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user72012
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Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $GN(N,p,q,r)=\sup\frac{∫|u|^{r}dx}{(∫|∇u|^{p}dx)^{a}(∫|u|^{q}dx)^{1-a}}$$(∫|u|^{r}dx)^{\frac{1}{r}} \leq GN(N,p,q,r)(∫|∇u|^{p}dx)^{\frac{a}{p}}(∫|u|^{q}dx)^{\frac{1-a}{q}}$. Can the supremumbest constant GN(N,p,q,r) be achieved in some cases? Does anyone know good references about this subject?

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $GN(N,p,q,r)=\sup\frac{∫|u|^{r}dx}{(∫|∇u|^{p}dx)^{a}(∫|u|^{q}dx)^{1-a}}$. Can the supremum be achieved in some cases? Does anyone know good references about this subject?

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $(∫|u|^{r}dx)^{\frac{1}{r}} \leq GN(N,p,q,r)(∫|∇u|^{p}dx)^{\frac{a}{p}}(∫|u|^{q}dx)^{\frac{1-a}{q}}$. Can the best constant GN(N,p,q,r) be achieved in some cases? Does anyone know good references about this subject?

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user72012
user72012
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Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $GN(N,p,q,r)=\sup\frac{∫|u|^{r}dx}{(∫|∇u|^{p}dx)^{a}(∫|u|^{q}dx)^{1-a})}$$GN(N,p,q,r)=\sup\frac{∫|u|^{r}dx}{(∫|∇u|^{p}dx)^{a}(∫|u|^{q}dx)^{1-a}}$. Can the supremum be achieved in some cases? Does anyone know good references about this subject?

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $GN(N,p,q,r)=\sup\frac{∫|u|^{r}dx}{(∫|∇u|^{p}dx)^{a}(∫|u|^{q}dx)^{1-a})}$. Can the supremum be achieved in some cases? Does anyone know good references about this subject?

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $GN(N,p,q,r)=\sup\frac{∫|u|^{r}dx}{(∫|∇u|^{p}dx)^{a}(∫|u|^{q}dx)^{1-a}}$. Can the supremum be achieved in some cases? Does anyone know good references about this subject?

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user72012
user72012
  • 233
  • 1
  • 7

Extremal functions for Gagliardo-Nirenberg inequality

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $GN(N,p,q,r)=\sup\frac{∫|u|^{r}dx}{(∫|∇u|^{p}dx)^{a}(∫|u|^{q}dx)^{1-a})}$. Can the supremum be achieved in some cases? Does anyone know good references about this subject?