Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 64 / 1 / iii / Solution

Apply the direct method in the calculus of variations to $J(u)=\Vert f-Ku{\Vert }_1+\alpha |Du|(\Omega )$. Since $J(0)=\Vert f{\Vert }_1$, take a minimizing sequence with $J(u_k)\le C:=\Vert f{\Vert }_1+1$. ThenThus the sequence is bounded in the bounded-variation space. The bounded-variation compactness gives a subsequence converging in the strong convergence sense in $L^1$ to $u\in BV(\Omega )$. Boundedness of the linear operator $K$ gives $K{u}_k\to Ku$ strongly in $L^1$, so the residual norm converges. The variation term is sequentially lower semicontinuous, henceTherefore $u$ attains the infimum. The useful inverse hypothesis is the lower bound $\Vert u{\Vert }_1\le C_K\Vert Ku{\Vert }_1$; a bounded inverse on the range already suffices for this proof. No uniqueness follows from the nonsquared $L^1$ fidelity.