It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex exponentials), or we compute a power series' radius of convergence by thinking in the complex plane (or use complex analytic properties in path integrals), or we evaluate $\int^\infty\_{-\infty} e^{-x^2}\ dx$ by squaring it...


So, are any 2D problems easier to solve in even higher dimensions? I can't think of any.



2D problems which are easier to solve in 3D
Of course, there is one such problem! This is the Cauchy problem for the wave equation
$$\frac{\partial^2u}{\partial t^2}=\Delta u,\qquad u(x,0)=f(x),\quad \frac{\partial u}{\partial t}(x,0)=g(x),$$
where $x\in{\mathbb R}^d$. To solve it, it is enough to know the case where $f\equiv0$.


If $d=3$, this problem is solved by using spherical means. We obtain
$$u(x,t)=tM\_{t,x}[g],$$
where $M\_{t,x}$ denotes the mean over the sphere of radius $t$ and center $x$.


The two-dimensional case is way more complicated. The formula can only be found by considering that a $2$D-solution is a special case of a $3$D-solution. Then the solution involves a complicated integral over the disk $D(x;t)$ instead of the circle. This is why the Huyghens principle holds true in $3$ space dimensions but not in $2$ space dimensions.


