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Nonself-dual Chern–Simons and Maxwell–Chern–Simons vortices on bounded domains. (English) Zbl 1080.35104

Let ${\Omega }$ be a smooth bounded simply connected domain in ${ℝ}^{2}$ and let ${{\Omega }}^{+}={ℝ}^{+}×{\Omega }\subset {ℝ}^{2,1}$, where ${ℝ}^{2,1}$ is the Minkowski space with the metric $\text{diag}\left(1,-1,-1\right)$. Then the Lagrangean of the Maxwell-Chern-Simons-Higgs (MCSH) model is given by

$\begin{array}{cc}\hfill {ℒ}_{\text{MCSH}}& =\left[{\left({D}^{q}\right)}_{\alpha }u\right]\overline{\left[{\left({D}^{q}\right)}^{\alpha }u\right]}-\frac{1}{4}\phantom{\rule{0.166667em}{0ex}}{F}_{\alpha \beta }{F}^{\alpha \beta }+\frac{\kappa }{4}\phantom{\rule{0.166667em}{0ex}}{\epsilon }^{\alpha \beta \gamma }\phantom{\rule{0.166667em}{0ex}}{A}_{\alpha }{F}_{\beta \gamma }\hfill \\ & +{\partial }_{\alpha }N{\partial }^{\alpha }N-{q}^{2}{|u|}^{2}{N}^{2}-\frac{1}{2}\left(\sqrt{\lambda }{q|u|}^{2}+\kappa N-\sqrt{\lambda }{q\right)}^{2},\hfill \end{array}$

where $q,\kappa ,\lambda >0$ are constants, ${\left({D}^{q}\right)}_{\alpha }={\partial }_{\alpha }-iq{A}_{\alpha }$, and $N:{{\Omega }}^{+}\to ℝ$ is the neutral scalar field which is introduced to attain self-duality.

The MCSH model was introduced as a unified system of the Abelian-Higgs (AH) model (Landau-Ginzburg model) and the Chern-Simons-Higgs (CSH) model [C. Lee, K. Lee and H. Min, Self-dual Maxwell-Chern-Simons solitons, Phys. Lett. B 252, No. 1, 79–83 (1990; Zbl 1079.58009)]. If $\lambda =1$, there exist global minimizers of the static energy functionals of the AH, CSH and MCSH models and they are achieved as solutions of self-dual equations. A static energy functional of the MCSH model reduces static energy functionals of the AH and CSH models by suitable choices of $q,\kappa ,N$ and ${A}_{0}$. The solutions of the energy minimizing self-dual equation of the MCSH model converge to the solutions of the energy minimizing equations of the AH and CSH models [D. Chae, O. Yu. Imanuvilov, J. Funct. Anal. 196, No. 1, 87–118 (2002; Zbl 1079.58009); D. Chae and N. Kim, J. Differ. Equations 134, No. 1, 154–182 (1997; Zbl 0869.35094)]. In this paper, these results are extended to nonself-dual cases. If the gauge field is absent in the model, the results are similar to the self-adjoint case (§2. Th. 2.4 and 2.6). But if the gauge field is present, the tangential current of the CSH limit may be different from the original one (§3. Th. 3.5 and 3.7). The authors mention this phenomenon is closely related with the energy loss in the gauge potential. The asymptotic behavior of solutions to the Euler Lagrange equations for the static energy functional of the static MCSH model when $\lambda \to \infty$ is also studied (§4. Th. 4.8).

To study AH and CSH limits, the function spaces ${𝒳}_{\right\}}^{\text{'}}={𝒫}_{\right\}}^{\text{'}}×ℋ$ in the case $A=0$ and ${𝒴}_{d}={𝒳}_{d}×ℋ$ in the case $A\ne 0$ are introduced. Here, $ℋ={H}_{0}^{1}\left({\Omega },ℝ\right)$,

${𝒫}_{g}^{0}=\left\{u\in {H}^{1}\left({\Omega },ℂ\right):u=g\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }\right\},\phantom{\rule{1.em}{0ex}}g:\partial {\Omega }\to {S}^{1},\phantom{\rule{4.pt}{0ex}}\text{deg}\phantom{\rule{0.166667em}{0ex}}g>0,$

and ${𝒳}_{d}={𝒫}_{d}×𝒱$, where

$\begin{array}{cc}\hfill {𝒫}_{d}& =\left\{u\in {H}^{1}\left({\Omega },ℂ\right):|u|=1\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },\phantom{\rule{4pt}{0ex}}\text{deg}\left(u,\partial {\Omega }\right)=d\right\},\hfill \\ \hfill 𝒱& =\left\{A\in {H}^{1}\left({\Omega },{ℝ}^{2}\right):\text{div}\phantom{\rule{0.166667em}{0ex}}A=0,\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}A·\nu =0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }\right\}·\hfill \end{array}$

It is noted that the norm ${\parallel A\parallel }_{{H}^{1}}$ is equivalent to ${\parallel F\parallel }_{{L}^{2}}$, because ${\Omega }$ is simply connected. The subspace ${𝒴}_{d,h}^{q}={𝒳}_{d,h}^{q}×ℋ$,

${𝒳}_{d,h}^{q}=\left\{\left(u,A\right)\in {𝒴}_{d}:i\left(\overline{u}{D}_{A}^{q}u-u\overline{{D}_{A}^{q}u}\right)/2·\tau =h\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }\right\},$

is also introduced. Then it is shown that the MCSH energy functionals achieve their minimum on ${𝒳}_{g}^{0}$ and on ${𝒴}_{d,h}^{q}$ (Lemma 2.1 and Th. 3.3). For the minimizer $\left(u,N\right)$ in the case $A=0$, the estimate $O\le N\le \frac{q}{\kappa \epsilon }{\left(1-|u|}^{2}\right),$ is shown (Th. 2.3). An estimate on the minimum value is also given (Lemma 2.5). By using these estimates, the Maxwell limit and Chern-Simons limit are computed. In the case $A\ne 0$, estimates of minimum value is done by using the properties of (Lemma. 3.4 and 3.6). Maxwell limit and Chern-Simons limit are also computed by using these estimates. Discussions on the asymptotics for minimizers in §4 are done following F. Bethuel, H. Brézis and F. Hélein, Calc. Var. Partial Differ. Equ. 1, No. 2, 123–148 (1993; Zbl 0834.35014)].

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 35B40 Asymptotic behavior of solutions of PDE 81T13 Yang-Mills and other gauge theories