# Calculus variation question

Assume that we have to minimize the integral $$I(y)=int_0^1 L(x,y,y'(x))dx$$ for smooth diffeomorphic mappings $$y:(0,1)to (0,1)$$ with $$y(0)=0$$ and $$y(1)=1$$, where $$Lin C^infty(Rtimes Rtimes R)$$, and $$L$$ satisfies convex condition wrt to $$y’$$ and coercivity condition. Is the minimiser of $$I$$ the only difeomorphic solution $$y:(0,1)to(0,1)$$ of Euler-Lagrange equation $$partial_t (partial_{y’} L)=partial_y L$$? Some reference I need.