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On a bounded Lipschitz domain, a sequence with bounded $L^1$ norms and bounded total variation seminorm has a subsequence converging strongly in $L^1$. The limit is in the BV space, and its variation does not exceed the limit inferior. This supplies the compactness step of the direct method in the calculus of variations for linearly growing gradient energies.

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 $u$.

The required functional, with the sign appropriate to a positive divergence operator, isIt is differentiable on $W_0^{1,2}$, with $D\mathcal{F}(u)[v]=Q(u,v)+\int fv$. A critical point consequently satisfies the weak equation. We now prove existence by the direct method in the calculus of variations, rather than invoke the representation theorem again.

The ellipticity and Poincare estimates implyThus the infimum is finite and any minimizing sequence is bounded in the Sobolev norm. A bounded sequence in a Hilbert space has a weakly convergent subsequence; write $u_j\rightharpoonup u$ in $W_0^{1,2}$. The equivalent $Q$ norm has the same continuous linear functionals and hence the same weak convergence. A Hilbert norm is Weak lower semicontinuity, giving $Q(u,u)\le {\mathrm{liminf}}_{j}Q(u_j,u_j)$. The term $\int f{u}_j$ converges to $\int fu$ because it is a bounded linear functional. ThereforeThe limit attains the infimum. Differentiating $t\mapsto \mathcal{F}(u+tv)$ at its minimum, for every $v$, gives $D\mathcal{F}(u)[v]=0$. This produces the desired weak solution. Strict positivity of $Q$ also makes the functional strictly convex, consistent with uniqueness.

Symmetry implies that all eigenvalues of $A(x)$ lie between $\lambda$ and $\Lambda$, so $\Vert A(x){\Vert }_{\mathrm{op}}\le \Lambda$. In particular the bilinear form $B$ from part (i) is bounded and symmetric under the printed assumptions. The symmetric elliptic Dirichlet energy with a nonpositive potential isAll its terms are finite. For an admissible variation $\phi \in H_0^1$, its first variation is $B(u,\phi )+\int f\phi$. Thus stationarity is exactly the weak formulation of $Lu=f$.

Here is the direct method in the calculus of variations. Put $u=\psi +w$. Expanding the symmetric quadratic form givesThe Poincaré inequality controls $\Vert w{\Vert }_2$ by $\Vert Dw{\Vert }_2$. Therefore $J$ is bounded below on the admissible class, and a minimizing sequence is bounded in $H^1$. By weak compactness in this reflexive Banach space, a subsequence converges weakly to $u$. The affine closed subspace $\psi +H_0^1$ is weakly closed, so $u$ remains admissible.

The symmetric square root $A^{1/2}(x)$ is measurable and bounded. Since $-q\ge 0$, the two bounded linear mapstake weak $H^1$ convergence to weak $L^2$ convergence. The weak lower semicontinuity of the Hilbert norm shows that $B(u,u)$ is weakly lower semicontinuous. The term $\int fu$ is weakly continuous. Hence $J(u)\le \mathrm{liminf}J(u_k)$ and $u$ attains the minimum. Varying in either sign yields the weak equation.

The sign $q\le 0$ was used both to make the potential energy nonnegative, hence weakly lower semicontinuous by this argument, and to obtain coercivity. Finally $J(u+z)-J(u)=B(z,z)/2\ge \lambda \Vert Dz{\Vert }_2^2/2$ for $z\in H_0^1$, since the first variation at $u$ vanishes. Therefore

