三维薄环上的Yamabe型问题
On a Yamabe Type Problem on Three Dimensional Thin Annulus
摘要：
本文考虑一类$\R^3$的环形域的$A_ε$上的Yamabe型问题，当$ε$趋于零时，$\R^3$的环形域变薄.我们证明了，对于任何给定的正常数$C$，存在$ε_0$，使得对于任何$ε< ε_0$，该问题没有解$u_ε$，其能量小于$C$。这样的结果扩展到三维，这是先前在更高维度中已知的结果。虽然证明这一结果的策略与在高维中相同，但我们需要对解$u_ε$在$ε$趋于零时的渐近分布进行更仔细、更细致的爆破分析。
英文摘要：
We consider a Yamabe type problem on a family $A_ε$ of annulus shaped domains of $\R^3$ which becomes "thin" as $ε$ goes to zero. We show that, for any given positive constant $C$, there exists $ε_0$ such that for any $ε< ε_0$, the problem has no solution $u_ε$ whose energy is less than $C$. Such a result extends to dimension three a result previously known in higher dimensions. Although the strategy to prove this result is the same as in higher dimensions, we need a more careful and delicate blow up analysis of asymptotic profiles of solutions $u_ε$ when $ε$ goes to zero.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
