Ngô Quốc Anh

December 31, 2011

A Hardy-Moser-Trudinger inequality: A conjecture by Wang and Ye

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 21:48

Let B denote the standard unit disk in \mathbb R^2. The famous Moser–Trudinger inequality says that

\displaystyle\int_B {\exp \left( {\frac{{4\pi {u^2}}}{{\left\| {\nabla u} \right\|_2^2}}} \right)dx} \leqslant C < \infty ,\quad\forall u \in H_0^1(B)\backslash \{ 0\}

holds. There is another important inequality in analysis, the Hardy inequality which claims that

\displaystyle H(u) = \int_B {|\nabla u{|^2}dx} - \int_B {\frac{{{u^2}}}{{{{(1 - |x{|^2})}^2}}}dx} \geqslant 0,\quad\forall u \in H_0^1(B)

holds. The one H is usuall called the Hardy functional. One can immediately see that

\displaystyle\frac{{4\pi {u^2}}}{{\left\| {\nabla u} \right\|_2^2}} \leqslant \dfrac{{4\pi {u^2}}}{{\displaystyle\int_B {|\nabla u{|^2}dx} - \int_B {\frac{{{u^2}}}{{{{(1 - |x{|^2})}^2}}}dx} }}

for any u \in H_0^1(B)\backslash \{ 0\}. Recently, in a paper accepted in Advances in Mathematics journal, Wang and Ye proved that there exists a constant C_0 >0 such that the following

\displaystyle\int_B {\frac{{4\pi {u^2}}}{{H(u)}}dx} \leqslant C_0 < \infty ,\quad\forall u \in \mathcal H(B^n)\backslash \{ 0\}

where B^n is the unit ball in \mathbb R^n, n \geqslant 2 and \mathcal H=\mathcal H(B^n) is the complement of C_0^\infty(B^n) with respect to the following norm \|u\|_{\mathcal H}=\sqrt{H(u)}.

Let us go back to the case n=2. They then defined

\displaystyle {H_d}(u) = \int_\Omega {|\nabla u{|^2}dx} - \frac{1}{4}\int_\Omega {\frac{{{u^2}}}{{d{{(x,\partial \Omega )}^2}}}dx} > 0,\quad \forall u \in H_0^1(\Omega )\backslash \{ 0\}

where \Omega is a regular, bounded and convex domain sitting in \mathbb R^2. They then conjectured that the following

\displaystyle\int_\Omega {\frac{{4\pi {u^2}}}{{{H_d}(u)}}dx} \leqslant C(\Omega ) < \infty ,\quad\forall u \in {\mathcal H_d}(\Omega )\backslash \{ 0\}

still holds for some constant C(\Omega)>0 where {\mathcal H_d}(\Omega ) denotes the completion of C_0^\infty (\Omega) with the corresponding norm associated with H_d. Apparently, the conjecture holds true for \Omega = B.

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