It is well known that the problem of minimizing
$ J[y] = \int_{0}^{1} \sqrt{y(x)^2 + \dot{y}(x)^2} dx $
with $y \in C^2[0,1]$ and $y(0) = 1$ and $y(1) = 0$ has no solutions. However, if we remove the condition $y(1) = 0$ and instead let the value of $y$ at $x = 1$ be free, then an optimal solution does exist.
An easy way to see this is to observe that $J[y]$ is really just the arc length of the plane curve with polar equation $r(\theta) = y(\theta)$. Clearly then, the function $y$ which traces out in that way the shortest line segment joining the point $(0,1)$ (given in polar coordinates) and the ray $\theta = 1$ is the (unique) solution to this new problem.
Inspired by this little example, I wonder: are there results regarding the existence of solutions to variational problems with freedom at one or both endpoints and similar integrands?