The affine class $C_{\psi }=\psi +H_0^1(\Omega )$ is nonempty, and part (i) gives a finite comparison energy $\mathcal{F}(\psi )$. Since $F(p)\ge \alpha |p{|}^2$, its energy infimum $m$ satisfies $0\le m\le \mathcal{F}(\psi )$. Choose a minimizing sequence $u_k\in C_{\psi }$ with $\mathcal{F}(u_k)\le \mathcal{F}(\psi )+1$. ThenThe gradient bounds also control the full Sobolev norm. Indeed $w_k=u_k-\psi \in H_0^1$, so the Poincaré inequality givesThus $u_k$ is bounded in the Hilbert space $H^1(\Omega )$. Its weak compactness yields a weakly convergent subsequence $u_k\rightharpoonup u$. The closed linear subspace $H_0^1$ is weakly closed, hence $u-\psi \in H_0^1$ and $u\in C_{\psi }$. By part (ii),The direct method in the calculus of variations therefore provesNeither a smooth boundary nor a classical trace theorem is needed: the boundary condition is encoded by the closed zero-boundary Sobolev space.

Positivity means $u>0$ almost everywhere, since the real logarithm must belong to $L^1$. A positivity-preserving operator gives $s=Tu\ge 0$. For fixed $x$, the scalar shifted Poisson data fidelity is $f_x(s)=s-g(x)\log (1+s)$, withComposition with the linear operator $T$ proves convexity in $u$, and strict convexity holds along pairs whose forward images differ on a set of positive measure. The admissible class itself is convex: the concavity of the logarithm gives $\log (\theta u+(1-\theta )v)\ge \theta \log u+(1-\theta )\log v$, controlling its negative part, while ${\log }^{+}w\le w$ controls its positive part.

Because $|\Omega |=1$ and $\log (1+s)\ge 0$, the Jensen inequality gives the requested bound:The final equality uses nonnegativity and the unit area. Since $t-G\log (1+t)\to \infty$, this controls the forward-image norm on energy sublevels.

The printed strictly positive problem has no minimizer. This is nonattainment under strict positivity for shifted Poisson fidelity, rather than a failure of the coercivity calculation. Indeed, with $0<g<1$ and $s>0$, $\log (1+s)<s$ implies $f_x(s)>0$. Also $Tu$ cannot vanish identically for a strictly positive $u$. To see this, let $E_n=\{u\ge 1/n\}$. If $Tu=0$, positivity and $0\le {\chi }_{E_n}\le nu$ give $T{\chi }_{E_n}=0$. But ${\chi }_{E_n}\to {\chi }_{\Omega }$ in $L^1$ and continuity of $T$ would imply $T{\chi }_{\Omega }=0$, contradicting the hypothesis. Thus every admissible $u$ has positive fidelity and hence positive total energy.

Conversely, the constants $u_{\epsilon }=\epsilon {\chi }_{\Omega }$ are admissible, have zero total variation, and satisfyTheir logarithms are integrable for each $\epsilon >0$, but the limit is excluded. ThereforeA bounded minimizing sequence and bounded-variation compactness do not repair a nonclosed positivity/logarithm constraint. In particular, a literal existence or uniqueness proof for that domain is impossible.

The natural correction is to minimize over $BV(\Omega )$ with $u\ge 0$, omitting the unnecessary $\log u\in L^1$ condition: the fidelity only contains $\log (1+Tu)$, which is already integrable. Here is the full existence for nonnegative shifted Poisson regularization argument, also valid for any bounded nonnegative data $g$. Let $G=\Vert g{\Vert }_{\infty }$, $m_G={\inf }_{t\ge 0}\{t-G\log (1+t)\}>-\infty$, and take a minimizing sequence of energy at most $C$. The displayed bound gives $\Vert T{u}_n{\Vert }_1\le C_1$ and $\alpha \mathrm{TV}(u_n)\le C-m_G$. Write $c_n={\int }_{\Omega }{u}_n\ge 0$. The Poincaré inequality for total variation and mean control for positive imaging operators yieldThe denominator is nonzero, so the full $BV$ norm is bounded. Bounded-variation compactness gives $u_n\to u$ in $L^1$ along a subsequence, with $u\ge 0$. Continuity gives $T{u}_n\to Tu$ in $L^1$. On $s\ge 0$, $|f_x^{\prime }(s)|\le 1+G$, so the fidelity converges in the integral; lower semicontinuity of variation completes the direct method in the calculus of variations.

