论文标题
与关键Sobolev指数的卷曲操作员的Sobolev型不平等和地面状态
A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation with critical Sobolev exponent
论文作者
论文摘要
令$ω\ subset \ mathbb {r}^3 $为Lipschitz域，让$ s_ \ mathrm {curl}（ω）$成为最大的常数，以至于$ \ int _ {\ int _ {\ mathbb {r}^3}^3}^3}^3} | s _ {\ mathrm {curl}}}（ω）\ inf _ {\ intack {w \ in W_0^6（\ Mathrm {curl {curl}; \ Mathbb {r}^3） w = 0}}} \ big（\ int _ {\ mathbb {r}^3} | u+w |^6 \，dx \ big） w_0^6（\ Mathrm {curl}; \ Mathbb {r}^3）$其中$ W_0^6（\ Mathrm {Curl};ω）$是$ \ Mathcal {c} _} _ _} _ _ _0^{\ infty}（\ infty}（ω，\ mathbb { l^6（ω，\ mathbb {r}^3）：\ nabla \ times u \ in l^2（ω，\ mathbb {r}^3）\} $相对于norm $ $ $ $（| U | _6^2+| \ nabla \ nabla \ nabla \ nabla \ times u | _2^2）我们表明，$ s _ {\ mathrm {curl}}（ω）$严格大于$ \ mathbb {r}^3 $的经典sobolev常数$ s $。此外，$ s _ {\ mathrm {curl}}（ω）$独立于$ω$，并通过基态解决方案来实现curl-curl问题$$ \ nabla \ times（\ nabla \ times u）（\ nabla \ times u）= | u | u | u |^4u $ $ $ $ $ $ $ $ $ $ $ω= \ mathbbbb {r}^3 $。借助这些结果，我们还研究了一个有限域$ω$ω$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ \ nabla \ times u = | $ \partialΩ$，其中$ν$是正常的外部到$ \partialΩ$。
Let $Ω\subset \mathbb{R}^3$ be a Lipschitz domain and let $S_\mathrm{curl}(Ω)$ be the largest constant such that $$ \int_{\mathbb{R}^3}|\nabla\times u|^2\, dx\geq S_{\mathrm{curl}}(Ω) \inf_{\substack{w\in W_0^6(\mathrm{curl};\mathbb{R}^3)\\ \nabla\times w=0}}\Big(\int_{\mathbb{R}^3}|u+w|^6\,dx\Big)^{\frac13} $$ for any $u$ in $W_0^6(\mathrm{curl};Ω)\subset W_0^6(\mathrm{curl};\mathbb{R}^3)$ where $W_0^6(\mathrm{curl};Ω)$ is the closure of $\mathcal{C}_0^{\infty}(Ω,\mathbb{R}^3)$ in $\{u\in L^6(Ω,\mathbb{R}^3): \nabla\times u\in L^2(Ω,\mathbb{R}^3)\}$ with respect to the norm $(|u|_6^2+|\nabla\times u|_2^2)^{1/2}$. We show that $S_{\mathrm{curl}}(Ω)$ is strictly larger than the classical Sobolev constant $S$ in $\mathbb{R}^3$. Moreover, $S_{\mathrm{curl}}(Ω)$ is independent of $Ω$ and is attained by a ground state solution to the curl-curl problem $$ \nabla\times (\nabla\times u) = |u|^4u $$ if $Ω=\mathbb{R}^3$. With the aid of those results, we also investigate ground states of the Brezis-Nirenberg-type problem for the curl-curl operator in a bounded domain $Ω$ $$\nabla\times (\nabla\times u) +λu = |u|^4u\quad\hbox{in }Ω$$ with the so-called metallic boundary condition $ν\times u=0$ on $\partialΩ$, where $ν$ is the exterior normal to $\partialΩ$.
