Proof. For each \(x_0\in\Omega\), choose \(r>0\) with \(\overline B_{2r}(x_0)\subset\Omega\). Continuity bounds \(u\) on this closed ball, and \(y\mapsto u(x_0+2ry)\) is infinity-harmonic on \(B_1\). Theorem 1 applies and gives the conclusion near \(x_0\). ◻

Proof. Suppose that \(\phi\) touches \(U\) from above at \((y_0,b_0)\). Restriction to the vertical line gives \(\phi_b(y_0,b_0)=1\). Put \(p=\nabla_y\phi(y_0,b_0)\). If \(p\ne0\), choose an affine function \(\ell\) with \(\ell(y_0)=b_0\) and \(\nabla\ell\cdot p=1\). The function \[\psi(y)=\phi(y,\ell(y))-\ell(y)\] touches \(u\) from above at \(y_0\). At that point \(\nabla\psi=p\), because \(\phi_b=1\), and \[D^2\psi[p] =\phi_{yy}[p]+2\phi_{yb}\cdot p+\phi_{bb} =D^2\phi[(p,1)]=\Delta_{\infty}\phi.\] The viscosity inequality for \(u\) gives the required sign. If \(p=0\), the vertical upper contact gives \(\phi_{bb}\ge0\), and this is \(\Delta_{\infty}\phi\) because \(\nabla\phi=(0,1)\). Lower tests are identical with the inequalities reversed. ◻

Proof. Fix \(\overline B_r(z)\subset D\) and \(L>L_r^+(w;z)\). Choose \(\gamma>0\) so small that \[\gamma r<L-L_r^+(w;z),\qquad 2\gamma r<L.\] On \(\partial B_r(z)\) the function \(w\) lies strictly below \[w(z)+L|Z-z|-\gamma|Z-z|^2.\] It lies below this function throughout the ball. Otherwise the positive maximum of their difference is attained at an interior point different from \(z\), and the displayed function, shifted by a constant, is an upper test there. Writing \(\rho=|Z-z|\), its infinity Laplacian is \[-2\gamma(L-2\gamma\rho)^2<0,\] contrary to the viscosity inequality. Letting \(\gamma\downarrow0\) and then \(L\downarrow L_r^+(w;z)\) gives \[w(Z)-w(z)\le L_r^+(w;z)|Z-z|\qquad (Z\in\overline B_r(z)).\] Applying the same argument to \(-w\) gives the lower bound with \(L_r^-(w;z)\). These bounds prove monotonicity in the radius.

For centers in a compact neighborhood, choose one small radius whose balls remain in \(D\). The two slopes at that radius are uniformly bounded by the oscillation on a larger compact set divided by the radius. The preceding inequalities then give a common local Lipschitz bound. Continuity in the center at fixed radius follows from uniform continuity of \(w\) on a compact neighborhood, taking the maximum over the fixed set \(\{|h|\le r\}\). ◻

Proof. The local Lipschitz bound and the exhaustion of \(\mathbb{R}^n\) by the rescaled domains give subsequential convergence on compact sets. By Lemma 4, the finite limits \[L_*^\pm=\lim_{r\downarrow0}L_r^\pm(w;z)\] exist. They are equal. Indeed, if \(h_r\) attains the positive maximum, then for each fixed small \(R>r\), once \(r\) is sufficiently small, \[L_r^+(w;z)\le L_r^-(w;z+h_r)\le L_R^-(w;z+h_r).\] First let \(r\downarrow0\), using continuity at fixed \(R\) and \(|h_r|\le r\), and then let \(R\downarrow0\). This gives \(L_*^+\le L_*^-\). Reversing signs gives equality; write the common value as \(L_*\).

Let \(V\) be a limit. Uniform convergence on balls shows that \[L_R^+(V;0)=L_R^-(V;0)=L_*\qquad(R>0),\] and that the slopes at every other center retain monotonicity in their radius. In particular \(|V(Y)|\le L_*|Y|\). For any center \(q\), \[L_T^\pm(V;q)\le\frac{L_*(T+|q|)+|V(q)|}{T}.\] For a fixed radius at \(q\), bound its slope by the slope at \(T\) and send \(T\to\infty\). Thus every slope of \(V\) is at most \(L_*\), so \(V\) is globally \(L_*\)-Lipschitz.

If \(L_*=0\), then \(V=0\). Otherwise let \(x_R\) and \(y_S\) attain the positive and negative extrema on the balls of radii \(R\) and \(S\) centered at zero. The equality of the origin slopes gives \[L_*(R+S)=V(x_R)-V(y_S) \le L_*|x_R-y_S|\le L_*(R+S).\] Equality throughout forces \(|x_R|=R\), \(|y_S|=S\), and these points to lie on opposite rays of one line. Comparing all choices of \(R,S\) gives a unit vector \(e\) such that \(V(te)=L_*t\) for every real \(t\). The Lipschitz bound then yields \[L_*t-L_*|Y-te|\le V(Y)\le L_*t+L_*|Y-te|.\] Letting \(t\to+\infty\) in the lower bound and \(t\to-\infty\) in the upper bound gives \(V(Y)=L_*e\cdot Y\). ◻

Proof. Put \(w_i=U_i-\ell_i\) and \(\varepsilon_i=\|w_i\|_{L^\infty(B_1)}\). The slopes of \(\ell_i\) are bounded: evaluating at \(e/2\) and \(-e/2\) for every unit vector \(e\) gives \(|\nabla\ell_i\cdot e|\le\mathop{\mathrm{Lip}}(U_i)+2\varepsilon_i\). Thus \(w_i\) have a common Lipschitz bound \(C\).