For the actual printed data $0<g<1$, the corrected problem has the unique minimizer $u=0$, even if $T$ is not injective. Zero attains energy zero. Any other zero-energy candidate would have both $Tu=0$ and $\mathrm{TV}(u)=0$; on the connected square, zero variation makes $u$ a nonnegative constant, and $T{\chi }_{\Omega }\ne 0$ forces that constant to vanish. For general positive bounded data, an injective $T$ is a sufficient uniqueness condition, because its fidelity is strictly convex; injectivity is not a necessary condition in every instance.

In the finite-dimensional interpretation, let ${\lambda }_{ij}=1+(Tu{)}_{ij}$. Independent Poisson observations with these intensities have negative log-likelihood ${\sum }_{ij}[{\lambda }_{ij}-g_{ij}\log {\lambda }_{ij}]+C(g)$. Removing the constant $\sum 1$ gives precisely the stated fidelity. Thus the model is Poisson counting noise with a unit background intensity, or an approximate version of it for rescaled/continuous grey values. Literal Poisson counts are integers; the constraint $0<g<1$ is a grey-value normalization, not a literal unscaled count sample.

An image segmentation separates an observed image signal into regions that are smooth within themselves and separated by meaningful image edges. Pure image smoothing blurs the very transitions that ought to define these regions. The Mumford–Shah segmentation model instead chooses the reconstruction and its discontinuity set together. For a bounded planar Lipschitz domain and bounded grey-value data $g$, one standard normalization isHere $K$ is the relatively closed edge set and $u\in H^1(\Omega \setminus K)$ can have different traces on its two sides. The first term is quadratic fidelity, the second penalizes variation within regions, and the Hausdorff measure term charges total edge length. It balances fitting, denoising and economical region geometry. The model was developed by David Mumford and Jayant Shah; their 1989 paper formulates this joint variational approach.

All three terms matter. Without fidelity, a constant reconstruction with no edge has zero energy. Without the gradient term, smooth data can be fitted exactly with no edge penalty. Without the length term, fine partitions with nearly constant region means can drive fidelity arbitrarily low while creating excessive boundaries. This is segmentation overfitting without an edge penalty. Increasing $\alpha$ favors flatter regions; increasing $\beta$ makes extra boundaries more expensive and can remove small features. These parameter effects describe a balance, not a guaranteed monotone evolution of every individual boundary.

For fixed $K$, varying $u$ gives the fixed-edge Euler-Lagrange equation for Mumford–Shah:The boundary condition is a separate one-sided Neumann boundary condition on each side of an edge, not continuity of $u$ across it. The fidelity makes this fixed-edge problem strictly convex, so its weak solution is unique by the Lax-Milgram theorem. Optimizing the edge set remains a different geometric problem. The nonconvexity of Mumford–Shah segmentation prevents a general uniqueness assertion or a guarantee that a numerical stationary point is globally optimal.

The piecewise-constant Mumford–Shah problem imposes $u=c_i$ on regions $E_i$. It minimizesThe factor $1/2$ counts each shared internal boundary once. The region means in piecewise-constant segmentation give $c_i=|E_i{|}^{-1}{\int }_{E_i}g$, provided $|E_i|>0$. Thus the remaining optimization concerns the partition. For a smooth interface between $i$ and $j$, moving it in the normal pointing out of $i$ changes fidelity by $[(c_i-g{)}^2-(c_j-g{)}^2]$ per unit displacement, while length changes by its curvature $\kappa$. The resulting segmentation interface curvature balance isWith equal isotropic interface costs, three freely meeting smooth edges satisfy the triple-junction angle in isotropic segmentation: force balance of their unit tangents gives angles of $120$ degrees. These are local stationarity conditions on regular interfaces, not a description of every singular edge configuration.

