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Optimizers of the Sobolev and Gagliardo–Nirenberg inequalities in $$ \dot{W}^{s,p} $$ W ˙ s , p
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-01-13 , DOI: 10.1007/s00526-021-01917-7
Yang Zhang

In this paper, we consider the existence of optimizers for the following Sobolev and Gagliardo–Nirenberg intepolation inequalities in \( \dot{W}^{s,p}({\mathbb {R}}^d) \) at the non-endpoint case:

$$\begin{aligned} ||u||_{L^{p^*}} \le C || u||_{ \dot{W}^{s,p}}, \qquad ||u||_{L^r} \le C || u||_{\dot{W}^{s_1,p}}^{\theta } ||u||_{L^p}^{1-\theta }, \end{aligned}$$

where \( \dot{W}^{s,p} ({\mathbb {R}}^d) \) is the fractional Sobolev space, defined by

$$\begin{aligned} \dot{W}^{s,p}({\mathbb {R}}^d) =\bigg \{ u\in L^1_{loc}: \frac{u(x)-u(y)}{|x-y|^{s+\frac{d}{p}}} \in L^p({\mathbb {R}}^d \times {\mathbb {R}}^d) \bigg \}, \end{aligned}$$

and \( 0<s,s_1<1, \) \(1<p,r <\infty , \) \( \frac{1}{p^*}=\frac{1}{p}-\frac{s}{d}>0, \) \( \frac{1}{r}=\theta (\frac{1}{p}-\frac{s_1}{d}) +(1-\theta )\frac{1}{p} .\) Comparing with the usual assumption, we don’t need \( s_1 < \frac{d}{p}, \) i.e., \( p<r < \frac{pd}{d-s_1p} \) if \( \frac{1}{p}-\frac{s_1}{d} >0, \) \( p<r <\infty , \) if \( \frac{1}{p}-\frac{s_1}{d} \le 0. \) To prove these, we establish a compactness up to symmetry lemma in Sobolev embedding for \( \dot{W}^{s,p} \) case by refined Sobolev inequalities in Morrey or Besov space, and we handle the two cases in a unified way. Also for existence of optimizers for Gagliardo–Nirenberg intepolation inequalities in \( \dot{W}^{s,p} \cap L^p, \) we give two alternative proofs: one follows the celebrated paper (Bellazzini et al. in Math Ann 360(3–4):653–673, 2014. https://doi.org/10.1007/s00208-014-1046-2), in which we establish a compactness up to translation lemma in \( \dot{W}^{s,p} \cap L^p; \) one follows the concentrarion compactness principle (Lions in Ann Inst Henri Poincaré Anal Non Linéaire 1(2):109–145, 1984), (Lions in Ann Inst Henri Poincaré Anal Non Linéaire 1(4):223–283 1984).



中文翻译:

$$ \ dot {W} ^ {s,p} $$ W˙s,p中的Sobolev和Gagliardo-Nirenberg不等式的优化器

在本文中,我们考虑了以下非Sobolev和Gagliardo-Nirenberg不等式在\(\ dot {W} ^ {s,p}({\ mathbb {R}} ^ d)\)上的优化器的存在端点情况:

$$ \ begin {aligned} || u || _ {L ^ {p ^ *}} \ le C || u || _ {\ dot {W} ^ {s,p}},\ qquad || u || _ {L ^ r} \ le C || u || _ {\ dot {W} ^ {s_1,p}} ^ {\ theta} || u || _ {L ^ p} ^ {1- \ theta},\ end {aligned} $$

其中\(\ dot {W} ^ {s,p}({\ mathbb {R}} ^ d)\)是分数Sobolev空间,定义为

$$ \ begin {aligned} \ dot {W} ^ {s,p}({\ mathbb {R}} ^ d)= \ bigg \ {u \ in L ^ 1_ {loc}:\ frac {u(x )-u(y)} {| xy | ^ {s + \ frac {d} {p}}} \ in L ^ p({\ mathbb {R}} ^ d \ times {\ mathbb {R}} ^ d )\ bigg \},\ end {aligned} $$

\(0 <s,s_1 <1,\) \(1 <p,r <\ infty,\) \(\ frac {1} {p ^ *} = \ frac {1} {p}-\ frac {s} {d}> 0,\) \(\ frac {1} {r} = \ theta(\ frac {1} {p}-\ frac {s_1} {d})+(1- \ theta) \ frac {1} {p}。\)与通常的假设相比,我们不需要\(s_1 <\ frac {d} {p},\),\(p <r <\ frac {pd} {d-s_1p} \)如果\(\ frac {1} {p}-\ frac {s_1} {d}> 0,\) \(p <r <\ infty,\)如果\(\ frac {1 } {p}-\ frac {s_1} {d} \ le0。\)为了证明这些,我们在Sobolev嵌入中为\(\ dot {W} ^ {s,p} \)建立了一个对称引理的紧致性在Morrey或Besov空间中由精细的Sobolev不等式构成的情况,我们以统一的方式处理这两种情况。另外,对于存在\(\ dot {W} ^ {s,p} \ cap L ^ p,\)中的Gagliardo–Nirenberg插值不等式的优化器,我们给出了两种替代证明:一种遵循著名的论文(Bellazzini等,2003年)。 Math Ann 360(3–4):653–673,2014. https://doi.org/10.1007/s00208-014-1046-2),其中我们在\(\ dot { W)^ {s,p} \ cap L ^ p; \)一个遵循集中度紧缩原理(Lion in Ann Inst HenriPoincaréAnal NonLinéaire1(2):109–145,1984),(Lions in Ann Inst Henri PoincaréAnal NonLinéaire1(4):223–283 1984)。

更新日期:2021-01-14
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