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BV Functions on Convex Domains in Wiener Spaces

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Abstract

We study functions of bounded variation defined in an abstract Wiener space X, relating the variation of a function u on a convex open set \({\Omega }\subset X\) to the behavior near t=0 of T(t)u, T(t) being the Ornstein–Uhlenbeck semigroup in Ω.

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References

  1. Ambrosio, L., Figalli, A.: Surface measure and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces. Ann. Fac. Sci. Toulouse Math 20, 407–438 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambrosio, L., Figalli, A., Runa, E.: On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl 24, 111–122 (2013)

    MATH  MathSciNet  Google Scholar 

  3. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs (2000)

  4. Ambrosio, L., Maniglia, S., Miranda Jr, M., Pallara, D.: BV functions in abstract Wiener spaces. J. Funct. Anal. 258, 785–813 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ambrosio, L., Miranda Jr, M., Pallara, D.: Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability, Discrete Contin. Dyn. Syst 28, 591–606 (2010)

    MATH  Google Scholar 

  6. Angiuli, L., Miranda Jr, M., Pallara, D., Paronetto, F.: BV functions and parabolic initial boundary value problems on domains, Ann. Mat. Pura Appl 188(4), 297–311 (2009)

    Article  MATH  Google Scholar 

  7. Bakry, D., Émery, M.: Diffusions hypercontractives Seminaire de probabilités, XIX, 1983/84, vol. 1123, pp 177–206. Springer, Berlin (1985)

    Google Scholar 

  8. Bertoldi, M., Fornaro, S.: Gradient estimates in parabolic problems with unbounded coefficients. Studia Math. 165, 221–254 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bogachev, V.I.: Gaussian Measures. American Mathematical Society (1998)

  10. Bogachev, V.I., Pilipenko, A.Yu., Shaposhnikov, A.V.: Sobolev functions on infinite-dimensional domains. J. Math. Anal. Appl. 419, 1023–1044 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bramanti, M., Miranda jr, M., Pallara, D.: Two characterization of BV functions on Carnot groups via the heat semigroup. Int. Math. Res. Not. 17, 3845–3876 (2012)

    Google Scholar 

  12. Caselles, V., Miranda Jr. M., Novaga, M.: Total variation and Cheeger sets in Gauss space. J. Funct. Anal. 259(6), 1491–1516 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Caselles, V., Lunardi, A., Miranda Jr, M., Novaga, M.: Perimeter of sublevel sets in infinite dimensional spaces. Adv. Calc. Var 5, 59–76 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Celada, P., Lunardi, A.: Traces of Sobolev functions on regular surfaces in infinite dimensions. J. Funct. Anal. 266, 1948–1987 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cianchi, A., Fusco, N., Maggi, F., Pratelli, A.: On the isoperimetric deficit in Gauss space. Amer. J. Math. 133(1), 131–186 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Da Prato, G., Lunardi, A.: Elliptic operators with unbounded drift coefficients and Neumann boundary condition. J. Differential Equations 198, 35–52 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dunford, N., Schwartz, J.T.: Linear operators I. Wiley (1958)

  18. Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93, 418–491 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fukushima, M.: BV functions and distorted Ornstein-Uhlenbeck processes over the abstract Wiener space. J. Funct. Anal. 174, 227–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fukushima, M., Hino, M.: On the space of BV functions and a Related Stochastic Calculus in Infinite Dimensions. J. Funct. Anal. 183, 245–268 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Güneysu, B., Pallara, D.: Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below, preprint

  22. Hino, M.: Sets of finite perimeter and the Hausdorff–Gauss measure on the Wiener space. J. Funct. Anal. 258, 1656–1681 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hino, M.: Dirichlet spaces on H-convex sets in Wiener space. Bull. Sci. Math 135, 667–683 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hino, M., Uchida, H.: Reflecting Ornstein-Uhlenbeck processes on pinned path spaces, Proceedings of RIMS Workshop on Analysis, Stochastic, 111–128, Applications, RIMS Kokyuroku Bessatsu, B6, Kyoto (2008)

  25. Miranda Jr, M., Novaga, M., Pallara, D.: An introduction to BV functions in Wiener spaces, Advanced Studies in Pure Mathematics, to appear

  26. Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. Revised and enlarged ed., Academic Press (1980)

  27. Qian, Z.: A gradient estimate on a manifold with convex boundary. Proc. Roy. Soc. Edinburgh Sect. A 127, 171–179 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. Savarè, G.: Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in \(RCD(K,\infty )\) metric measure spaces Disc. Cont. Dyn. Sist. A 34, 1641–1661 (2014)

    Article  MATH  Google Scholar 

  29. Wang, F.-Y.: On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups, Probab. Theory Related Fields 108, 87–101 (1997)

    Article  MATH  Google Scholar 

  30. Wang, F.-Y., Yan, L.: Gradient estimate on convex domains and applications. Proc. Amer. Math. Soc 141, 1067–1081 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Yosida, K.: Functional analysis. Sixth ed., Springer (1980)

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Parma, Parma, Italy

    Alessandra Lunardi

  2. Dipartimento di Matematica e Informatica, Università di Ferrara, Ferrara, Italy

    Michele Miranda Jr.

  3. Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Lecce, Italy

    Diego Pallara

Authors
  1. Alessandra Lunardi
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  2. Michele Miranda Jr.
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  3. Diego Pallara
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Correspondence to Diego Pallara.

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Lunardi, A., Miranda, M. & Pallara, D. BV Functions on Convex Domains in Wiener Spaces. Potential Anal 43, 23–48 (2015). https://doi.org/10.1007/s11118-015-9462-9

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  • DOI: https://doi.org/10.1007/s11118-015-9462-9

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