World Scientific
  • Search
    Quick Search in Journals
    Access type:
    Quick Search anywhere
    Access type:
    Advanced Search
  • My Cart
  •   
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×
No Access

On anisotropic Sobolev spaces

    https://doi.org/10.1142/S0219199718500177Cited by:8 (Source: Crossref)
    Previous
    Next

    Abstract

    We investigate two types of characterizations for anisotropic Sobolev and BV spaces. In particular, we establish anisotropic versions of the Bourgain–Brezis–Mironescu formula, including the magnetic case both for Sobolev and BV functions.

    AMSC: 46E35, 28D20, 82B10, 49A50

    References

    • 1. J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J. 45 (1978) 847–883. Crossref, Web of Science, Google Scholar
    • 2. J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan’s 60th Birthday, eds. J. L. Menaldi, E. Rofman and A. Sulem (IOS Press, Amsterdam, 2001), pp. 439–455. Google Scholar
    • 3. J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for Ws,p when s ↑ 1 and applications, J. Anal. Math. 87 (2002) 77–101. Crossref, Web of Science, Google Scholar
    • 4. J. Bourgain, H. Brezis and H.-M. Nguyen, A new estimate for the topological degree, C. R. Math. Acad. Sci. Paris 340 (2005) 787–791. Crossref, Web of Science, Google Scholar
    • 5. J. Bourgain and H.-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris 343 (2006) 75–80. Crossref, Web of Science, Google Scholar
    • 6. H. Brezis, How to recognize constant functions. Connections with Sobolev spaces, Russian Math. Surveys 57 (2002) 693–708. Crossref, Web of Science, Google Scholar
    • 7. H. Brezis and H.-M. Nguyen, On a new class of functions related to VMO, C. R. Acad. Sci. Paris 349 (2011) 157–160. Crossref, Web of Science, Google Scholar
    • 8. H. Brezis and H.-M. Nguyen, The BBM formula revisited, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 515–533. Crossref, Google Scholar
    • 9. H. Brezis and H.-M. Nguyen, Two subtle convex nonlocal approximations of the BV-norm, Nonlinear Anal. 137 (2016) 222–245. Crossref, Web of Science, Google Scholar
    • 10. H. Brezis and H.-M. Nguyen, Non-convex, non-local functionals converging to the total variation, C. R. Acad. Sci. Paris 355 (2017) 24–27. Crossref, Web of Science, Google Scholar
    • 11. H. Brezis and H.-M. Nguyen, Non-local functionals related to the total variation and connections with Image Processing; http://arxiv.org/abs/1608.08204. Google Scholar
    • 12. J. Davila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations 15 (2002) 519–527. Crossref, Web of Science, Google Scholar
    • 13. L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1992). Google Scholar
    • 14. B. Fleury, O. Guédon and G. Paouris, A stability result for mean width of Lp-centroid bodies, Adv. Math. 214 (2007) 865–877. Crossref, Web of Science, Google Scholar
    • 15. C. Haberl and F. Schuster, General Lp affine isoperimetric inequalities, J. Differential Geom. 83 (2009) 1–26. Crossref, Web of Science, Google Scholar
    • 16. T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, in Mathematical Physics, Spectral Theory and Stochastic Analysis, Operator Theory Advances and Applicattions, Vol. 232 (Birkhäuser/Springer, Basel, 2013), pp. 247–297. Crossref, Google Scholar
    • 17. M. Ludwig, Ellipsoids and matrix valued valuations, Duke Math. J. 119 (2003) 159–188. Crossref, Web of Science, Google Scholar
    • 18. M. Ludwig, Anisotropic fractional Sobolev norms, Adv. Math. 252 (2014) 150–157. Crossref, Web of Science, Google Scholar
    • 19. M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom. 96 (2014) 77–93. Crossref, Web of Science, Google Scholar
    • 20. H.-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689–720. Crossref, Web of Science, Google Scholar
    • 21. H.-M. Nguyen, Further characterizations of Sobolev spaces, J. Eur. Math. Soc. 10 (2008) 191–229. Crossref, Web of Science, Google Scholar
    • 22. H.-M. Nguyen, Some inequalities related to Sobolev norms, Calc. Var. Partial Differential Equations 41 (2011) 483–509. Crossref, Web of Science, Google Scholar
    • 23. H.-M. Nguyen, Γ-convergence, Sobolev norms, and BV functions, Duke Math. J. 157 (2011) 495–533. Crossref, Web of Science, Google Scholar
    • 24. H.-M. Nguyen, A. Pinamonti, M. Squassina and E. Vecchi, New characterizations of magnetic Sobolev spaces, Adv. Nonlinear Anal. 7 (2018) 227–245. Crossref, Web of Science, Google Scholar
    • 25. A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain–Brezis–Mironescu formula, Adv. Calc. Var. (2017), to appear. Crossref, Web of Science, Google Scholar
    • 26. A. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations 19 (2004) 229–255. Crossref, Web of Science, Google Scholar
    • 27. R. Schneider and W. Weil, Stochastic and Integral Geometry, Probability and its Applications (Springer-Verlag, Berlin, 2008). Crossref, Google Scholar
    • 28. M. Squassina and B. Volzone, Bourgain–Brezis—Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris 354 (2016) 825–831. Crossref, Web of Science, Google Scholar