Edge-free Mumford–Shah limit ($u-\alpha \Delta u=g$)

As the jump-length weight tends to infinity with the gradient weight fixed, the limiting problem minimizes $\Vert u-g{\Vert }_2^2+\alpha \Vert \nabla u{\Vert }_2^2$ on $H^1$. It has a unique minimizer by the Lax-Milgram theorem and strict convexity. Without prescribed boundary values, its weak equation is $u-\alpha \Delta u=g$ with natural Neumann boundary conditions.