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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.

Put $\delta =\mathrm{dist}({\Omega }^{\prime },\partial \Omega )>0$. Choose a smooth cutoff $0\le \eta \le 1$, equal to one on ${\Omega }^{\prime }$, supported within its $\delta /2$ neighborhood, with $|D\eta |\le C_n/\delta$. Such a cutoff is obtained by smoothing a distance cutoff; its support is relatively compact in $\Omega$. Let $0<|h|<\delta /8$, fix a coordinate $k$, and put $z={\Delta }_k^{h}u$. It lies in $W^{1,2}$ on the cutoff neighborhood because both translated copies of $u$ do. Extend ${\eta }^{2}z$ by zero, and use the test functionThe weak equation is $\int Du\cdot D\varphi =-\int f\varphi$. The discrete integration-by-parts identity $\int a{\Delta }_k^{-h}b=-\int ({\Delta }_k^{h}a)b$, valid by translation with these supports, turns it intoWrite $X=\Vert \eta Dz{\Vert }_2$, $U=\Vert z{\Vert }_{L^2(\mathrm{supp}\eta )}$, $A=\Vert D\eta {\Vert }_{\infty }$ and $F=\Vert f{\Vert }_2$. Expanding the left side gives $X^2$ plus a cross term of absolute value at most $2AUX$. Applying part (a) on the whole space to the zero extension of ${\eta }^{2}z$ bounds the right side byConsequentlywhere the last step is the elementary Young inequality. Part (a), on a slightly larger cutoff neighborhood if needed, gives $U\le \Vert D_{k}u{\Vert }_{L^2(\Omega )}$. We conclude thatuniformly as $h\to 0$.

To obtain actual second weak derivatives, choose $h_j\to 0$. For each $i,k$, the bounded sequence ${\Delta }_k^{h_j}{D}_{i}u$ has a weakly convergent subsequence in $L^2({\Omega }^{\prime })$, say to $g_{ik}$. For every smooth compactly supported $\psi$ in ${\Omega }^{\prime }$,Thus $g_{ik}=D_k{D}_{i}u$ is the second weak derivative. weak lower semicontinuity gives the same uniform bound for these derivatives. Adding the already known zeroth and first derivative norms and summing over the finitely many coordinates provesAll derivative existence and estimates have been established by the difference-quotient argument; no second derivatives were assumed in choosing the test. This proves the Interior second-derivative estimate for the Poisson equation in the stated setting.

For $u\in H_0^1(\Omega )$, both $u$ and its weak gradient are square integrable. The assumed bounds give $|F(u,x)|\le K(1+|u{|}^2)$, so the potential term is integrable and $E(u)$ is finite. Alsowhich is bounded below and coercive in $H_0^1$ by the Poincaré inequality.

For distinct $u,v$ and $0<t<1$, convexity of $F$ and expansion of the squared gradient giveThe last norm is nonzero: a zero weak gradient makes $u-v$ constant, and its zero trace forces that constant to vanish. Thus $E$ is strictly convex.

Weak lower semicontinuity means that $u_j\rightharpoonup u$ in $H_0^1$ implies $E(u)\le {\mathrm{liminf}}_{j}E(u_j)$. A minimizing sequence is bounded by coercivity. Reflexivity of this Hilbert space gives a weakly convergent subsequence with limit still in $H_0^1$. The assumed weak lower semicontinuity makes that limit a minimizer, and strict convexity makes it unique. Hence $\boxed{E\text{ has exactly one minimizer in }{H}_0^1(\Omega )}$.

On a real Hilbert space, for a proper lower-semicontinuous convex function $E$, the proximal operator isThe quadratic makes this objective strongly convex and coercive; the affine lower bound available for a proper closed convex function and weak lower semicontinuity give existence. The minimizer is unique. The subdifferential optimality condition isFor a positive parameter $t$, ${\mathrm{prox}}_{tE}$ uses $tE(x)$ in the same formula. On a general Banach space, the analogous norm-squared minimization is Banach-space proximal minimization; it can be set-valued and the identity-valued Hilbert derivative must be replaced by the duality mapping.

