Towards an asymptotic analysis of the anisotropic Ginzburg–Landau model

We develop a set of tools for the asymptotic analysis of minimizers of the anisotropic Ginzburg–Landau energy functional among the admissible competitors with Dirichlet boundary datum of negative degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> . As a byproduct of our analysis, we prove that the energy of a minimizer is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo form="prefix">ln</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mo form="prefix">ln</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> depends only on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> and the material constants that enter into the expression for the energy.