# Article

 Title: Essential norms of a potential theoretic boundary integral operator in $L\sp 1$ (English) Author: Král, Josef Author: Medková, Dagmar Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 123 Issue: 4 Year: 1998 Pages: 419-436 Summary lang: English . Category: math . Summary: Let $G \subset\Bbb R^m$ $(m \ge2)$ be an open set with a compact boundary $B$ and let $\sigma\ge0$ be a finite measure on $B$. Consider the space $L^1(\sigma)$ of all $\sigma$-integrable functions on $B$ and, for each $f \in L^1(\sigma)$, denote by $f \sigma$ the signed measure on $B$ arising by multiplying $\sigma$ by $f$ in the usual way. $\Cal N_{\sigma}f$ denotes the weak normal derivative (w.r. to $G$) of the Newtonian (in case $m >2$) or the logarithmic (in case $n=2$) potential of $f\sigma$, correspondingly. Sharp geometric estimates are obtained for the essential norms of the operator $\Cal N_{\sigma} - \alpha I$ (here $\alpha\in\Bbb R$ and $I$ stands for the identity operator on $L^1(\sigma)$) corresponding to various norms on $L^1(\sigma)$ inducing the topology of standard convergence in the mean w.r. to $\sigma$. (English) Keyword: single layer potential Keyword: weak normal derivative Keyword: essential norm MSC: 31A10 MSC: 31B10 MSC: 31B20 MSC: 31B25 idZBL: Zbl 0936.31007 idMR: MR1667114 DOI: 10.21136/MB.1998.125966 . Date available: 2009-09-24T21:33:55Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/125966 . Reference: [1] T. S. Angell R. E. Kleinman J. Král: Layer potentials on boundaries with corners and edges.Časopis Pěst. Mat. 113 (1988), 387-402. MR 0981880 Reference: [2] Yu. D. Burago V. G. Maz'ya: Some problems of potential theory and function theory for domains with nonregular boundaries.Zapiski Naučnych Seminarov LOMI 3 (1967). (In Russian.) Reference: [3] N. Dunford J. T Schwartz W. G. Bade R. G. Barth: Linear Operators, Part I.Interscience Publishers, New York, 1958. MR 0117523 Reference: [4] H. Federer: The Gauss-Green theorem.Trans. Amer. Math. Soc. 58 (1945), 44-76. Zbl 0060.14102, MR 0013786, 10.1090/S0002-9947-1945-0013786-6 Reference: [5] H. Federer: Geometric Measure Theory.Springer-Verlag, 1969. Zbl 0176.00801, MR 0257325 Reference: [6] I. Gohberg R. Markus: Some remarks on topologically equivalent norms.Izv. Mold. Fil. Akad. Nauk SSSR 10(76) (1960), 91-95. (In Russian.) Reference: [7] J. Král: Integral Operators in Potential Theory.Lecture Notes in Mathematics vol. 823, Springer-Verlag, 1980. MR 0590244, 10.1007/BFb0091035 Reference: [8] J. Král: Problème de Neumann faible avec condition frontière dans $L^1$.Séminaire de Théorie du Potentiel (Université Paris VI) No. 9. Lecture Notes in Mathematics 1393, Springer-Verlag, 1989, pp. 145-160. Reference: [9] J. Král: The Fredholm method in potential theory.Trans. Amer. Math. Soc. 125 (1996), 511-547. MR 0209503, 10.2307/1994580 Reference: [10] J. Král D. Medková: Angular limits of double layer potentials.Czechoslovak Math. J. 45 (1995), 267-292. MR 1331464 Reference: [11] J. Král W. Wendland: Some examples concerning applicability of the Fredholm-Radon method in potential theory.Apl. Mat. 31 (1986), 293-308. MR 0854323 Reference: [12] V. G. Maz'ya: Boundary Integral Equations.Encyclopaedia of Mathematical Sciences 27, Analysis IV, Springer-Verlag, 1991. Zbl 0778.00012, MR 1098507 Reference: [13] I. Netuka: Generalized Robin problem in potential theory.Czechoslovak Math. J. 22 (1970), 312-324. MR 0294673 Reference: [14] I. Netuka: The third boundary value problem in potential theory.Czechoslovak Math. J. 22 (1972), 554-580. Zbl 0242.31007, MR 0313528 Reference: [15] J. Neveu: Bases Mathématiques du Calcul des Probabilités.Masson et Cie, Paris, 1964. Zbl 0137.11203, MR 0198504 Reference: [16] L. C. Young: A theory of boundary values.Proc. London Math. Soc. 14A (1965), 300-314. Zbl 0147.07802, MR 0180891 .

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