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On a bounded regular domain, a bounded subset of $C^{k,\alpha }$ is relatively compact in $C^{l,\beta }$ whenever $k+\alpha >l+\beta$. This is an Arzela-Ascoli theorem argument applied successively to the derivatives.

For a fixed bounded measurable uniformly elliptic operator in divergence form, solutions bounded by one have uniform local Caccioppoli inequalities and Hölder continuity estimates. A diagonal subsequence argument yields a local weak $W^{1,2}$ limit, while the Arzela-Ascoli theorem gives uniform convergence on a fixed smaller closed Euclidean ball. The weak equation passes to the limit because the coefficients are fixed and bounded. A global supremum normalization does not force the interior limit to be nonzero.

On a fixed smooth bounded domain, zero-boundary solutions with forcing in a bounded subset of $C^{0,\mu }$ are uniformly bounded in $C^{2,\mu }$ by a maximum-principle barrier and the global Schauder estimate. The Arzela-Ascoli theorem gives convergence in $C^2$. A closed Hölder-norm constraint passes to the limit by lower semicontinuity of a Hölder seminorm, so the constrained solution set is compact rather than merely relatively compact.

For smooth vorticity, parts (b)--(c) and the transport equation give $\omega (t,S_{0,t}(y))={\omega }_{\mathrm{in}}(y)$ and measure preservation. The change of variables formula therefore conserves $\Vert \omega (t){\Vert }_1$ and $\Vert \omega (t){\Vert }_{\infty }$. Parts (d)--(e) give a uniformly bounded velocity and a uniform log-Lipschitz modulus; applying the time-dependent version of part (f) constructs unique global characteristic curves.

For a rough Yudovich characteristic flow, state the needed temporal hypothesis explicitly:The usual weak Euler solution class provides these conditions and the initial time trace. On $[0,T]$, put $A(t)=\Vert \omega (t){\Vert }_1+\Vert \omega (t){\Vert }_{\infty }$. The velocity bounds are $|u(t,x)|\le CA(t)$ and $|u(t,x)-u(t,y)|\le CA(t)\mu (|x-y|)$, with $A$ integrable on this interval.

For completeness, mollify $u$ spatially. The resulting smooth velocities have trajectories and obey the same speed and modulus bounds, up to one common constant. Their speeds have the integrable majorant $CA(t)$, giving uniform boundedness and equicontinuity of trajectories. Arzela-Ascoli theorem supplies a uniformly convergent subsequence. The modulus estimate controls $u(t,X_{\epsilon })-u(t,X)$, while the spatial mollification error tends to zero with an integrable majorant. Thus the limit solvesReplace $Ct$ by $C{\int }_0^{t}A(s)ds$ in the Osgood uniqueness criterion proof. It gives uniqueness and prevents finite-time escape. The characteristics are globally uniquely defined for each given Euler solution in the usual Yudovich class. This constructs the flow for a fixed solution; it is not by itself the entire uniqueness proof for the nonlinear Euler equation.

Merely saying that both spatial norms are finite separately at each time does not state the local temporal bound or measurability used in this argument. The printed formulation must be interpreted in the usual solution class, or supplemented with these temporal conditions. Also, an initial datum here is a function of $x\in {\mathbb{R}}^2$; the extra time variable in the displayed initial-data space is extraneous.

We prove Kantorovich duality theorem here by a positive-functional extension argument; it produces the minimizing transport plan at the same time as equality of the values. Let $Y=X\times X$ and $c(x,y)=d(x,y)$. This is a bounded continuous function because $X$ is a compact metric space. Write $C(Y)$ for the real space of continuous functions on a compact space, and letOn this vector subspace define $\ell (s)=\int fd\mathbf{P}+\int gd\mathbf{Q}$. It is well-defined: two representations differ by a constant in the first variable and the opposite constant in the second, and both measures have mass one. It is positive, since $s\ge 0$ implies $\min f+\min g\ge 0$ and hence $\ell (s)\ge 0$. Also $\ell (1)=1$.

Define the majorant envelope, which is a sublinear functional:Constants majorize every $h$, and positivity gives $\min h\le p(h)\le \max h$, so this quantity is finite. Taking approximate minimizing majorants proves subadditivity; rescaling majorants proves positive homogeneity. Moreover $p(s)=\ell (s)$ for $s\in S$, and subtracting $s$ from a majorant provesThe dual value in the question is exactlySubadditivity at $c+(-c)=0$ gives $m_d\le p(c)$.

