The Equivalence Of Gagliardo-Nirenberg-Sobolev Inequality?Nash Inequality And Isoperimetric Inequality

In this paper,we study three important inequalities in partial differential e-quations:Gagliardo-Nirenberg-Sobolev inequality,Nash inequality and Isoperimetric inequality.For the proof of these three inequalities,we mainly apply partial differ-ential equations' ideas.To be more precise,we use fundamental solution of Laplace equation and potential theory to prove Gagliardo-Nirenberg-Sobolev inequality,ap-ply the Poisson formula of heat equation to prove Nash inequality and use a Pois-son equation's solution to approximate Isoperimetric inequality.On the other hand,these three inequalities are equivalent in some certain sense.Firstly,we prove that Gagliardo-Nirenberg-Sobolev inequality in Ll is equivalent to Isoperimetric inequali-ty.Secondly,we prove Gagliardo-Nirenberg-Sobolev inequality in L2 is equivalent to Nash inequality.Further-more,we find a generalized Nash inequality,that is Nash inequality in LP(1<p<n).At last we prove the generalized Nash inequality is equivalent to Gagliardo-Nirenberg-Sobolev inequality in 1<p<n.