Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ defined over the class $\mathcal{C}=\{u\in H^1([0,2\pi],R^2\setminus\{0\}): u(0)=u(2\pi), A(u)=1, \mbox{ind}(u)=\pm1\}$ with $A(u)$ the area functional associated to the curve with parametrization $u$, $\mbox{ind}(u)$ the winding number associated to the curve with parametrization $u$ calculated in the origin, and $m\colon R^2\setminus\{0\}$ a vector field.
I know that exists a minimum $u$ in $\mathcal{C}_1=\{u\in H^1([0,2\pi],R^2\setminus\{0\}): u(0)=u(2\pi), A(u)=1\}$ for any choice of the vector field $m$, and i want to construct some condition on $m$ to prove that exists also a minimum in $\mathcal{C}$: the problem is given a minimizing sequence for $\mathcal{F}(u)$ such that $\mbox{ind}(u)$ is well defined and $\mbox{ind}(u)=\pm1$, i have to prove that also for the limit $u$ the winding number is well defined, i.e that $\lvert u(t)\rvert\neq 0$ for all $t\in[0,2\pi]$. I know that the sequence $u_n\rightharpoonup u$ in $H^{1}_{per}$ and that $u\in Lip([0,2\pi])$. I'm asking if there is some technic to manage winding number or similar topic.
I prove that if $m(z)=m_1(z)+\nabla M(z)$ where, $\lvert M(z)\rvert \to \infty$ for $\lvert z \rvert \to 0$ and $m_1$ is bounded there is solution in $\mathcal{C}$, and i was also asking if there is some results such that if i consider a minimizing sequence $(u_\epsilon)$ associated to the problem with $m_\epsilon=m+\epsilon\nabla M$, i can said ''something'' about the case $\epsilon\to 0$, maybe adding other hyphotesis on $M$.