Suppose that (1) fails. Passing to a subsequence and replacing \(U_i,\ell_i\) by their negatives if necessary, there are endpoint pairs in \(B_{1/4}\) with positive midpoint defect greater than \(\delta|x-y|\) for some fixed \(\delta>0\). For triples \(x,y,z\in\overline B_{3/4}\) with \(x\ne y\), write \[c=\frac{x+y}{2},\qquad h=z-c,\qquad r=|x-y|, \qquad f(r)=r-\frac18r^{3/2}.\] Here \(0<r\le3/2\), so \(\nu r\le f(r)\le r\) with \(\nu=1-\sqrt{3/2}/8>0\). Choose a smooth function \(B(c)\) that equals \(\delta/2\) on \(\overline B_{1/4}\), is at least \(\delta/2\) everywhere, and equals a sufficiently large constant \(B_{\mathrm{out}}\) when \(|c|\ge1/2\). Consider \[ w_i(x)+w_i(y)-2w_i(z)-B(c)f(r)-\frac{|h|^2}{r^2}. \tag{2}\] Allowing the third point to move permits the use of Lemma 6, while the quadratic penalty keeps it close to the midpoint. The factor \(B(c)\) localizes the maximum, and the concavity of \(f\) will supply the dominant negative second variation. The violating pair with \(z=c\) makes this expression positive. On the other hand it is at most \[Cr+2C|h|-B(c)f(r)-\frac{|h|^2}{r^2} \le Cr+C^2r^2-B(c)f(r).\] Its upper limit as \(r\downarrow0\) is therefore at most zero, uniformly in the other variables. A positive maximum is attained at some triple with \(r>0\).

At these maximizing triples, \(r\to0\) as \(i\to\infty\): positivity and \(Bf\ge\delta\nu r/2\) give \(r\le8\varepsilon_i/(\delta\nu)\). Comparing the maximizing value with the admissible choice \(z=c\) also gives \[\frac{|h|^2}{r^2}\le2\bigl(w_i(c)-w_i(z)\bigr)\le2C|h|, \qquad |h|\le2Cr^2.\] Choose \(B_{\mathrm{out}}>(C+\tfrac32C^2)/\nu\). The preceding upper bound for (2) is then nonpositive when \(|c|\ge1/2\). Hence \(|c|<1/2\) at a positive maximum. Since \(r\to0\) and \(h=O(r^2)\), the maximizing triples are interior to \((B_{3/4})^3\) for all large \(i\).

At one such triple set \(q=\nabla\ell_i\) and \(e=(x-y)/r\). Apply Lemma 6 to the functions \(U_i,U_i,-2U_i\) and the test \[\Psi=\ell_i(x)+\ell_i(y)-2\ell_i(z)+B(c)f(r)+\frac{|h|^2}{r^2}.\] This test is \(C^2\) near the triple. Define \(p_1=D_x\Psi\), \(p_2=D_y\Psi\), and \(p_0=-D_z\Psi/2\). Direct differentiation gives \[\begin{align*} p_1&=q+\tfrac12f(r)DB(c)-h/r^2+a e,\\ p_2&=q+\tfrac12f(r)DB(c)-h/r^2-a e,\\ p_0&=q-h/r^2,\qquad a=B(c)f'(r)-2|h|^2/r^3. \end{align*}\] These vectors and \(a\) are bounded. Since \(f'(r)=1-3\sqrt r/16\) and \(|h|=O(r^2)\), we also have \(a\ge\delta/4\) for all sufficiently large \(i\).

Let \(\sigma=(p_1,p_2,p_0)\). The third jet supplied by Lemma 6 has gradient \(-2p_0\). The function \(-2U_i\) is infinity-harmonic, so the viscosity inequalities for the three superjets give \[X_1[p_1]\ge0,\qquad X_2[p_2]\ge0,\qquad X_3[-2p_0]\ge0.\] In particular \(X_3[p_0]\ge0\). If \(A=D^2\Psi\) at the maximum, the matrix inequality evaluated at \(\sigma\) gives \[0\le\operatorname{diag}(X_1,X_2,X_3)[\sigma] \le A[\sigma]+\eta|A\sigma|^2.\] The maximum, \(A\), and \(\sigma\) are fixed while \(\eta\downarrow0\), although the jets may vary. It follows that \(A[\sigma]\ge0\).