On $S+\mathbb{R}c$, assign the value $\ell (s)+t{m}_d$ to $s+tc$. If $c\notin S$ the representation is unique. If $c\in S$, then $m_d=\ell (c)$, so the assignment is still well-defined. It is dominated by $p$: for $t\ge 0$, use $t{m}_d\le tp(c)=p(tc)$; for $t<0$, positive homogeneity gives $p(tc)=(-t)p(-c)=t{m}_d$. Together with the translation identity, these verify domination in both cases.

The real Hahn-Banach theorem extends this functional to $T:C(Y)\to \mathbb{R}$ with $T\le p$. If $h\ge 0$, then $p(-h)\le \ell (0)=0$, so $T(h)\ge 0$. Thus $T$ is a positive linear functional, with $T(1)=1$ and $T(c)=m_d$. Positivity also gives $|T(h)|\le \Vert h{\Vert }_{\infty }$, so $T$ is continuous. The Riesz-Markov-Kakutani representation theorem supplies a Borel probability measure ${\pi }_0$ on the compact space $Y$, satisfying $T(h)={\int }_{Y}hd{\pi }_0$.

For every $f\in C(X)$, its pullback $f(x)$ belongs to $S$, so $\int f(x)d{\pi }_0=\int fd\mathbf{P}$. Likewise $\int g(y)d{\pi }_0=\int gd\mathbf{Q}$. Uniqueness in the Riesz-Markov-Kakutani representation theorem shows that these are exactly the two marginal distributions. Finally, every feasible pair $f+g\le c$ gives a lower bound for the cost of every transport plan, by integration. Our constructed plan achieves that bound:This Kantorovich duality by positive extension establishes the requested existence and equality without assuming an optimal plan in advance.

The metric structure additionally gives the Kantorovich–Rubinstein theorem formulation. For a feasible pair define $h(x)={\inf }_y\{d(x,y)-g(y)\}$. The triangle inequality for $d$ makes $h$ one-Lipschitz continuous, $h\ge f$, and $h(y)\le -g(y)$. Hence the dual objective is at most $\int hd\mathbf{P}-\int hd\mathbf{Q}$. Conversely $(h,-h)$ is feasible for every one-Lipschitz $h$. ThereforeNormalizing $h(x_0)=0$ for a fixed $x_0$ gives a uniformly bounded equicontinuous family; it is closed and compact in the uniform topology by the Arzela-Ascoli theorem. The objective is uniformly continuous on this family, so the supremum is attained as well. In particular, $m_d$ is the first Wasserstein distance between the two measures.

Use a bounded BV extension operator for the bounded Lipschitz domain. Extend $u_k$ to $w_k\in BV({\mathbb{R}}^n)$, supported in one fixed compact set, with $\Vert w_k{\Vert }_{BV({\mathbb{R}}^n)}\le C_{\Omega }\Vert u_k{\Vert }_{BV(\Omega )}$. Such an extension is obtained by reflection in Lipschitz boundary charts, a partition of unity and a fixed cutoff function. A naive zero extension must also charge its boundary-trace jump; it cannot discard that contribution.

Let $M$ bound these norms and let $w_{k,ϵ}=w_k\ast {\rho }_{ϵ}$ be their mollifications. The BV mollification error estimate givesThe variation measure here is on ${\mathbb{R}}^n$. The printed lemma's $|Dw|(\Omega )$ needs this correction unless it also assumes all the derivative mass lies in $\Omega$: a nonconstant bump supported outside $\overline{\Omega }$ disproves its literal wording. The correct estimate follows by averaging the BV translation estimate $\Vert w(\cdot -h)-w{\Vert }_1\le |h||Dw|({\mathbb{R}}^n)$ over a mollifier supported in $|h|\le ϵ$.

For each fixed $ϵ>0$, convolution gives uniform boundsThe mollifications have common compact support and are uniformly equicontinuous. The Arzela-Ascoli theorem supplies a uniformly convergent subsequence at each scale $ϵ=1/m$. A diagonal subsequence converges at every one of those scales. For two late members of that subsequence,First choose large $m$, then late $k,\ell$; the sequence is a Cauchy sequence in $L^1$. Its limit $w$ belongs to the BV space because the total variation seminorm on a domain is sequentially lower semicontinuous. Uniform derivative bounds and integration by parts give $D{w}_k\stackrel{\ast }{\rightharpoonup }Dw$. Restriction to $\Omega$ proves the required weak-star convergence in BV.

