Potential theory and applications in a constructive method for finding critical points of Ginzburg--Landau type equations
Let \(H_1\) and \(H_2\) two Hilbert spaces with \(H_1\) dense in \(H_2\), \(H_1\subset H_2\), \(\| x\|_{H_2}\leq\| x\|_{H_1}\). The authors define the transformation \(M : H_2\to H_1\) by \[ (x,y)_{H_2}= (x, My)_{H_1}\quad\forall x\in H_1,\;y\in H_2, \] and prove that \(M^{-1}\) exists, \(M\) has square roots \(\sqrt{M}: H_2\to H_2\), \(\sqrt{M}_{H_1}: H_1\to H_1\), the range of \(\sqrt{M}\) is \(H_1\) and that \(M:H_2\to H_2\) is compact if \(M:H_2\to H_1\) is compact. The authors use their general results to obtain critical points of some energy functionals of Ginzburg--Landau type.
scientific article; zbMATH DE number 5324026