Use the standard inverse-problem setting of a bounded linear $K:{\ell }^2\to {\ell }^2$ and exact data having a finite-penalty solution. Let $u^{†}$ denote the unique $J$-minimizing exact solution for $J(u)=\Vert u{\Vert }_1+\beta \Vert u{\Vert }_2^2$, $\beta >0$. This is the limiting target of elastic net regularization in Hilbert sequence space; it need not be the Moore–Penrose minimum-norm solution unless additional hypotheses identify them, for example injectivity of $K$.

For each $\alpha >0$, the reconstruction objective is weakly lower semicontinuous, coercive by $\alpha \beta \Vert u{\Vert }_2^2$, and strictly convex. The direct method on the reflexive l2 sequence space gives existence, and strict convexity gives uniqueness. For two data sets, monotonicity of the one-norm subgradients and the quadratic penalty give, with $d=u_1-u_2$ and $g=f_1-f_2$,Completing the square yields $\Vert d\Vert \le \Vert g\Vert /(2\sqrt{2\alpha \beta })$, so each reconstruction map is continuous.

Minimality against $u^{†}$ givesChoose $\alpha \to 0$ and ${\delta }^2/\alpha \to 0$, for example $\alpha =\delta$. This bounds the Hilbert norms, drives the data residual to zero and gives $\mathrm{limsup}J(u_{\alpha }^{\delta })\le J(u^{†})$. Every weakly convergent subsequence has an exact solution as its limit, by boundedness of $K$; weak lower semicontinuity and uniqueness of the penalty minimizer identify that limit with $u^{†}$. Separately applying weak lower semicontinuity to the one-norm givesWeak convergence and convergence of Hilbert norms imply strong convergence by the Radon-Riesz theorem. This proves norm convergence of elastic net regularization:The argument applies to every sequence of noise levels and admissible data, hence gives the usual uniform noise-ball convergence for a fixed admissible exact datum.

For nonlinearity of elastic net reconstruction, zero is the minimizer exactly when $\Vert K^{\ast }{f}^{\delta }{\Vert }_{\infty }\le \alpha$. This includes a neighborhood of zero. If $K\ne 0$, choose $e_j$ with $K{e}_j\ne 0$ and $f^{\delta }=tK{e}_j$ for $t\Vert K{e}_j{\Vert }^2>\alpha$; zero then fails its optimality condition. A linear map vanishing on a neighborhood would vanish everywhere, so the elastic-net reconstruction is nonlinear whenever $K\ne 0$. For $K=I$ the explicit formula is $[R_{\alpha }f{]}_j=\mathrm{sign}(f_j)(|f_j|-\alpha {)}_{+}/(1+2\alpha \beta )$, exhibiting soft thresholding.

A concrete distinction from Moore–Penrose convergence is $K(u)=u_1+2u_2$ (embedded in the first output coordinate of ${\ell }^2$), exact datum one, and $\beta =1$. Its minimum-norm solution is $(1/5,2/5,0,\dots )$, with penalty $4/5$, while the unique penalty-minimizing exact solution is $(0,1/2,0,\dots )$, with penalty $3/4$. For $0<\alpha <2$ the exact-data elastic-net reconstruction is $(0,(2-\alpha )/(4+2\alpha ),0,\dots )$, converging to the latter. Thus it is not a Moore–Penrose-consistent regularization for every noninjective operator under the earlier definition.

The PDF supplies neither a nonzero assumption on $K$ nor a target/domain qualification here. Its unconditional nonlinearity claim is false for $K=0$, whose reconstruction is the zero linear map. The convergence proof above states the finite-penalty exact-solution hypotheses explicitly; it establishes convergence to the penalty-minimizing solution, not an automatically identical minimum-norm least-squares solution.