We compute the same quadratic form by moving \((x,y,z)\) along the straight line with velocity \(\sigma\) and parameter \(\tau\), with \(\tau=0\) at the maximum. The endpoint relative velocity \(p_1-p_2=2ae\) is parallel to \(x-y\), and the third velocity differs from the mean endpoint velocity by \(O(r)\). Write \(c(\tau),h(\tau),r(\tau)\) for the induced quantities, and use dots for their derivatives at zero. The formulas above give \[\dot r=2a,\qquad \ddot r=0,\qquad \dot c=\tfrac12(p_1+p_2)=O(1),\qquad \dot h=p_0-\dot c=-\tfrac12f(r)DB(c)=O(r).\] Since \(h=O(r^2)\), \[\left.\frac{d^2}{d\tau^2}\frac{|h(\tau)|^2}{r(\tau)^2}\right|_{\tau=0} =\frac{2|\dot h|^2}{r^2} -\frac{8\dot r\,h\cdot\dot h}{r^3} +\frac{6\dot r^{\,2}|h|^2}{r^4}=O(1).\] Also \(f''(r)=-3/(32\sqrt r)\), and therefore \[\begin{align*} \left.\frac{d^2}{d\tau^2}\bigl(B(c(\tau))f(r(\tau))\bigr)\right|_{\tau=0} &=B(c)f''(r)\dot r^{\,2}+2f'(r)\dot r\,DB(c)\cdot\dot c\\ &\quad+f(r)D^2B(c)[\dot c]\\ &=-\frac{3B(c)a^2}{8\sqrt r}+O(1)\longrightarrow-\infty. \end{align*}\] The affine part of \(\Psi\) has zero Hessian. Thus \(A[\sigma]\to-\infty\), contradicting its nonnegativity and proving (1). ◻

Proof. The cylinder contains \(\overline B_{\sqrt\lambda r}(z)\). Evaluating the fit on this ball and using the Lipschitz bound gives the first inequality in (8). The exact increment of \(U\) in direction \(e_b\) equals the increment length, which gives the second.

Suppose that (7) fails. There are \(s_i\to0\), functions \(U_i\in\mathcal U\), centers \(z_i\in G\), radii \(0<r_i\le s_iD(z_i)\), and parameters \(\lambda_i\) that are not good at these triples, with \(\lambda_i\ge\delta>0\). Passing to a subsequence, the common bounds give \[z_i\longrightarrow z_*\in\overline G,\qquad U_i\longrightarrow U_\circ \quad\hbox{uniformly on }\overline B_{7/8}^n.\] The limit \(U_\circ\) is infinity-harmonic on \(B_{7/8}^n\) by the usual viscosity stability argument [8]: add a positive fourth-order term to an upper test to make its contact strict, take nearby interior maxima for the approximating functions minus that test, and pass their inequalities to the limit. The added term leaves the first two derivatives at contact unchanged. A negative fourth-order term and minima treat lower tests. This argument also applies to the varying continuous jet operators below, since they converge uniformly on compact jet sets. Choose scales \(R_i\downarrow0\) so slowly that \[ r_i+|z_i-z_*|+ \|U_i-U_\circ\|_{L^\infty(\overline B_{7/8}^n)}=o(R_i). \tag{9}\] Such a decreasing sequence can be obtained after passage to a subsequence, since all three terms tend to zero. The point \(z_*\) is interior to \(B_{7/8}^n\). Proposition 5, applied to the one function \(U_\circ\) at \(z_*\), gives a further subsequence on which its \(R_i\) blow-ups converge locally uniformly to a linear function. By (9), the functions \[\widehat U_i(X)= \frac{U_i(z_*+R_iX)-U_i(z_*)}{R_i}\] converge to that same linear function uniformly on \(B_1^n\). They are infinity-harmonic and uniformly Lipschitz there. Proposition 7 therefore gives numbers \(\varepsilon_i\to0\) such that \[ |\widehat U_i(X)+\widehat U_i(Y) -2\widehat U_i((X+Y)/2)|\le\varepsilon_i|X-Y| \qquad (X,Y\in B_{1/4}^n). \tag{10}\]

Return now to the smaller radii \(r_i\). The functions \[W_i(H)=\frac{U_i(z_i+r_iH)-U_i(z_i)}{r_i}\] are uniformly bounded and Lipschitz on \(\overline B_2^n\), so a further subsequence converges uniformly there to \(W\). The slow-scale choice ensures that \(z_i+r_i\overline B_2^n\subset B_{R_i/4}(z_*)\) for all large \(i\). Substituting \[X=\frac{z_i-z_*+r_iH}{R_i},\qquad Y=\frac{z_i-z_*+r_iK}{R_i}\] into (10) shows that the midpoint defect of \(W_i\) is at most \(\varepsilon_i|H-K|\). Thus \(W\) satisfies the midpoint identity on \(\overline B_2^n\). Continuity and dyadic subdivision on segments make \(W\) affine on this ball. Since \(W(0)=0\) and \(W(e_b)=1\), it has the form \(W(H)=P\cdot H\) with \(P\cdot e_b=1\).

