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PAPER: On a class of N-dimensional anisotropic Sobolev inequality and related results

REF. DATE:24 March 2010 CREATED/MODIFIED: 14 January 2016 344 views No Comment

On a class of N-dimensional anisotropic Sobolev inequality and related results (with J. Chen), pp.31, preprint.


ABSTRACT: We determine the best (smallest) constant \alpha in the Anisotropic Sobolev inequality of the form

    \[\|u\|_p^p \leq  \alpha\left\|u\right\|_2^{\frac{2(2N-1)+(3-2N)p}{2}}\,\|u_x\|_2^{\frac{N(p-2)}{2}} \,\prod_{k=1}^{N-1}\|D_x^{-1}\partial_{y_k}u\|_2^{\frac{p-2} {2}}\]

and the best (smallest) constant \beta in the inequality

    \[\|u\|_{p_*}^{p_*} \leq \beta \|u_x\|_2^{\frac{2N}{2N-3}}\, \prod_{k=1}^{N-1}\|D_x^{-1}\partial_{y_k}u\|_2^{\frac{2}{2N-3}},\]

where (x, y_1, \cdots, y_{N-1})\in \mathbb{R}^N with N\geq 3 and 2 < p < p_* = {\frac{2(2N-1)}{2N-3}}. These best constants are obtained by introducing a new method and using variational techniques. The method introduced here seems to have independent interest. We also use this method to find the best constant of the Gagliardo-Nirenberg interpolation inequality involving the m-Laplacian

    \[\|u\|_q^q \leq C_{Gm}\:\|\nabla u\|_m^{\frac{N(q-m)}{m}}\:\|u\|_m^{\frac{Nm-(N-m)q}{m}},\qquad u\in W^{1,m}(\mathbb{R}^N),\]

where 1 < m < q < m^\ast and m^\ast = Nm/(N-m) for 1 < m < N and m^\ast = \infty for m \geq N.


Extra info: (P074)

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