Is an exact solution to the following minimization problem known? Find $f$ such that:
$$\sum_{k=1}^m\int_{\mathbb{R}^2}(g_k(\mathbf{x})-H(f(\mathbf{x})))^2+\int_{\mathbb{R}^2}|\nabla H(f(\mathbf{x}))|^2$$
is minimized, where $g_k:\mathbb{R}^2\rightarrow\{0,1\}$ are known binary functions on the plane. $H$ is the Heaviside step function.