Sobolev-type lower bounds on $\parallel \nabla \psi \parallel ^{2}$ for arbitrary regions in two-dimensional Euclidean space

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