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
$$ \ begin {aligned} \ dot {W} ^ {s，p}（{\ mathbb {R}} ^ d）= \ bigg \ {u \ in L ^ 1_ {loc}：\ frac {u（x ）-u（y）} {| xy | ^ {s + \ frac {d} {p}}} \ in L ^ p（{\ mathbb {R}} ^ d \ times {\ mathbb {R}} ^ d ）\ bigg \}，\ end {aligned} $$和\（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}，\），即\（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）。
京公网安备 11010802027423号