A simple piecewise-constant segmentation contrast threshold explains why small objects can disappear. On a domain of area $A$, suppose two constant intensities differ by $d$, occupying areas $a$ and $A-a$ with internal boundary length $L$. Keeping that boundary fits the data exactly and costs $\beta L$. Merging both regions costs the within-region squared error $a(A-a)d^2/A$. Among these two candidates, splitting wins precisely when $\beta L<a(A-a)d^2/A$. Other partitions may beat either candidate, so this is not a universal global segmentation formula.

This original synthetic example compares those two candidate geometries using their least-squares means. It illustrates the edge cost; it does not claim to compute the globally optimal Mumford–Shah segmentation.

Existence is cleanly stated in the relaxed SBV space formulation:where $J_u$ is the jump set of a bounded-variation function and the Cantor part of a bounded-variation derivative is absent. Clip a minimizing sequence to the bounded data range: this cannot increase fidelity, gradient energy or jump length. Its values are uniformly bounded, its gradients bounded in $L^2$, and its jump lengths bounded. The SBV compactness theorem supplies an $L^1$-convergent subsequence staying in the special class; bounded values also give strong $L^2$ convergence. The fidelity then converges, while gradient energy and jump length are lower semicontinuous. The direct method in the calculus of variations produces a minimizer. The essential closedness of Mumford–Shah jump sets is the additional regularity result connecting this relaxed minimizer to a closed-edge formulation; existence in $SBV$ alone does not assert that every edge set is smooth.

A practical continuous approximation is the Ambrosio–Tortorelli approximation, introduced in Ambrosio and Tortorelli's 1990 paper. An auxiliary field $v\in [0,1]$ is near one in regions and near zero at edges. One normalized energy isThe edge field weakens smoothing across a narrow transition, and its own energy approximates interface length. The Gamma-convergence result, together with the required compactness, relates global minimizing sequences to the limiting segmentation energy; it does not make the finite-parameter problem jointly convex. Alternating the two fields solves quadratic elliptic subproblems, but initialization and stopping can affect which local stationary configuration is found. The strengths of the model are joint denoising and segmentation, sharp transitions and a geometric cost. Limitations include competing local minima, sensitivity to scale parameters, loss of fine texture, and boundaries driven by intensity rather than semantic object identity.

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.

A one-dimensional Sobolev representative is absolutely continuous. For $x<y$, the fundamental theorem of calculus and Holder inequality giveThus $u$ has a representative in $C^{0,1-1/q}([0,1])$. The qualification about representatives matters because a Sobolev space element is an almost-everywhere equivalence class.

In two dimensions, the Sobolev fundamental theorem of calculus on lines and Fubini's theorem imply that almost every horizontal and vertical slice belongs to $W^{1,q}(0,1)$ and has this one-dimensional Hölder continuity. The slice seminorm depends on the slice; this does not give one uniform pointwise estimate on the square. For $q>2$, Morrey's inequality additionally gives a globally Hölder continuous representative of exponent $1-2/q$. For $1<q\le 2$, global continuity need not hold. For example, with a smooth cutoff around an interior point, $u(x)=|x-x_0{|}^{-a}$ lies in $W^{1,q}$ when $0<a<2/q-1$, but is unbounded. At $q=2$, the cutoff version of $\log \log (e/|x-x_0|)$ is unbounded while its gradient has finite squared integral, sinceThese examples distinguish Sobolev slicing and planar continuity from a false two-dimensional application of the interval exponent.

