JournalsaihpcVol. 35, No. 5pp. 1355–1376

Gagliardo–Nirenberg inequalities and non-inequalities: The full story

  • Petru Mironescu

    Université de Lyon, Université Lyon 1, CNRS UMR 5208 Institut Camille Jordan, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France, Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Griviţei 21, 010702 Bucureşti, Romania
  • Haïm Brezis

    Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA, Mathematics Department, Technion - Israel Institute of Technology, Haifa, 32000, Israel
Gagliardo–Nirenberg inequalities and non-inequalities: The full story cover
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Abstract

We investigate the validity of the Gagliardo–Nirenberg type inequality

fWs,p(Ω)fWs1,p1(Ω)θfWs2,p2(Ω)1θ,\|f\|_{W^{s,p}(\mathrm{\Omega })}≲\|f\|_{W^{s_{1},p_{1}}(\mathrm{\Omega })}^{\theta }\|f\|_{W^{s_{2},p_{2}}(\mathrm{\Omega })}^{1−\theta },

with ΩRN\mathrm{\Omega } \subset \mathbb{R}^{N}. Here, 0s1ss20 \leq s_{1} \leq s \leq s_{2} are non negative numbers (not necessarily integers), 1p1,p,p21 \leq p_{1},p,p_{2} \leq \infty , and we assume the standard relations

s=θs1+(1θ)s2,1/p=θ/p1+(1θ)/p2 for some θ(0,1).s = \theta s_{1} + (1−\theta )s_{2},\:1/ p = \theta / p_{1} + (1−\theta )/ p_{2}\text{ for some }\theta \in (0,1).

By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when s1,s2,ss_{1},s_{2},s are integers. It turns out that (1) holds for “most” of values of s1,,p2s_{1},…,p_{2}, but not for all of them. We present an explicit condition on s1,s2,p1,p2s_{1},s_{2},p_{1},p_{2} which allows to decide whether (1) holds or fails.

Cite this article

Petru Mironescu, Haïm Brezis, Gagliardo–Nirenberg inequalities and non-inequalities: The full story. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1355–1376

DOI 10.1016/J.ANIHPC.2017.11.007