Union Symp., Tbilisi, 1982) [in Russian], Tbilis. Gos. Univ., Tbilisi (1986).
I. Radon, “On boundary-value problems for the logarithmic potential,” Uspekhi Mat. Nauk,1, No. 3–4, 96–124 (1946) [translated from Akad. Wiss. Wien, S.-B. IIa,128, 1123–1167 (1920)].
J. Král, “The Fredholm method in potential theory,” Trans. Amer. Math. Soc,125, No. 3, 511–547 (1966).
Yu. D. Burago and V. G. Maz'ya, “Certain questions of potential theory and functions theory for domains with irregular boundaries,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,3, 152–162 (1967).
V. Yu. Shelepov, “On the index of an integral operator of potential type in the space Lp, ” Dokl. Akad. Nauk SSSR,186, No. 6, 1266–1268 (1969).
V. G. Maz'ya, “Boundary integral equations,” Sovrem. Probl. Mat. Fundam. Napravl., 130–256 (1988).
N. V. Grachev and V. G. Maz'ya, “On the Fredholm radius of operators of double layer potential type on piecewise smooth surfaces,” Vestnik Leningrad. Univ., Mat. Mekh. Astronom., Ser. 1, Issue 4, 60–64 (1986).
V. Zaionchkovskii (Z. Wojciech) and V. A. Solonnikov, “On the Neumann problem for second-order elliptic equations in domains with edges on the boundary,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,127, 7–48 (1983).
V. A. Kondrat'ev, “Boundary value problems for elliptic equations in domains with conical or angular points,” Trudy Moskov. Mat. Obshch.,16, 209–292 (1967).
V. G. Maz'ya and B. A. Plamenevskii, “Lp-estimates of solutions of elliptic boundary value problems in domains with edges,” Trudy Moskov. Mat. Obshch.,37, 49–93 (1978).
V. G. Maz'ya and B. A. Plamenevskii, “Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points,” in: Elliptische Differentialgleichungen, Vortrage der Tagung in Rostock, Sekt. Math., 1977, Wilhelm-Pieck Univ., Rostock (1978), pp. 161–190.
V. G. Maz'ya and B. A. Plamenevskii, “On boundary value problems for a second order elliptic equation in a domain with edges,” Vestnik Leningrad. Univ., Mat. Mekh. Astronom., No. 1, Issue 1, 102–108 (1975).
I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1, Properties and Operations, Academic Press, New York (1964).
V. A. Solonnikov, “Estimates of the solutions of the Neumann problem for second-order elliptic equations in domains with edges on the boundary,” Preprint LOMI, R-43-83 (1983).
V. G. Maz'ya and B. A. Plamenevskii, “Estimates of the Green functions and Schauder estimates of the solutions of elliptic boundary value problems in a two-sided angle,” Sib. Mat. Zh.,19, No. 5, 1065–1082 (1978).
V. G. Maz'ya and B. A. Plamenevskii, “The first boundary value problem for classical equations of mathematical physics in domains with piecewise-smooth boundaries. I; II,” Z. Anal. Anwendungen,2, No. 4, 335–359 (1983);2, No. 6, 523–551 (1983).
S. Prössdorf, Some Classes of Singular Equations, North-Holland, Amsterdam (1978).
F. V. Atkinson, “The normal solvability of linear equations in normed spaces,” Mat. Sb.,28 (70), 3–14 (1951).
Additional information
Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 109–133, 1990.
Rights and permissions
About this article
Cite this article
Grachev, N.V., Maz'ya, V.G. On the Fredholm radius of integral operators of potential theory. J Math Sci 64, 1297–1313 (1993). https://doi.org/10.1007/BF01098022
Issue Date:
DOI: https://doi.org/10.1007/BF01098022