A Sobolev-Type Inequality for the Curl Operator and Ground States for the Curl–Curl Equation with Critical Sobolev Exponent,Archive for Rational Mechanics and Analysis

当前位置： X-MOL 学术 › Arch. Rational Mech. Anal. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

A Sobolev-Type Inequality for the Curl Operator and Ground States for the Curl–Curl Equation with Critical Sobolev Exponent
Archive for Rational Mechanics and Analysis ( IF 2.5 ) Pub Date : 2021-06-10 , DOI: 10.1007/s00205-021-01684-x
Jarosław Mederski , Andrzej Szulkin

Let $\Omega \subset \mathbb {R}^3$ be a Lipschitz domain and let $S_\mathrm {curl}(\Omega )$ be the largest constant such that

$$\begin{aligned} \int _{\mathbb {R}^3}|\nabla \times u|^2\, \mathrm{d}x\ge S_{\mathrm {curl}}(\Omega ) \inf _{\begin{array}{c} w\in W_0^6(\mathrm {curl};\mathbb {R}^3)\\ \nabla \times w=0 \end{array}}\Big (\int _{\mathbb {R}^3}|u+w|^6\,\mathrm{d}x\Big )^{\frac{1}{3}} \end{aligned}$$

for any u in $W_0^6(\mathrm {curl};\Omega )\subset W_0^6(\mathrm {curl};\mathbb {R}^3)$, where $W_0^6(\mathrm {curl};\Omega )$ is the closure of $\mathcal {C}_0^{\infty }(\Omega ,\mathbb {R}^3)$ in $\{u\in L^6(\Omega ,\mathbb {R}^3): \nabla \times u\in L^2(\Omega ,\mathbb {R}^3)\}$ with respect to the norm $(|u|_6^2+|\nabla \times u|_2^2)^{1/2}$. We show that $S_{\mathrm {curl}}(\Omega )$ is strictly larger than the classical Sobolev constant S in $\mathbb {R}^3$. Moreover, $S_{\mathrm {curl}}(\Omega )$ is independent of $\Omega $ and is attained by a ground state solution to the curl–curl problem

$$\begin{aligned} \nabla \times (\nabla \times u) = |u|^4u \end{aligned}$$

if $\Omega =\mathbb {R}^3$. With the aid of these results we also investigate ground states of the Brezis–Nirenberg-type problem for the curl–curl operator in a bounded domain $\Omega $

$$\begin{aligned} \nabla \times (\nabla \times u) +\lambda u = |u|^4u\quad \hbox {in }\Omega , \end{aligned}$$

with the so-called metallic boundary condition $\nu \times u=0$ on $\partial \Omega $, where $\nu $ is the exterior normal to $\partial \Omega $.

中文翻译：

具有临界 Sobolev 指数的卷曲-卷曲方程的卷曲算子和基态的 Sobolev 型不等式

令$\Omega \subset \mathbb {R}^3$是一个 Lipschitz 域，并让$S_\mathrm {curl}(\Omega )$是最大的常数，使得

对于任何ü在\（W_0 ^ 6（\ mathrm {卷曲}; \欧米茄）\子集W_0 ^ 6（\ mathrm {卷曲}; \ mathbb {R} ^ 3）\），其中\（W_0 ^ 6（\ mathrm {curl};\Omega )\)是$\mathcal {C}_0^{\infty }(\Omega ,\mathbb {R}^3)$在$\{u\in L ^6(\Omega ,\mathbb {R}^3): \nabla \times u\in L^2(\Omega ,\mathbb {R}^3)\}$相对于范数$(| u|_6^2+|\nabla \times u|_2^2)^{1/2}$。我们表明，\（S _ {\ mathrm {卷曲}}（\欧米茄）\）是严格比经典的Sobolev常数大小号在\（\ mathbb {R} ^ 3 \）。此外，$S_{\mathrm {curl}}(\Omega )$独立于$\Omega $ 并且通过卷曲-卷曲问题的基态解获得