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Sobolev Conjugate

Nov 14, 20241 min read

PDEsFunctionalAnalysis Given a function whose gradient is in with , we may ask for what values of is the function in ? In other words when does the Gagliardo-Nirenberg-Sobolev Inequality hold? Such a will be called the Sobolev conjugate of p. Taking we see that

Misplaced &\left(\int |u(\lambda x)|^q\space dx\right)^{1/q} &= \left(\frac1{\lambda^n}\int |u(y)|^q\space dy\right)^{1/q} \\ \left(\int ||\nabla u(\lambda x)||^p\space dx\right)^{1/p} &= \left(\frac{\lambda^p}{\lambda^n}\int ||\nabla u(y)||^p\space dy\right)^{1/p} \\ \lambda^{-\frac{n}{q}}||f||_{L^q(\mathbb{R}^n)} &\leq C \lambda^{1-\frac{n}{p}} ||\nabla f||_{L^p(\mathbb{R}^n)} \end{aligned}$$ so that the inequality can only hold when $1 - \frac{n}{p} + \frac{n}{q} = 0$ holds (otherwise one can take $\lambda \to 0$ or $\lambda \to \infty$ for a contradiction) and the Sobolev conjugate of p is $$q := \frac{np}{n-p}$$

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