Uniform convergence now gives, for all large \(i\), \[\sup_{|H|\le2}|U_i(z_i+r_iH)-U_i(z_i)-r_iP\cdot H| <\delta r_i.\] The nonzero slope \(P\) therefore gives a fit with error at most \(\lambda_i r_i\) on \(Z_{\lambda_i}(z_i,r_i;P)\subset B_{2r_i}(z_i)\). This contradicts the choice of \(\lambda_i\) and proves (7). ◻

Proof. Choose any sequence \(\gamma_i\downarrow0\). The function \(b(s)s^{-\gamma_i}\) is bounded on intervals separated from zero and is unbounded near zero. Taking a factor-two approximate maximum on a sufficiently long truncated interval \([h_i,1/8]\) gives a point \(s_i\) such that \(s_i\to0\) and \[ \frac{b(s)}{s^{\gamma_i}} \le2\frac{b(s_i)}{s_i^{\gamma_i}}\qquad(s_i\le s\le1/8). \tag{12}\] For example, make the approximate maximum larger than the supremum on \([1/i,1/8]\) to ensure \(s_i<1/i\). Pass to a subsequence to make \(s_i\) decreasing. We have \(b(s_i)>0\). Choose \(U_i\in\mathcal U\), \(z_i\in G\), and \(r_i\le s_iD(z_i)\) with \(\beta_{U_i}(z_i,r_i)>b(s_i)/2\), and then choose a parameter \(\lambda_i>\beta_{U_i}(z_i,r_i)/2\) that is not good for \(U_i\) at this center and radius. This uses only approximate suprema and gives \[ b(s_i)/4<\lambda_i\le b(s_i). \tag{13}\] In particular \(\lambda_i\to0\); pass to a further subsequence if needed to make it decreasing.

Fix \(M\ge2\). For large \(i\), we have \(Mr_i\le Ms_iD(z_i)<D(z_i)\), so every \(z\) with \(|z-z_i|\le Mr_i\) belongs to \(G\). The distance to the boundary then satisfies \[D(z)\ge D(z_i)-Mr_i\ge\tfrac12D(z_i)>0.\] For \(r\le Mr_i\) we have \(r/D(z)\le2Ms_i\le1/8\), so the center and radius are admissible. By (12) and (13), uniformly over all these choices, \[\beta_{U_i}(z,r)\le b(2Ms_i) \le2b(s_i)(2M)^{\gamma_i} \le8\lambda_i(2M)^{\gamma_i} <32\lambda_i<1\] for all sufficiently large \(i\). Here \((2M)^{\gamma_i}\to1\) for each fixed \(M\), so the same \(\Lambda=32\) works for every fixed \(M\); only the threshold in \(i\) depends on \(M\). The parameter \(\Lambda\lambda_i\) is strictly greater than every bad parameter and is therefore good. ◻

Proof. In the common-center case both cylinders contain \(B_{\theta H}(z)\). Evaluation of the difference of the fits on this ball gives \(|P-P'|\le3\theta\). Normalization of the vectors, using the lower norm bound, gives \(|n-n'|\le C\theta\). Choose a fixed small \(c>0\). The point \(z+cHn\) belongs to both cylinders: relative to \(n'\) its transverse displacement is at most \(cH|n-n'|\le Cc\theta H\), and its axial displacement is at most \(cH\). Taking \(c\) sufficiently small, the fit errors at that point give \[|(P-P')\cdot n|\le C\theta^2.\] Since \(1-n'\cdot n=|n-n'|^2/2\), \[\bigl||P|-|P'|\bigr| \le |(P-P')\cdot n|+|P'|(1-n'\cdot n)\le C\theta^2.\]

For displaced centers, a ball \(B_{c\theta H}(z')\) with fixed small \(c\) belongs to both cylinders by the quarter-radius margins. Subtracting the first fit’s value at \(z'\) makes it an estimate for increments from \(z'\) with error \(2\theta^2H\); the second fit has error \(\theta^2H\). Evaluation on the common ball again gives the vector and direction bounds. The segment \(z'+s n'\) for \(|s|\le c'H\) lies in both cylinders when \(c'>0\) is sufficiently small: the first cylinder’s initial offsets use a quarter of its radii, and the additional transverse offset is at most \(c'H|n-n'|\). Evaluation on this segment and the same norm calculation give the last bound in (14). ◻

Proof. Set \(\theta_i=\sqrt{\Lambda\lambda_i}\). A good fit of radius \(Rr_i\) has error \(\theta_i^2Rr_i\) and transverse radius \(\theta_i Rr_i\). Its slope obeys uniform upper and lower norm bounds by (8), once \(i\) is large enough.