Formal self-adjointness gives symmetry of $B$, and strict positivity gives positive definiteness, so $B$ is an inner product. Bounded coefficients give $B[u,u]\le C\Vert u{\Vert }_{H^1}^2$. To prove the lower bound, uniform ellipticity givesIf no lower bound by the $H^1$ norm existed, choose $u_j\in H_0^1$ with $\Vert u_j{\Vert }_{H^1}=1$ and $B[u_j,u_j]\to 0$. By weak convergence in a Hilbert space and the Rellich-Kondrachov compactness theorem, a subsequence has $u_j\rightharpoonup u$ in $H_0^1$ and $u_j\to u$ in $L^2$. The principal energy has weak lower semicontinuity: it is the squared $L^2$ norm of the bounded multiplication operator $a^{1/2}Du$. The potential term converges because $c$ is bounded. HenceStrict positivity forces $u=0$. The preceding ellipticity estimate now gives $\Vert D{u}_j{\Vert }_2\to 0$, while $\Vert u_j{\Vert }_2\to 0$, contradicting $\Vert u_j{\Vert }_{H^1}=1$. Thus strict positivity implies coercivity for an elliptic Dirichlet form, andThe new norm is equivalent to the original norm, so the space remains complete.

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.

Minimize $\Vert u{\Vert }_{\Sigma }^2$ subject to $\Vert u{\Vert }_2=1$. A minimizing sequence is bounded in $\Sigma$, and part 2 supplies a subsequence converging strongly in $L^2$ and weakly in $\Sigma$. The constraint survives the strong convergence, while weak lower semicontinuity of the norm shows that the limit $\psi$ attains the minimum. Since $\Vert |\psi |{\Vert }_2=\Vert \psi {\Vert }_2$ and $|\nabla |\psi ||\le |\nabla \psi |$ almost everywhere, we may take $\psi \ge 0$.

The Lagrange multiplier equation iswhere testing with $v=\psi$ shows that $\lambda =\Vert \psi {\Vert }_{\Sigma }^2>0$. ThusThis is the ground-state eigenfunction of a confining Schrödinger operator.

By the two invariances from part 1, normalize a minimizing sequence $(u_n)$ so thatReplacing $u_n$ by $|u_n|$ and then by its symmetric decreasing rearrangement preserves its $L^2$ and $L^p$ norms and does not increase the gradient norm. A harmless dilation restores the normalization, so we may take the sequence nonnegative, radial, and radially decreasing.

The sequence has a weakly convergent subsequence in $H^1$. Radial compactness and the fixed scale give strong convergence in $L^p$; in particular the limit is nonzero because $\Vert u_n{\Vert }_p^p\to I^{-1}$. Weak lower semicontinuity of the $L^2$ and gradient norms then shows that the limit attains the infimum. This is the existence of a Weinstein-functional minimizer.

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.

Let $B[u,v]={\int }_U{a}^{ij}{\partial }_{i}u{\partial }_{j}v+cuv$ on $H_0^1(U)$ for a bounded smooth domain, smooth symmetric uniformly positive coefficients, and bounded real $c$. If $B[u,u]>0$ for every nonzero $u$, then $B[u,u]\ge \alpha \Vert u{\Vert }_{H^1}^2$ for some $\alpha >0$. If this failed, choose $\Vert u_j{\Vert }_{H^1}=1$ with $B[u_j,u_j]\to 0$. The Rellich-Kondrachov compactness theorem gives a subsequence weakly converging in $H_0^1$ and strongly in $L^2$. The principal energy has weak lower semicontinuity, and the potential term converges, so strict positivity forces the limit to be zero. Uniform ellipticity then gives $\Vert D{u}_j{\Vert }_2\to 0$, contradicting normalization. Merely being a positive semidefinite operator is insufficient: $-\Delta -{\lambda }_1$ has a nonzero Dirichlet eigenfunction of zero energy.

For $g\in L^2$ on a bounded domain, total variation denoising has a unique minimizer. A bounded minimizing sequence has a weakly convergent $L^2$ subsequence; variation is a supremum of weakly continuous test-function pairings and the fidelity has weak lower semicontinuity. The strictly convex fidelity proves uniqueness. Bounded-variation contraction under clipping proves that data range bounds pass to the minimizer.