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.