For each fixed \(R\ge2\), Lemma 11, applied with any fixed \(M>R\), provides fits at \(z_i\) through successive doublings from \(r_i\), followed if necessary by a final step to \(Rr_i\) of ratio at most two, for all sufficiently large \(i\). Comparing them by Lemma 12 gives, for a slope \(P_R\) at the latter radius and its unit vector \(n_R\), \[ |n_R-P_i/c_i|\le C\theta_i(1+\log R),\qquad \bigl||P_R|-c_i\bigr|\le C\theta_i^2(1+\log R). \tag{18}\] Fix a box \(|x|,|t|\le K\) in the coordinates of (16) and a radius factor \(S>0\). Choose \(R\ge2\) so large that \(R>16K\) and \[K\bigl(\Lambda^{-1/2}+C(1+\log R)\bigr)<R/4.\] Such a choice is possible because \(R/\log R\to\infty\). Next fix \(M>\max\{2,2K,R,S\}\). The physical centers in the box are within \(2Kr_i\) of \(z_i\), and all radii in the comparisons below are at most \(\max\{R,S\}r_i\). Lemma 11 therefore supplies these fits uniformly once \(i\) is sufficiently large. The physical image of the box then lies inside the cylinder at \(z_i\) for \(P_R\), with axial and transverse offsets at most one quarter of the respective radii. For the axial offset use \(|\sqrt{\lambda_i}x|+|t|\le2K\). For the transverse offset, (18) gives the bound \[K\bigl(\sqrt{\lambda_i}+C\theta_i(1+\log R)\bigr)<\theta_i R/4\] in units of \(r_i\).

Compare a fit centered at any point of this box, first at radius \(Rr_i\) with the enclosing fit, and then through successive doublings or halvings, with a final ratio at most two, to radius \(Sr_i\). The number of comparisons depends on \(K,S\) but not on \(i\). If \(P=(P_x,P_t)\) is any such slope in the coordinates of (16), Lemma 12 gives \[|P/|P|-P_i/c_i|\le C_{K,S}\theta_i, \qquad \bigl||P|-c_i\bigr|\le C_{K,S}\theta_i^2.\] Since \(P_i/c_i\) is the last coordinate vector, \[|P_x|\le C_{K,S}\theta_i,\qquad |P_t-c_i|\le\bigl||P|-c_i\bigr| +\tfrac12|P|\,|P/|P|-P_i/c_i|^2 \le C_{K,S}\theta_i^2.\] It follows that the normalized coefficients \[ p_i'=\frac{P_x}{c_i\sqrt{\lambda_i}},\qquad q_i'=\frac{P_t/c_i-1}{\lambda_i} \quad\text{satisfy}\quad |p_i'|+|q_i'|\le C_{K,S}. \tag{19}\] In particular, using the enclosing fit at \(z_i\) in (16) gives on the box \[|v_i(x,t)-p_{i,R}'\cdot x-q_{i,R}'t|\le\Lambda R/c_i.\] Thus \(v_i\) are uniformly bounded on each fixed box.

We next express the domain of a fit in these coordinates. Suppress the index \(i\) temporarily, write \(\lambda=\lambda_i\), and set \[A=1+\lambda q_i',\qquad N=(A^2+\lambda|p_i'|^2)^{1/2}, \qquad n=\frac{(\sqrt\lambda p_i',A)}{N}.\] For fixed \(K,S\), (19) gives \(A>0\), \(N/A=1+O_{K,S}(\lambda)\), and \(|n_x|=O_{K,S}(\sqrt\lambda)\) for large \(i\). The physical axis has ratio \(\sqrt\lambda\,p_i'/A\) of transverse to axial displacement, which becomes the finite shear \(p_i'/A\) after the transverse stretch. Figure 1 illustrates this change of geometry. The next calculation identifies a sheared box contained in the cylinder’s image in normalized coordinates, with a fixed margin at the cylinder’s axial ends. The exact decomposition \[ (\sqrt\lambda h,\ell) =\frac{\ell N}{A}\,n +\left(\sqrt\lambda\left(h-\frac{p_i'}A\ell\right),0\right) \tag{20}\] shows that the mapped increment lies in the radius-\(Sr_i\) cylinder whenever \[ |\ell|\le S/2,\qquad \left|h-\frac{p_i'}A\ell\right|\le S, \tag{21}\] for all large \(i\). Indeed the residual in (20) has transverse projection at most \(\sqrt\lambda S\le\theta_i S\), and its axial projection is \(O_{K,S}(\lambda S)\). The first term has axial length at most \((1+O_{K,S}(\lambda))S/2\), leaving a strict axial margin. Dividing (5) by \(r_i c_i\lambda_i\) therefore gives \[ |v_i(x+h,t+\ell)-v_i(x,t)-p_i'\cdot h-q_i'\ell| \le\frac{\Lambda}{c_i}S \tag{22}\] on (21). The coefficient of \(S\) is independent of \(K\) and \(S\).

