= Existence for nonnegative shifted Poisson regularization
Let $\Omega$ be a bounded connected <Lipschitz domain>, $\alpha>0$, $g\ge0$ bounded, and $T:L^1\to L^1$ bounded and positivity preserving with $T1\ne0$. Minimize $\alpha\operatorname{TV}(u)+D_g(u)$ over nonnegative <BV space> functions. The logarithmic <Jensen inequality> bounds $\|Tu\|_1$ 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 $s\ge0$, 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 $u$.
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