A Sobolev-Type Inequality for the Curl Operator and Ground States for the Curl–Curl Equation with Critical Sobolev Exponent
A Sobolev-Type Inequality for the Curl Operator and Ground States for the Curl–Curl Equation with Critical Sobolev Exponent
Abstract Let $$\Omega \subset \mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> be a Lipschitz domain and let $$S_\mathrm {curl}(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>curl</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the largest constant such that $$\begin{aligned} \int …