Put $\Omega =(0,1{)}^2$. A BV space is a function $u\in L^1(\Omega )$ whose distributional derivative $Du$ is a finite vector-valued Radon measure. Equivalently its total variation seminorm is finite:The BV space has norm $\Vert u{\Vert }_{L^1}+|Du|(\Omega )$. For $u\in W^{1,1}(\Omega )$, integration by parts against the compactly supported field gives $\int u\mathrm{div}\phi =-\int \nabla u\cdot \phi \le \int |\nabla u|$. Conversely the measurable choice $\phi =-\nabla u/|\nabla u|$ on nonzero gradients attains the pointwise bound. Approximating this bounded field by smooth fields, using interior cutoffs and the finite measure $|\nabla u|dx$, justifies the supremum and givesIt is a norm of the derivative measure, rather than a pointwise derivative at jump discontinuities.

There is a genuine mismatch in the printed definition of the next functional. Its constraints on ${\phi }_0$ and $\phi$ are independent. Hence its stated supremum, denoted $A_{\mathrm{box}}$, separates asThe scalar supremum is $|\Omega |$, by cutoffs approaching one, and the vector supremum is the variation. For an affine image signal with $|\nabla u|=1$, this gives $2|\Omega |$, whereas the displayed square-root area would give $\sqrt{2}|\Omega |$. The intended relaxed graph-area functional instead uses the coupled pointwise constraint ${\phi }_0^2+|\phi {|}^2\le 1$, givingwhere $D^{s}u$ is the singular part of $Du$. Both readings have a minimizer, but their equations are different.

Here is the direct method in the calculus of variations for either reading. Let $A_{\bullet }$ be the literal $A_{\mathrm{box}}$ or the corrected $A$, and define the energy on $BV(\Omega )\cap L^2(\Omega )$, assigning infinity elsewhere. A minimizing sequence has bounded energy by comparison with $u=0$. Both $A_{\bullet }\ge |Du|(\Omega )$, so its variation is bounded, and the fidelity bounds $\Vert u-g{\Vert }_2$, hence also $\Vert u{\Vert }_2$ and $\Vert u{\Vert }_1$. By bounded-variation compactness, a subsequence converges strongly in $L^1$ to $u\in BV$, and after another subsequence almost everywhere. Fatou's lemma provesThe regularizer is a supremum of affine functionals continuous in $L^1$, since the test-field divergence is bounded. It is therefore lower semicontinuous. Combining the two lower bounds proves existence of a minimizer. In fact the convex regularizer and the strictly convex squared fidelity make the minimizer unique up to null sets. This does not assert that the minimizer must belong to $W^{1,1}$.

For the intended graph area, conditionally assume that the minimizer is in $W^{1,1}$. For $\eta \in C_c^{\infty }(\Omega )$, differentiate at $u+t\eta$. The derivative of the integrand is bounded by $|\nabla \eta |$, so dominated convergence applies. The weak equation isthat is,This is the graph-area Euler-Lagrange equation. Compactly supported variations impose no boundary condition in this statement.

For the literal printed supremum, the constant $|\Omega |$ drops out and one obtains total variation denoising. Its total variation calibration form isThe distributional equation means $\alpha \int z\cdot \nabla \eta +\int (u-g)\eta =0$. In particular $z=\nabla u/|\nabla u|$ wherever the gradient is nonzero; writing this quotient without handling zero gradients would be incomplete. Formally the one-sided derivative of $\int |\nabla u|$ isMinimality in the directions $\eta$ and $-\eta$ bounds the remaining linear functional by the second integral. The Hahn-Banach theorem extends it on that zero-gradient set to a bounded vector field of magnitude at most one, furnishing $z$ and the displayed weak equation. Thus the literal definition has a nonsmooth subgradient equation, not the square-root equation above.

Work with real-valued functions. The screened sine-Gordon energy is well defined: $0\le 1-\cos u\le u^2/2$, and $fu$ is integrable by the Cauchy-Schwarz inequality. It is coercive, becauseA minimizing sequence is bounded in the Hilbert space $H^1$. Weak sequential compactness of bounded sequences in a reflexive Banach space supplies a weakly convergent subsequence. The squared H1 space norm is weakly lower semicontinuous, the source pairing is weakly continuous, and the nonlinear term is covered by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. The direct method in the calculus of variations therefore gives a minimizer $\varphi$.