This proves equicontinuity on compact sets. Given an accuracy \(\varepsilon>0\), first fix \(S>0\) so small that \(2\Lambda S<\varepsilon/3\). For a fixed box, (19) now bounds its coefficients by some finite \(C_{K,S}\). Choose the increments sufficiently small, depending on this bound and \(S\), that (21) holds and the affine increment in (22) is less than \(\varepsilon/3\). For all sufficiently large \(i\), (15) and (22) then bound the value increment by \(2\varepsilon/3\). Discard an initial segment so that all domains contain the compact box. The threshold in \(i\) may depend on \(\varepsilon\); the finitely many remaining preceding continuous functions can be included by decreasing the increment size. The local bounds and domain exhaustion now give a locally uniformly convergent subsequence, with limit \(v\) and \(v(0,0)=0\).

The jet operators \(X[(p_x,1+\lambda_i p_t)]\) converge locally uniformly to \(X[(p_x,1)]\). Viscosity stability gives (3); equivalently, strictify a smooth upper or lower test for \(v\), choose nearby touching points for \(v_i\), and pass to the limit in (17). To pass the fits, fix a center and a target radius \(R>0\), and take \(S=4R\) in (22). After a further subsequence its bounded coefficients converge to \(p,q\). The shears \(p_i'/(1+\lambda_i q_i')\) converge to \(p\). Every increment with \(|\ell|\le R\) and \(|h-p\ell|\le R\) then satisfies (21), uniformly, for all sufficiently large \(i\). Passing to the limit in (22) yields (4) with \[a=q+|p|^2/2,\qquad C_0=8\Lambda=256.\] The additional subsequence may depend on the chosen center and radius: the limit function \(v\) stays fixed, and (4) asserts existence separately for each choice.

Finally suppose that \(v\) is affine. Since \(v(0,0)=0\), write \(v(x,t)=p\cdot x+qt\). In the physical coordinates consider \[\widetilde P_i=c_i(\sqrt{\lambda_i}p,1+\lambda_i q).\] These slopes are nonzero for large \(i\). The cylinders \(Z_{\lambda_i}(z_i,r_i;\widetilde P_i)\) lie in one fixed box of the rescaled coordinates. In fact, if \[N_i=\bigl(\lambda_i|p|^2+(1+\lambda_i q)^2\bigr)^{1/2}\longrightarrow1,\] a point in such a cylinder, in units of \(r_i\), is \(s(\sqrt{\lambda_i}p,1+\lambda_i q)/N_i+\sqrt{\lambda_i}\xi\) with \(|s|\le1\), \(|\xi|\le1\), and \(\xi\) perpendicular to the axis. Its rescaled coordinates satisfy \(|x|\le |p|/N_i+1\) and \(|t|\le1+\sqrt{\lambda_i}\). On this fixed box the residual for the proposed fit is exactly \[r_i c_i\lambda_i\bigl(v_i(x,t)-p\cdot x-qt\bigr)=o(\lambda_i r_i)\] uniformly, by local uniform convergence and the upper bound for \(c_i\). It is eventually at most \(\lambda_i r_i\) on the whole cylinder. This makes \(\lambda_i\) good for \(U_i\) at \((z_i,r_i)\), contrary to Lemma 11. Hence \(v\) is nonaffine. ◻

Proof of Proposition 10. If no power bound held, Lemma 11 and Proposition 13 would produce a nonaffine entire solution satisfying all hypotheses of Theorem 8, with \(m=d\) and \(C_0=256\). The theorem makes that solution affine, a contradiction. Hence a power bound holds. Reducing its exponent if necessary gives \(\gamma\le1\). Since the family \(\mathcal U\) and the function \(b\) were fixed solely by \(d\), the constants in (11) depend only on \(d\). ◻

Proof of Theorem 1, assuming Theorem 8. Take \(B,\gamma,s_0\) from Proposition 10. At every center \(z\in\overline B_{1/2}^n\) one has \(D(z)\ge d_0:=1/4\). Choose \(A_0>Bd_0^{-\gamma}\). There is \(r_0>0\), depending only on \(d\), such that for all \(0<r\le r_0\), \[r/d_0\le\min\{s_0,1/8\},\qquad A_0r^\gamma<1.\] For every \(U\in\mathcal U\) and each such center and radius, \[\beta_U(z,r)\le b(r/D(z)) \le B(r/d_0)^\gamma<A_0r^\gamma.\] Thus \(\lambda=A_0r^\gamma\) is good, with the same constants for all functions and centers.

