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# Article

 Title: Essential norms of the Neumann operator of the arithmetical mean (English) Author: Král, Josef Author: Medková, Dagmar Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 126 Issue: 4 Year: 2001 Pages: 669-690 Summary lang: English . Category: math . Summary: Let $K\subset \mathbb{R}^m$ ($m\ge 2$) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let ${C}_0^{(1)}$ be the class of all continuously differentiable real-valued functions with compact support in $\mathbb{R}^m$ and denote by $\sigma _m$ the area of the unit sphere in $\mathbb{R}^m$. With each $\varphi \in {C}_0^{(1)}$ we associate the function $W_K\varphi (z)={1\over \sigma _m}\underset{\mathbb{R}^m \setminus K}{\rightarrow }\int \mathop {\mathrm grad}\nolimits \varphi (x)\cdot {z-x\over |z-x|^m}\ x$ of the variable $z\in K$ (which is continuous in $K$ and harmonic in $K\setminus B$). $W_K\varphi$ depends only on the restriction $\varphi |_B$ of $\varphi$ to the boundary $B$ of $K$. This gives rise to a linear operator $W_K$ acting from the space ${C}^{(1)}(B)=\lbrace \varphi |_B; \varphi \in {C}_0^{(1)}\rbrace$ to the space ${C}(B)$ of all continuous functions on $B$. The operator ${T}_K$ sending each $f\in {C}^{(1)}(B)$ to ${T}_Kf=2W_Kf-f \in {C}(B)$ is called the Neumann operator of the arithmetical mean; it plays a significant role in connection with boundary value problems for harmonic functions. If $p$ is a norm on ${C}(B)\supset {C}^{(1)}(B)$ inducing the topology of uniform convergence and $G$ is the space of all compact linear operators acting on ${C}(B)$, then the associated $p$-essential norm of ${T}_K$ is given by $\omega _p {T}_K=\underset{Q\in {G}}{\rightarrow }\inf \sup \bigl \lbrace p[({T}_K-Q)f]; \ f\in {C}^{(1)}(B), \ p(f)\le 1\bigr \rbrace .$ In the present paper estimates (from above and from below) of $\omega _p {T}_K$ are obtained resulting in precise evaluation of $\omega _p {T}_K$ in geometric terms connected only with $K$. (English) Keyword: double layer potential Keyword: Neumann’s operator of the arithmetical mean Keyword: essential norm MSC: 31B10 MSC: 45P05 MSC: 47A30 MSC: 47G10 idZBL: Zbl 0998.31003 idMR: MR1869461 DOI: 10.21136/MB.2001.134114 . Date available: 2009-09-24T21:55:56Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/134114 . Reference:  T. S. Angell, R. E. Kleinman, J. Král: Layer potentials on boundaries with corners and edges.Čas. Pěst. Mat. 113 (1988), 387–402. MR 0981880 Reference:  M. Balavadze, I. Kiguradze, V. Kokilashvili (eds.): Continuum Mechanics and Related Problems of Analysis. Proceedings of the Internat. Symposium Dedicated to the Centenary of Academician N. Muskhelishvili.Tbilisi, 1991. Reference:  Yu. D. Burago, V. G. Maz’ ya: Potential theory and function theory for irregular regions.Zapiski Naučnyh Seminarov LOMI 3 (1967), 1–152. (Russian) Reference:  M. Chlebík: Tricomi potentials. Thesis.Mathematical Institute of the Czechoslovak Academy of Sciences. Praha, 1988. (Slovak) Reference:  H. Federer: Geometric Measure Theory.Springer, Berlin, 1969. Zbl 0176.00801, MR 0257325 Reference:  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:  I. Gohberg, A. Marcus: Some remarks on topologically equivalent norms.Izvestija Mold. Fil. Akad. Nauk SSSR 10 (1960), 91–95. (Russian) Reference:  J. Král: The Fredholm method in potential theory.Trans. Amer. Math. Soc. 125 (1966), 511–547. MR 0209503, 10.1090/S0002-9947-1966-0209503-0 Reference:  J. Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics 823.Springer, Berlin, 1980. MR 0590244 Reference:  J. Král: The Fredholm-Radon method in potential theory.Continuum Mechanics and Related Problems of Analysis. Proceedings of the Internat. Symposium Dedicated to the Centenary of Academician N. Muskhelishvili, Tbilisi, 1991, pp. 390–397. MR 1379845 Reference:  J. Král, D. Medková: Angular limits of double layer potentials.Czechoslovak Math. J. 45 (1995), 267–291. MR 1331464 Reference:  J. Král, D. Medková: Essential norms of a potential theoretic boundary integral operator in $L^1$.Math. Bohem. 123 (1998), 419–436. MR 1667114 Reference:  J. Král, W. L. Wendland: Some examples concerning applicability of the Fredholm -Radon method in potential theory.Aplikace matematiky 31 (1986), 293–308. MR 0854323 Reference:  J. Lukeš, J. Malý: Measure and Integral.Matfyzpress, 1994. MR 2316454 Reference:  V. G Maz’ya: Boundary Integral Equations. Encyclopaedia of Mathematical Sciences vol. 27, Analysis IV.Springer, 1991. 10.1007/978-3-642-58175-5_2 Reference:  L. C. Young: A theory of boundary values.Proc. London Math. Soc. 14A (1965), 300–314. Zbl 0147.07802, MR 0180891 Reference:  W. P. Ziemer: Weakly Differentiable Functions.Springer, 1989. Zbl 0692.46022, MR 1014685 .

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