Taking its first variation in any $v\in H^1$ givesThus the Euler-Lagrange equation isas a weak solution, equivalently in distributions when tested against smooth compactly supported functions. The derivative of the nonlinear term is justified by $|\sin \varphi |\le |\varphi |$ and the second-order remainder bound $|\cos |\le 1$.

Now $G=f-\sin \varphi \in L^2$, so the supplied elliptic regularity estimate puts $\varphi$ in $H^2$. Applying the Sobolev inequality to $\varphi$ and each first weak derivative gives $\varphi \in W^{1,6}({\mathbb{R}}^3)$. Morrey's inequality and uniformly continuous integrable functions vanish at infinity prove that its continuous representative tends to zero.

The final printed supremum estimate is false in general. The maximum bound for a monotone reaction term involves $h(s)=s+\sin s$, which is odd and strictly increasing: $h^{\prime }=1+\cos s\ge 0$, and its zeros are isolated. If $M=\Vert f{\Vert }_{\infty }<\infty$, a positive maximum $m$ of $\varphi$ satisfies $h(m)\le f\le M$; apply the same argument to $-\varphi$. The valid general estimate isThe bound by $M$ is valid if $M\le \pi$, but $h(s)$ can be smaller than $s$ for larger positive $s$.

For an explicit failure of the source-size bound for the screened sine-Gordon equation, take $a=3\pi /2$, $L=10$, $\varphi (x)=a{e}^{-|x{|}^2/L^2}$ and $f=-\Delta \varphi +\varphi +\sin \varphi$. These are smooth functions in the required spaces. With $q=|x{|}^2/L^2$,Hence $\Vert f{\Vert }_{\infty }\le a-1+6a/L^2<a=\Vert \varphi {\Vert }_{\infty }$. Moreover $V(s)=s^2/2+1-\cos s$ has $V^{\prime \prime }=1+\cos s\ge 0$, so the energy is convex and this critical point is a minimizer. The example therefore satisfies even the minimizing and $C^2$ hypotheses of the printed claim.

The direct method in the calculus of variations gives the following existence theorem. Suppose bounded sequences in the Banach space $\mathcal{U}$ have $\tau$-convergent subsequences, and $E$ is a proper extended-real function with coercivity and is $\tau$-sequentially lower semicontinuous. Then $E$ has a finite-valued global minimizer.

To prove this, properness makes $m=\inf E<+\infty$. First rule out $m=-\infty$: a sequence with $E(u_n)\to -\infty$ eventually lies in a fixed sublevel set, which is bounded by coercivity. A $\tau$-convergent subsequence would then have a limit $u$ with $E(u)\le -\infty$, contradicting the codomain. Hence $m$ is finite.

Choose a minimizing sequence with $E(u_n)\to m$. It eventually belongs to the bounded sublevel set $\{E\le m+1\}$. Extract $u_{n_j}\stackrel{\tau }{\to }\bar{u}$. By sequential lower semicontinuity,Thus $E(\bar{u})=m$. In particular, a reflexive Banach space with the weak topology supplies the required subsequence property by weak sequential compactness of bounded sequences in a reflexive Banach space. A strictly convex function has at most one minimizer; this is an additional property, not part of the existence theorem.

Take the weak topology on the reflexive Banach space $\mathcal{U}$. A bounded linear operator is weak-to-weak continuous: $u_n\rightharpoonup u$ implies $⟨K{u}_n,v⟩\to ⟨Ku,v⟩$ for every $v\in \mathcal{V}$, by its adjoint operator. The weak lower semicontinuity of the Hilbert norm therefore makes $D(u)=\Vert Ku-f\Vert$ weakly sequentially lower semicontinuous.