Fix \(U\in\mathcal U\). Set \[r_j=2^{-j}r_0,\qquad \rho_j=\sqrt{A_0}\,r_j^{1+\gamma/2},\qquad j\ge0,\] and at each center \(z\in\overline B_{1/2}^n\) choose a fitted slope \(P_j(z)\) for the parameter \(A_0r_j^\gamma\). The fit holds on the inscribed ball \(\overline B_{\rho_j}(z)\), with error at most \(A_0r_j^{1+\gamma}\). Comparing consecutive fits on their common ball \(\overline B_{\rho_{j+1}}(z)\) gives \[|P_j(z)-P_{j+1}(z)| \le\frac{A_0(r_j^{1+\gamma}+r_{j+1}^{1+\gamma})}{\rho_{j+1}} \le C r_j^{\gamma/2}.\] The geometric tail converges uniformly in \(U\) and \(z\). Consequently there is a vector \(P_\infty(z)\) with \[ |P_j(z)-P_\infty(z)|\le C r_j^{\gamma/2}. \tag{23}\] If \(0<|h|\le\rho_0\), choose \(j\) so that \(\rho_{j+1}<|h|\le\rho_j\). The fit at radius \(r_j\) and (23) give \[\begin{align*} \frac{|U(z+h)-U(z)-P_\infty(z)\cdot h|}{|h|} &\le\frac{A_0r_j^{1+\gamma}}{\rho_{j+1}} +|P_j(z)-P_\infty(z)|\\ &\le C r_j^{\gamma/2}\le C'|h|^{\alpha_d}, \end{align*}\] where \[ \alpha_d=\frac{\gamma}{2+\gamma}\in(0,1/3]. \tag{24}\] The final inequality follows from \(\rho_{j+1}=\sqrt{A_0}\,2^{-(1+\gamma/2)}r_j^{1+\gamma/2}\) and \((1+\gamma/2)\alpha_d=\gamma/2\). Thus \(U\) is differentiable at every such \(z\), with gradient \(P_\infty(z)\), and \[ |U(z+h)-U(z)-\nabla U(z)\cdot h| \le C|h|^{1+\alpha_d}\qquad(|h|\le\rho_0). \tag{25}\] All increments here remain in \(G\), since each lies in one of the admissible inscribed balls. The original Lipschitz bound gives \(|\nabla U(z)|\le L\).

To compare gradients, take \(z,z'\in\overline B_{1/2}^n\) with \(\ell=|z-z'|\le\rho_0/2\) and \(\ell>0\), and fix any unit vector \(e\in\mathbb{R}^n\). The evaluation point \(z'+\ell e\) need not lie in \(B_{1/2}^n\); the Taylor estimates centered at \(z\) and \(z'\) still apply. Applying (25) at \(z'\) to the increment \(\ell e\) gives \[U(z'+\ell e)-U(z') =\ell\nabla U(z')\cdot e+O(\ell^{1+\alpha_d}).\] Applying the same estimate at \(z\) to the increments \(z'-z\) and \(z'+\ell e-z\), whose lengths are at most \(2\ell\), gives \[U(z'+\ell e)-U(z') =\ell\nabla U(z)\cdot e+O(\ell^{1+\alpha_d}).\] The constants are independent of \(e,U,z,z'\). Subtracting and taking the supremum over unit vectors proves \[|\nabla U(z)-\nabla U(z')|\le C\ell^{\alpha_d}.\] For larger distances the same estimate follows, after increasing \(C\), from \(|\nabla U|\le L\). Restriction to points \((y,0)\) therefore gives the claimed uniform gradient norm and Hölder seminorm for every normalized \(u\).

Finally, if \(u\) is as in Theorem 1 and its oscillation is \(M>0\), then \((u-u(0))/M\) is a normalized solution: positive scaling and addition of constants preserve the viscosity equation by transformation of tests. Multiplying the normalized estimates by \(M\) gives the theorem. If \(M=0\), the function is constant and the conclusion is immediate. ◻

Proof. Set \[B=\begin{pmatrix}I_m&p_0\\0&1\end{pmatrix}.\] For smooth functions, at corresponding points, \(\nabla_y\bar v=\nabla_xv-p_0\) and \(D^2\bar v=R B^{\mathsf T}D^2vB\). Since \(B(\nabla_y\bar v,1)=(\nabla_xv,1)\), this gives \(F[\bar v]=R F[v]\). Pulling back an upper or lower smooth test gives the same implication in the viscosity sense. For the fits, an increment \((h,\ell)\) in the new variables corresponds to \(R(h+p_0\ell,\ell)\). The old affine increment, divided by \(R\) and with the affine part of (26) subtracted, is \[(p-p_0)\cdot h+ \left(a-a_0-\frac{|p-p_0|^2}{2}\right)\ell.\] The old box conditions become \(|\ell|\le\rho\) and \(|h-(p-p_0)\ell|\le\rho\), and the error is \(C_0\rho\). ◻

Proof. We prove the estimates for centering at the origin. Translation by one unit in time gives the other case after enlarging the boxes by a fixed factor.

