论文标题
Sobolev不平等的稳定性：最小化的存在
Stability for the Sobolev inequality: existence of a minimizer
论文作者
论文摘要
我们证明，稳定性不平等与Sobolev的不等式及其一组优化器$ \ MATHCAL M $相关，并由\ [\ frac {\ frac {\ | \ | \ nabla f \ | _ | _ {l^2（\ Mathbb r^d） \ | f \ | _ {l^\ frac {2d} {d -2}（\ Mathbb r^d）}^2} {\ inf_ { \ qquad \ text {对于每个} f \ in \ dot {h}^1（\ mathbb r^d），\]，\]，\ \]，\]是bianchi和egnell引起的，每$ d \ geq 3 $都承认一个最小化器。我们的证明包括适当的改进，可以追溯到布雷兹和利布。作为一种关键成分，我们建立了严格的不等式$ c_ {be} <2-2^\ frac {d-2} {d} $，这意味着不能最小化两个渐近性非相互作用气泡的序列。我们的论点实际上涵盖了对任意分数指数$ s \ in（0，d/2）$和dimension $ d \ geq 2 $的分数sobolev不平等的类似稳定性不平等。
We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $\mathcal M$ and given by \[ \frac{\|\nabla f\|_{L^2(\mathbb R^d)}^2 - S_d \|f\|_{L^\frac{2d}{d-2}(\mathbb R^d)}^2}{ \inf_{h \in \mathcal M} \|\nabla (f - h)\|_{L^2(\mathbb R^d)}^2 } \geq c_{BE} > 0 \qquad \text{ for every } f \in \dot{H}^1(\mathbb R^d),\] which is due to Bianchi and Egnell, admits a minimizer for every $d \geq 3$. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality $c_{BE} < 2 - 2^\frac{d-2}{d}$, which means that a sequence of two asymptotically non-interacting bubbles cannot be minimizing. Our arguments cover in fact the analogous stability inequality for the fractional Sobolev inequality for arbitrary fractional exponent $s \in (0, d/2)$ and dimension $d \geq 2$.