This functional is finite everywhere and nonnegative. If it has coercivity, the direct method in the calculus of variations supplies a global minimizer $\bar{u}$. Squaring the residual does not change its global minimizers. Here the adjoint operator has codomain ${\mathcal{U}}^{\ast }$, since the source is a Banach space. Differentiating the squared residual along every real direction $h\in \mathcal{U}$ gives $\mathrm{Re}⟨K^{\ast }(K\bar{u}-f),h⟩=0$ in the dual pairing, hence the normal equation for a linear inverse problem $K^{\ast }(K\bar{u}-f)=0$. HenceThe converse fails. For $K:{\mathbb{R}}^2\to \mathbb{R}$, $K(x,y)=x$, and $f=1$, the datum is in the operator range, but $D(1,n)=0$ while $\Vert (1,n)\Vert \to \infty$. Thus admissible data do not imply coercivity of the residual; an unpenalized null space already prevents it.

The Iterated Tikhonov regularization update is the proximal operator stepThe objective has coercivity and is weakly sequentially lower semicontinuous and strongly convex, so the direct method in the calculus of variations and uniqueness of a minimizer of a strictly convex function give existence and uniqueness. Its gradient equation iswhich is exactly $(I+\tau K^{\ast }K)u=u_{\delta }^{(k)}+\tau K^{\ast }{f}^{\delta }$. The second term penalizes the change from the previous iterate, rather than the norm of $u$ relative to zero.

First interpret the stated well-definedness as including uniqueness of the global minimizer for each positive parameter. Under that assumption, the parameter continuity of variational regularization follows from the direct method in the calculus of variations argument below.

Let $M=D(0)=\Vert f{\Vert }^2/2$. For large $n$, ${\alpha }_n\ge \beta :=\alpha /2$. Comparison with zero, using $J(0)=0$, givesCoercivity of ${\Phi }_{\beta ,f}$ therefore bounds the sequence. Take a $\tau$-convergent subsequence with limit $\bar{u}$. For any $v\in \mathrm{dom}J$, optimality givesThe last two terms vanish. Sequential lower semicontinuity at the fixed parameter $\alpha$ implies ${\Phi }_{\alpha ,f}(\bar{u})\le {\Phi }_{\alpha ,f}(v)$. Thus every cluster point is a global minimizer at $\alpha$. Uniqueness makes it $u_{\alpha }$, and the subsequence property provesIndeed, a subsequence staying outside a neighborhood of $u_{\alpha }$ would have a further convergent subsequence, whose limit must be $u_{\alpha }$, a contradiction.

The listed existence hypotheses alone do not imply uniqueness or convergence of an arbitrary selection. For example, on ${\mathbb{R}}^2$, set $K(x,y)=x$, $f=1$, andThis is nonnegative, convex, continuous and has $J(0)=0$; every ${\Phi }_{\alpha ,f}$ has coercivity. Its minimizers are $(1/(1+2\alpha ),y)$ for all $|y|\le 1$. Along ${\alpha }_n=\alpha +1/n$, selecting $y_n=(-1{)}^n$ prevents convergence. Without uniqueness, the valid conclusion is that every cluster point minimizes the limiting objective.

Assume $g\in L^2(\Omega )$, as required for the finite-valued data fidelity; extend the total variation seminorm on a domain by $+\infty$ outside $BV(\Omega )$. The zero function is a finite-energy competitor. A minimizing sequence $f_n$ has bounded $\Vert f_n-g{\Vert }_2$ and bounded variation, hence bounded $L^2$ norm. Take a weakly convergent subsequence in $L^2$, with limit $f$. For each fixed test vector field $\zeta$, the pairing $\int f\mathrm{div}\zeta$ is weakly continuous because $\mathrm{div}\zeta \in L^2$. Its supremum has weak lower semicontinuity. The squared Hilbert space norm also has weak lower semicontinuity. Thus the direct method in the calculus of variations gives a minimizer in $BV\cap L^2$.

