Abstract
Note that the Gaussian perimeter element dPγ = γdP exists as the (n − 1)-dimensional area element dP with the weight γ. So, \(\mathbb{G}^{n}\) merits a geometric capacity analysis on the functions of bounded variation which are differentiable in the weakest measure theoretic sense. In this chapter we utilize four sections to deal with a Gaussian analogue of the bounded variation capacity of a subset of \(\mathbb{R}^{n}\).
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D.R. Adams, Choquet integrals in potential theory. Publ. Math. 42, 3–66 (1998)
D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory (Springer, Berlin/Heidelberg, 1996)
D.R. Adams, J. Xiao, Morrey spaces in harmonic analysis. Ark. Mat. 50, 201–230 (2012)
L. Ambrosio, M. Miranda Jr, S. Maniglia, D. Pallara, BV functions in abstract Wiener spaces. J. Funct. Anal. 258, 785–813 (2010)
S.G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25, 206–214 (1997)
E.A. Carlen, C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement. Calc. Var. Partial Differ. Equ. 13, 1–18 (2001)
V. Caselles, M. Miranda Jr, M. Novaga, Total variation and Cheeger sets in Gauss space. J. Funct. Anal. 259, 1491–1516 (2010)
L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (CRC, Boca Raton, 1992)
H. Hakkarainen, J. Kinnunen, The BV-capacity in metric spaces. Manuscripta Math. 132, 51–73 (2010)
J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations (Oxford University Press, New York, 1993)
R. Latala, On some inequalities for Gaussian measures. Proc. ICM 2, 813–822 (2002)
M. Ledoux, Isoperimetry and Gaussian analysis. Lect. Notes Math. 1648, 165–294 (1996)
J. Martín, M. Milman, Isoperimetry and symmetrization for logarithmic Sobolev inequalities. J. Funct. Anal. 256, 149–178 (2009)
V. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd, revised and augmented edn. (Springer, Heidelberg, 2011), xxviii+866 pp.
S.-T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Scient. Éc. Norm. Sup. 8, 487–507 (1975)
W.P. Ziemer, Weakly Differentiable Functions (Springer, New York, 1989)
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Liu, L., Xiao, J., Yang, D., Yuan, W. (2018). Gaussian BV -Capacity. In: Gaussian Capacity Analysis. Lecture Notes in Mathematics, vol 2225. Springer, Cham. https://doi.org/10.1007/978-3-319-95040-2_6
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DOI: https://doi.org/10.1007/978-3-319-95040-2_6
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