A \(\texttt {p}(\cdot )\)-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups

Abstract

We prove a \(\texttt {p}(\cdot )\)-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. We then establish the existence and uniqueness (up to a set of zero \(\texttt {p}(\cdot )\)-capacity) of a minimizer to the Dirichlet energy integral for the variable exponent case.

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Change history

  • 03 January 2019

    In this short note, we remark on the main theorem of Theorem?5.1].

  • 03 January 2019

    In this short note, we remark on the main theorem of Theorem��5.1].

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Correspondence to Thomas Bieske.

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Bieske, T., Freeman, R.D. A \(\texttt {p}(\cdot )\)-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. Anal.Math.Phys. 8, 289–308 (2018). https://doi.org/10.1007/s13324-018-0235-7

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Keywords

  • Poincaré inequality
  • Sobolev spaces with zero boundary values
  • Dirichlet energy minimizer

Mathematics Subject Classification

  • Primary 46E35
  • 35J66
  • 53C17
  • 31C45 35H20
  • Secondary 22E25
  • 31E05