Existence for nonnegative shifted Poisson regularization

Let $\Omega$ be a bounded connected Lipschitz domain, $\alpha >0$, $g\ge 0$ bounded, and $T:L^1\to L^1$ bounded and positivity preserving with $T1\ne 0$. Minimize $\alpha \mathrm{TV}(u)+D_g(u)$ over nonnegative BV space functions. The logarithmic Jensen inequality bounds $\Vert Tu{\Vert }_1$ and variation on sublevels; mean control for positive imaging operators then bounds the full $BV$ norm. Bounded-variation compactness gives strong $L^1$ convergence, including nonnegativity. The scalar fidelity has bounded derivative on $s\ge 0$, so continuity of $T$ makes its integral continuous in that limit. Lower semicontinuity of variation proves existence by the direct method in the calculus of variations. If $g>0$ almost everywhere and $T$ is injective, strict convexity of the fidelity gives uniqueness. For $0<g<1$, zero is the unique minimizer even without injectivity, since zero energy forces both $Tu=0$ and a constant nonnegative $u$.