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<p>We establish a Gagliardo–Nirenberg type inequality valid for log-concave functions. Key ingredients of the proof are the logarithmic Sobolev inequality and an entropy estimate for log-concave functions. It turns out that our result provides an almost optimal estimate when the dimension and the exponents are appropriately chosen. Moreover, as an application of our Gagliardo–Nirenberg type inequality, we study two kinds of eigenvalue problems for nonlinear elliptic operators. Namely, a fully nonlinear uniformly elliptic operator and the <i>p</i>-Laplacian. For both the problems we derive lower bounds for eigenvalues.</p>

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Gagliardo–Nirenberg type inequality for log-concave functions and its application to nonlinear elliptic eigenvalue problems

  • 期刊论文

  • 发表日期：

    2026年5月8日

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  • Yasuhiro Fujita,
  • Nao Hamamuki,
  • Ren Igarashi

摘要

We establish a Gagliardo–Nirenberg type inequality valid for log-concave functions. Key ingredients of the proof are the logarithmic Sobolev inequality and an entropy estimate for log-concave functions. It turns out that our result provides an almost optimal estimate when the dimension and the exponents are appropriately chosen. Moreover, as an application of our Gagliardo–Nirenberg type inequality, we study two kinds of eigenvalue problems for nonlinear elliptic operators. Namely, a fully nonlinear uniformly elliptic operator and the p-Laplacian. For both the problems we derive lower bounds for eigenvalues.