Ngô Quốc Anh

Tháng Hai 28, 2008

BĐT tích phân

Chuyên mục: Các Bài Tập Nhỏ, Giải Tích 2 — Ngô Quốc Anh @ 13:40

Let [0,1]->R a differentiable function with f' continous so that :

int_{0}^{1}f(x)dx= int_{0}^{1}xf(x)dx=1.

Prove that int_{0}^{1}(f'(x))^{2}dx geq 30.

Solution. Integrating by parts we get that

intlimits_{0}^{1}xf'(x)dx=f(1)-1=alpha,

and

intlimits_{0}^{1}x^{2}f'(x)dx=f(1)-2=beta.

Thus

(beta-alpha)^{2}=1=left(intlimits_{0}^{1}(x-x^{2})f'(x)dxright)^{2}leq intlimits_{0}^{1}(x-x^{2})^{2}dxcdot intlimits_{0}^{1}f'(x)^{2}dx.

It follows that intlimits_{0}^{1}f'(x)^{2}dxgeq 30.

Tháng Hai 25, 2008

KQ6 – Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p-Laplacian type

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 22:10

This paper deals with the multiplicity of weak solutions in W_0^1 left( Omega right) to a class of nonuniformly elliptic equations of the form

- operatorname{div}left( {aleft( {x,nabla u} right)} right) = hleft( x right)left| u right|^{r - 1} u + gleft( x right)left| u right|^{s - 1} u

in a bounded domain Omega of Rset^N. Here a satisfies left| {aleft( {x,xi } right)} right| leqq c_0 left( {h_0 left( x right) + h_1 left( x right)left| xi right|^{p - 1} } right) for all xi in Rset^N, a.e. x in Omega, h_0 in L^{frac{p}{{p - 1}}} left( Omega right), h_1 in L_{loc}^1 left( Omega right), h_1(x) geqq 1 for a.e. x in Omega, 1<r<p-1<s<(Np-N+p)/(N-p).

http://dx.doi.org/10.1016/j.na.2008.02.033

KQ5 – On the Inverse of an Integral Inequality

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 22:02

http://eureka.vu.edu.au/~rgmia/v10n4/Kq24_en.pdf

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