Two-dimensional flat-target Derrick obstruction

For smooth fields into ${\mathbb{R}}^l$ with nonnegative differentiable potential and finite energy, Derrick scaling in two spatial dimensions forces the potential integral to vanish. Thus the field takes values where $U=0$ and $\nabla U=0$, so its static field equation makes every component harmonic. Each first derivative is an entire harmonic function in L2 space; the mean-value inequality on arbitrarily large balls forces it to vanish. Hence the field is constant. The scale identity alone does not exclude a pure-gradient model; this additional argument uses the flat unconstrained target. Curved-target harmonic maps, for example the O3 nonlinear sigma model, can instead have finite-energy nonconstant solutions.