Reilly used PDEs to give a very elegant proof that spheres are the only embedded hypersurfaces of constant mean curvature in $\mathbb{R}^n$.

Let $\Sigma$ be such a hypersurface, bounding a region $\Omega$. He showed that any solutions to the PDE $\Delta u = -1$ in $\Omega$ with $u=0$ on $\Sigma$ must be a second order polynomials with leading term proportional to $|x|^2$. One sees that level sets of this function are spheres by completing the square.