Geodesic
Approximate calculation of the defect of a Lipschitz cylindrical condenser
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 906-926.

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We introduce the notion of defect of a Lipschitz cylindrical condenser. It is the difference between the capacity of the condenser and its Ahlfors integral. We calculate the defect approximately for condensers over arbitrary open sets. For a condenser over an inner uniform domain the quantity obtained is comparable to the sum of the squares of the seminorms of the plates in a weighted homogeneous Slobodetskii space. This uses the characterization of inner uniform domains by the following property: every inner metric ball is a centered John domain.
Keywords: Ahlfors integral, capacity, defect, Lipschitz domain.
Mots-clés : condenser, inner uniform domain 
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A. I. Parfenov. Approximate calculation of the defect of a Lipschitz cylindrical condenser. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 906-926. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a132/

[1] V. Anandam, Harmonic Functions and Potentials on Finite or Infinite Networks, Springer, Heidelberg, 2011 | MR | Zbl

[2] G.D. Anderson, “Derivatives of the conformal capacity of extremal rings”, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 10 (1985), 29–46 | DOI | MR | Zbl

[3] N. Arcozzi, “Capacity of shrinking condensers in the plane”, J. Funct. Anal., 263 (2012), 3102–3116 | DOI | MR | Zbl

[4] N. Arcozzi, R. Rochberg, E.T. Sawyer, B.D. Wick, “Potential theory on trees, graphs and Ahlfors-regular metric spaces”, Potential Anal., 41 (2014), 317–366 | DOI | MR | Zbl

[5] Z. Balogh, A. Volberg, “Geometric localization, uniformly John property and separated semihyperbolic dynamics”, Ark. Mat., 34 (1996), 21–49 | DOI | MR | Zbl

[6] Yu.A. Brudnyi, S.G. Krein, E.M. Semenov, “Interpolation of linear operators”, Itogi Nauki Tekh., Ser. Mat. Anal., 24, 1986, 3–163 (Russian) | MR

[7] A. Colesanti, K. Nyström, P. Salani, J. Xiao, D. Yang, G. Zhang, “The Hadamard variational formula and the Minkowski problem for $p$-capacity”, Adv. Math., 285 (2015), 1511–1588 | DOI | MR | Zbl

[8] A.R. Elcrat, K.G. Miller, “Variational formulas on Lipschitz domains”, Trans. Amer. Math. Soc., 347 (1995), 2669–2678 | DOI | MR | Zbl

[9] I. Fragalà, F. Gazzola, M. Pierre, “On an isoperimetric inequality for capacity conjectured by Pólya and Szegö”, J. Differ. Equations, 250 (2011), 1500–1520 | DOI | MR | Zbl

[10] P. Gyrya, L. Saloff-Coste, Neumann and Dirichlet Heat Kernels in Inner Uniform Domains, Astérisque, 336, Société Mathématique de France, Paris, 2011 | MR | Zbl

[11] P. Hajłasz, “Sobolev inequalities, truncation method, and John domains”, Papers on analysis, 83, eds. J. Heinonen et al., Univ. Jyväskylä, Jyväskylä, 2001, 109–126 | MR | Zbl

[12] F. John, “Rotation and strain”, Comm. Pure Appl. Math., 14 (1961), 391–413 | DOI | MR | Zbl

[13] R. Laugesen, “Extremal problems involving logarithmic and Green capacity”, Duke Math. J., 70 (1993), 445–480 | DOI | MR | Zbl

[14] R.D. Lyons, Y. Peres, Probability on Trees and Networks, Cambridge University Press, New York, 2016 | MR | Zbl

[15] O. Martio, J. Sarvas, “Injectivity theorems in plane and space”, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 4 (1978/79), 383–401 | DOI | MR

[16] V.G. Maz'ya, Sobolev Spaces. With Applications to Elliptic Partial Differential Equations, Springer, Berlin, 2011 | MR | Zbl

[17] A.I. Parfenov, “Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem”, Sib. Adv. Math., 27 (2017), 274–304 | DOI | MR | Zbl

[18] Yu.G. Reshetnyak, “Integral representations of differentiable functions in domains with non-smooth boundary”, Sib. Math. J., 21, 108–116 (Russian) | MR | Zbl

[19] B. Rodin, S.E. Warschawski, “Extremal length and the boundary behavior of conformal mappings”, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 2 (1976), 467–500 | DOI | MR | Zbl

[20] B. Rodin, S.E. Warschawski, “A necessary and sufficient condition for the asymptotic version of Ahlfors' distortion property”, Trans. Amer. Math. Soc., 276 (1983), 281–288 | MR | Zbl

[21] P.M. Soardi, Potential Theory on Infinite Networks, Springer, Berlin, 1994 | MR | Zbl

[22] H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983 | MR | Zbl

[23] J. Väisälä, “Uniform domains”, Tôhoku Math. J., 40 (1988), 101–118 | DOI | MR | Zbl