Sobolev-type lower bounds on ∥∇𝜓∥² for arbitrary regions in two-dimensional Euclidean space

This note reports the derivation of lower bounds of the Sobolev type on ‖ ∇ ψ ‖ 2 ≡ ∫ R ( ∂ ψ / ∂ x 1 ) 2 + ( ∂ ψ / ∂ x 2 ) 2 ) d x 1 d x 2 {\left \| {\nabla \psi } \right \|^2} \equiv \smallint {}_R{(\partial \psi /\partial {x_1})^2} + {(\partial \psi /\partial {x_2})^2})d{x_1}d{x_2} for generic real scalar ψ = ψ ( x 1 , x 2 ) \psi = \psi ({x_1},{x_2}) of function class C 0 {C^0} piecewise C 2 {C^2} which vanish over the boundary of the (bounded or unbounded) region R R in Euclidean 2-space.