The variation term is convex, and the squared fidelity term is strictly convex. Their sum is strictly convex on its effective domain, so the minimizer is unique.

Let $T(t)=\min (b,\max (a,t))$. The bounded-variation contraction under clipping gives $|D(T\circ f)|(\Omega )\le |Df|(\Omega )$. If $g(x)\in [a,b]$, then $|T(f(x))-g(x)|\le |f(x)-g(x)|$, strictly whenever $f(x)\notin [a,b]$. A positive-measure set outside the interval would therefore strictly decrease the fidelity integral, while not increasing variation, contradicting minimality. HenceNo compactness theorem for the $BV$ embedding is required for this weak-$L^2$ argument.

Take a minimizing sequence $(X_k,Y_k)$ in the affine Sobolev class $W$. The standing bounds $C^{-1}\le X_k\le C$ make the energy uniformly equivalent toThe fixed boundary values and the Poincaré inequality therefore bound the sequence in $H^1(\Omega ;{\mathbb{R}}^2)$. By weak compactness in a reflexive Banach space, a subsequence converges weakly to $(X,Y)$. The assumed weak closedness keeps the limit in $W$, and the assumed weak lower semicontinuity givesThus the direct method in the calculus of variations produces a minimizer. Its first variation vanishes in every compactly supported direction, so part (b) makes it a weak solution of the system.

The boundary data are encoded by the Affine Sobolev spaceA function $u\in {\mathcal{A}}_{\phi }$ is a weak solution of the homogeneous p-Laplacian equation when

For existence, take a minimizing sequence for the p-energy on ${\mathcal{A}}_{\phi }$. The Poincaré inequality bounds $u-\phi$ in $W^{1,p}$ by its gradient, so the sequence is bounded in the reflexive Banach space $W^{1,p}$. A weakly convergent subsequence remains in the weakly closed affine space, and convexity of $|\eta {|}^p$ gives weak lower semicontinuity. The direct method in the calculus of variations therefore produces a minimizer, whose first variation is precisely the displayed weak equation.

Take a minimizing sequence for the variational regularization functionalIts coercivity makes the sequence bounded. Since $X$ is a reflexive Banach space, a subsequence converges weakly to some $u\in X$. A convex norm-lower-semicontinuous functional has weak lower semicontinuity, so this applies to both $J$ and the convex continuous map $u\mapsto \Vert Au-f{\Vert }_Y^2$. Thereforeand $u$ is a minimizer. This is the direct method in the calculus of variations.

A minimizing sequence in ${\mathrm{Lip}}_g(\Omega ,k)$ is uniformly bounded and equi-Lipschitz. The Arzelà-Ascoli theorem gives a uniformly convergent subsequence with limit $u$ having the same boundary data and Lipschitz constant at most $k$. Its gradients have a weak-star convergent subsequence in $L^{\infty }$, with limit $Du$. Convexity of $F$ makes the integral functional weak-star lower semicontinuous, soThus $u$ attains the infimum by the direct method in the calculus of variations.

For real $f\in L^2({\mathbb{R}}^3)$, this energy functional on $H^1({\mathbb{R}}^3)$ is coercive and attains its minimum by the direct method in the calculus of variations. The nonnegative potential term is weakly lower semicontinuous by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. Its Euler-Lagrange equation is $-\Delta u+u+\sin u=f$ in the weak solution sense. The extra linear restoring term screens the Sine-Gordon equation nonlinearity. The scalar potential $V(s)=s^2/2+1-\cos s$ is convex because $V^{\prime \prime }(s)=1+\cos s\ge 0$; hence any weak critical point is a minimizer. The elliptic regularity estimate for $-\Delta +1$, together with $|\sin u|\le |u|$, gives $u\in H^2$. The Sobolev inequality then puts $u$ in $W^{1,6}$, so Morrey's inequality and uniformly continuous integrable functions vanish at infinity give a continuous representative tending to zero.