Previous |  Up |  Next


single layer potential; weak normal derivative; essential norm
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 $\sigma$.
[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
[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.)
[3] N. Dunford J. T Schwartz W. G. Bade R. G. Barth: Linear Operators, Part I. Interscience Publishers, New York, 1958. MR 0117523
[4] H. Federer: The Gauss-Green theorem. Trans. Amer. Math. Soc. 58 (1945), 44-76. DOI 10.1090/S0002-9947-1945-0013786-6 | MR 0013786 | Zbl 0060.14102
[5] H. Federer: Geometric Measure Theory. Springer-Verlag, 1969. MR 0257325 | Zbl 0176.00801
[6] I. Gohberg R. Markus: Some remarks on topologically equivalent norms. Izv. Mold. Fil. Akad. Nauk SSSR 10(76) (1960), 91-95. (In Russian.)
[7] J. Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics vol. 823, Springer-Verlag, 1980. MR 0590244
[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.
[9] J. Král: The Fredholm method in potential theory. Trans. Amer. Math. Soc. 125 (1996), 511-547. DOI 10.2307/1994580 | MR 0209503
[10] J. Král D. Medková: Angular limits of double layer potentials. Czechoslovak Math. J. 45 (1995), 267-292. MR 1331464
[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
[12] V. G. Maz'ya: Boundary Integral Equations. Encyclopaedia of Mathematical Sciences 27, Analysis IV, Springer-Verlag, 1991. MR 1098507 | Zbl 0778.00012
[13] I. Netuka: Generalized Robin problem in potential theory. Czechoslovak Math. J. 22 (1970), 312-324. MR 0294673
[14] I. Netuka: The third boundary value problem in potential theory. Czechoslovak Math. J. 22 (1972), 554-580. MR 0313528 | Zbl 0242.31007
[15] J. Neveu: Bases Mathématiques du Calcul des Probabilités. Masson et Cie, Paris, 1964. MR 0198504 | Zbl 0137.11203
[16] L. C. Young: A theory of boundary values. Proc. London Math. Soc. 14A (1965), 300-314. MR 0180891 | Zbl 0147.07802
Partner of
EuDML logo