We use a blow-up compactness proof of an interior Schauder estimate, with Taylor normalization in elliptic blow-up arguments. Fix $n,\mu ,\delta$ and suppose no suitable constant exists. There would be solutions $u_j$ with forcing $f_j$ for whichChoose distinct $x_j,y_j\in B_{1/2}$ such that, for $r_j=|y_j-x_j|$,Then $S_j/A_j<2/\delta$, $N_j/A_j<2/j$, andIn particular $r_j\to 0$. Subtract the quadratic Taylor polynomial of $u_j$ at $x_j$ and rescale:Its domain $\{z:x_j+r_{j}z\in B_1\}$ exhausts ${\mathbb{R}}^n$. The normalization givesTaylor theorem, or integration along line segments from zero, consequently bounds $D^2{v}_j,\nabla v_j,v_j$ on every fixed ball by constants times $|z{|}^{\mu },|z{|}^{1+\mu },|z{|}^{2+\mu }$, respectively. The Hessian matrices are equicontinuous. The Arzela-Ascoli theorem and a diagonal subsequence therefore give $v_j\to v$ in $C^2$ on compact subsets of ${\mathbb{R}}^n$.

The rescaled equation ison each compact set. Thus $v$ is harmonic and is smooth. Every entry of its Hessian matrix is harmonic and has a global Hölder seminorm at most $2/\delta$. The allowed Liouville lemma for globally Hölder harmonic functions makes each entry constant. Since $D^{2}v(0)=0$, all these constants are zero.

On the other hand, $e_j=(y_j-x_j)/r_j$ is a unit vector and $|D^2{v}_j(e_j)|=1$ by construction. A further subsequence has $e_j\to e$; the local $C^2$ convergence implies $|D^{2}v(e)|=1$, a contradiction. Hence the desired constant exists, andThe global Hölder seminorm on the limiting Hessian matrix, rather than a bound on its magnitude throughout space, is what makes this Liouville argument work.

We prove compactness of zero-boundary Poisson solutions. Here the printed $\partial \Omega$ must mean $\partial B_1$; otherwise the boundary condition refers to an unspecified set. Use this intended boundary condition throughout. Let $f=\Delta u$. A maximum-principle barrier gives a uniform bound on $u$: with $\psi (x)=(1-|x{|}^2)/(2n)$, we have $\Delta \psi =-1$ and $\psi =0$ on $\partial B_1$. Since $|f|\le 1$,The weak maximum principle gives $-\psi \le u\le \psi$, so $\Vert u{\Vert }_{\infty }\le 1/(2n)$.

Standard elliptic regularity on the smooth ball first gives $u\in C^{2,\mu }(\overline{B_1})$. The global Schauder estimate then givesThus the functions and all their derivatives through order two are uniformly bounded and equicontinuous on the closed ball. The Arzela-Ascoli theorem gives a subsequence converging uniformly along with all these derivatives. Integration along line segments identifies the limiting derivatives, so the convergence is in $C^2(\overline{B_1})$.

It remains to keep the limit in the set, rather than merely proving relative compactness. If $u_j\to u$ in $C^2$, then the zero boundary values pass to the limit and $f_j=\Delta u_j\to f=\Delta u$ uniformly. For each pair of distinct points,Taking suprema gives $[f{]}_{\mu }\le {\mathrm{liminf}}_j[f_j{]}_{\mu }$, while $\Vert f_j{\Vert }_{\infty }\to \Vert f{\Vert }_{\infty }$. Hence $\Vert f{\Vert }_{C^{0,\mu }}\le 1$. The limit belongs to the set, proving sequential compactness in $C^2(\overline{B_1})$. The fixed positive Hölder exponent supplies the compactness; a bound on the supremum norm of $\Delta u$ alone would not supply this argument.

For the inclusion, $\Vert Tf{\Vert }_{\infty }\le \Vert f{\Vert }_{\infty }+\Vert f^{\prime }{\Vert }_{\infty }$, with equality for a nonzero constant. Thus $\Vert T\Vert =1$. The domain unit ball is uniformly bounded and has $\Vert f^{\prime }{\Vert }_{\infty }\le 1$, so it is equicontinuous. The Arzela-Ascoli theorem makes its image relatively compact in $C[0,1]$. Therefore $T$ is compact.