Ngô Quốc Anh

June 7, 2008

Some important functional inequalities

Hardy’s inequality: Nếu p>1, f(x) \geq 0 and F(x) = \int_0^x f(t) dt, thì

\displaystyle\int_0^{+\infty} \left( \frac{F(x)}{x} \right)^p dx < \left( \frac{p}{p - 1} \right)^p \int_0^{+\infty} f^p( t )dt

trừ trường hợp hàm f(x) \equiv 0. Hằng số ở vế phải là tốt nhất.

Opial‘s inequality: Giả sử y(x) thuộc lớp C^1 trên đoạn [0, h] với y(0)=y(h)=0 and y(x) >0 với mọi 0<x<h. Khi đó ta có

\displaystyle\int_0^h {|y(x)y'(x)|dx} \leqq \frac{h}{4}\int_0^h {|y'(x){|^2}dx}.

Hằng số \frac{h}{4} ở đây là tốt nhất.

Rellich‘s inequality: Giả sử hàm u khả vi vô hạn với giá compắc trong \mathbb R^N trừ điểm gốc. Khi đó ta có bất đẳng thức

\displaystyle\int_{\mathbb{R}^N } {\left| {\Delta u} \right|^2 dx} \geqq \frac{{n^2 \left( {n - 4} \right)^2 }} {{16}}\int_{\mathbb{R}^N } {\left| x \right|^{ - 4} \left| u \right|^2 dx} , \quad n \ne 2.

Serrin‘s inequality: Giả sử hàm u khả vi vô hạn với giá compắc triệt tiêu trên biên \Omega, khi đó

\displaystyle\left( {\int_\Omega {u^{\frac{n} {{n - 1}}} dx} } \right)^{\frac{{n - 1}} {n}} \leqq \frac{1} {{\sqrt {4n} }}\int_\Omega {\left| {\nabla u} \right|dx} .

Caffarelli–Kohn–Nirenberg‘s inequality: Giả sử hàm u khả vi vô hạn với giá compắc trong \mathbb R^N trừ điểm gốc. Khi đó ta có bất đẳng thức

\displaystyle\frac{{\left| {N - \left( {a + b + 1} \right)} \right|}} {2}\int_{\mathbb{R}^N } {\frac{{\left| u \right|^2 }} {{\left| x \right|^{a + b + 1} }}dx} \leqq \sqrt {\int_{\mathbb{R}^N } {\frac{{\left| u \right|^2 }} {{\left| x \right|^{2a} }}dx} } \sqrt {\int_{\mathbb{R}^N } {\frac{{\left| u \right|^2 }} {{\left| x \right|^{2b} }}dx} } .

Gagliardo-Nirenberg-Sobolev‘s inequality: Giả sử hàm u khả vi liên tục với giá compắc trong \mathbb R^N1 \leq p < N. Khi đó ta có bất đẳng thức

\displaystyle\left( {\int_{\mathbb{R}^N } {\left| u \right|^{\frac{{Np}} {{N - p}}} dx} } \right)^{\frac{{N - p}} {{Np}}} \leqq C\left( {p,N} \right)\left( {\int_{\mathbb{R}^N } {\left| {\nabla u} \right|^p dx} } \right)^{\frac{1} {p}} .

Horgan‘s inequality: Giả sử hàm u trơn, khi đó với miền đang xét là bị chặn với biên đủ trơn thì

\displaystyle\int_\Omega {\left| u \right|^3 dx} \leqq \frac{1} {{\sqrt {4\pi } }}\left( {\int_\Omega {\left| u \right|^2 dx} } \right)^{\frac{3} {4}} \left( {\int_\Omega {\left| {\nabla u} \right|^2 dx} } \right)^{\frac{3} {4}} .

8 Comments »

  1. excuse me, how to prove the Trudinger inequality on the torus \mathbb T:

    \displaystyle \parallel u \parallel_{L^p} \leq C \sqrt p \parallel u \parallel_{H^{\frac{1}{2}}}

    thank you very much.

    Comment by xaboblog — January 26, 2013 @ 2:38

  2. Hello, it is great to see your website! I have benefited much from it.

    About Horgan‘s inequality, I am interested in it. Do you have any references about Horgan’s inequality? I cannot find any related literature on the internet. Thank you!

    Comment by Li-Chang hung — February 5, 2014 @ 17:07

    • Hi there, for the Horgan inequality, let me know if the following paper is not useful, see http://dx.doi.org/10.1007/BF01595590.

      Comment by Ngô Quốc Anh — February 5, 2014 @ 20:46

      • Thanks for the quick reply:)

        I have found this paper, where the Horgan inequality is mentioned. It also cites some related references. This inequality is of Sobolev type and is useful in PDEs.

        Comment by Li-Chang hung — February 6, 2014 @ 14:20

      • Do you still need other resources?

        Comment by Ngô Quốc Anh — February 6, 2014 @ 22:35

      • You mean references related to the Horgan inequality?

        Comment by Li-Chang hung — February 9, 2014 @ 14:49

      • Yep, somehow. If you are still not happy with the paper I gave above, just let me know.

        Comment by Ngô Quốc Anh — February 9, 2014 @ 19:04

      • The first paper you showed me is enough. If you have more related references to provide me, I will be very appreciated:)

        Comment by Li-Chang hung — February 9, 2014 @ 22:47


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