I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$.
My boundary conditions are:
$$\lim_{r\rightarrow\infty} v(x,y)=0$$ $$v(r=a,\theta)=( \cos(\theta/2), \sin(\theta/2) )$$
Normally, without the constraints, I would have
$$\boldsymbol v = \frac{a^{1/2}}{r^{1/2}} ( \cos(\theta/2), \sin(\theta/2) )$$
But how could I use Lagrange multipliers in this situation, if I want to impose the condition $|\boldsymbol{\hat{x}}+\boldsymbol{v}|=1$? Are there any other straightforward methods?