First compare two fits \((p,a),(p',a')\) at the same center, with radii \(R\) and \(R'\in[R,2R]\). On the horizontal slice \(\ell=0\), both estimates hold for \(|h|\le R\), so \(|p-p'|\le 3C_0\le4C_0\). Write \[\ell_{p,a}(h,\ell)=p\cdot h+ \left(a-\frac{|p|^2}{2}\right)\ell.\] With \(c=(2+8C_0)^{-1}\), the point \((p cR,cR)\) lies in both sheared boxes. At this point \[(\ell_{p,a}-\ell_{p',a'})(p cR,cR) =\left(a-a'+\frac{|p-p'|^2}{2}\right)cR.\] The two fitting errors sum to at most \(3C_0R\). It follows that \(|a-a'|\le 3C_0/c+8C_0^2\). Thus the differences of both parameters are bounded by one constant whenever the radii differ by at most a factor of two. Comparing along successive dyadic radii with the centered fit of radius \(1\) gives, for every fit at the origin of radius \(r\), \[ |p|+|a|\le C(1+|\log r|). \tag{32}\]

Fix \(K\ge2\). Choose once and for all a sufficiently large constant \(A=A(m,C_0)\) and set \(R=A K L(K)\). If \((p_R,a_R)\) is any fit of radius \(R\) at the origin, then \(\mathcal B_K\) lies in its sheared box with \[|t|\le R/4,\qquad |x-p_Rt|\le R/4.\] Indeed, (32) bounds the latter offset by \(K(1+C(1+\log R))\), while \(\log R\le\log A+2L(K)\). Since \(L(K)\ge1\), the asserted two inequalities hold for all \(K\ge2\) once \(A\) is chosen larger than a fixed multiple of \(1+\log A\). Such a choice is possible because \(\log A/A\to0\). The fit at radius \(R\) now gives \[|v(x,t)| \le C_0R+|p_R|K+ \left(|a_R|+\frac{|p_R|^2}{2}\right)K \le C K L(K)^2,\] which is (27).

Let \(z=(x,t)\in\mathcal B_K\), and compare a fit \((p_z,a_z)\) of radius \(R\) at \(z\) with the enclosing fit \((p_R,a_R)\). For horizontal increments of length at most \(R/2\) from \(z\), the enclosing estimate can be subtracted at the two endpoints, with error at most \(2C_0R\). Comparison with the fit at \(z\) therefore gives \(|p_z-p_R|\le6C_0\). Set \(c'=(4(1+6C_0))^{-1}\). The increments \((p_z c'R,c'R)\) from \(z\) remain in both boxes: in the enclosing box the transverse offset is at most \(R/4+6C_0c'R\le R/2\), and the time offset is at most \(R/2\). The same affine-increment identity used above gives a bound on \(|a_z-a_R|\) depending only on \(C_0\). Hence \(|p_z|+|a_z|\le C L(K)\). Dyadic comparisons at \(z\), from radius \(R\) to an arbitrary radius \(r\), give \[|p|+|a| \le C L(K)+C(1+|\log(r/R)|) \le C\bigl(L(K)+|\log r|\bigr),\] as claimed. These comparisons apply to every choice of fit.

Using a fit of radius \(r\) on the horizontal slice proves (29). For a time increment \(\tau\) with \(0<|\tau|\le1/2\), put \(B=L(K)+|\log|\tau||\) and choose the fit radius \(r=A'|\tau|B\). Here \(A'\) is a constant to be fixed, independent of \(K\) and \(\tau\). Since \(B\ge1\), \[|\log r|\le|\log|\tau||+\log A'+\log B \le2B+\log A'.\] The fit-parameter bound consequently gives \(|p|\le C(3+\log A')B\). Choose \(A'\ge1\) sufficiently large that \(A'\ge C(3+\log A')\). This choice is possible and is made before \(K\) and \(\tau\) vary. Then \(|\tau|\le r\) and \(|p\tau|\le r\), so the vertical increment lies in the fitted box. In that estimate, the affine time coefficient and error are bounded by \[\left(|a|+\frac{|p|^2}{2}\right)|\tau|+C_0r \le C|\tau|B^2.\] This proves (30).

The two moduli, applied successively to a spatial and a time increment in a slightly larger box, give equicontinuity on compact sets. The growth bound gives local uniform boundedness. Arzelà–Ascoli and a diagonal choice of subsequence therefore give local uniform convergence. To pass an upper viscosity test to the limit, add a positive fourth-order term at its contact, preserving its first and second derivatives and making the contact strict on a small closed ball. Maxima of the sequence minus this modified test on that ball converge to the contact, so they are interior for large indices. The viscosity inequalities there pass to the limit by continuity of \(F\). Subtracting the fourth-order term and using minima treats a lower test. To pass the fits, fix one center and one radius \(r\). By (28), pass to a further subsequence of the corresponding fits with \((p_j,a_j)\to(p,a)\). For a point with \(|\ell|\le r\) and \(|h-p\ell|\le r\), use in the \(j\)th fitted box the increment \(h_j=p_j\ell+(h-p\ell)\). It converges to \(h\) and belongs to that closed box. Local uniform convergence gives the limiting inequality. This subsequence may depend on the fixed center and radius, which is sufficient for existence of a fit at each such pair. The centered fit passes directly to the limit. Finally, for \(t\) in a fixed compact interval, (27) gives a bound of the form \(C_J(1+|x|)(1+\log(2+|x|))^2\), uniformly in the centered class. Dividing by \(|x|^2\) proves (31). ◻

Proof. Fix a compact range of centers and of allowed positive lengths. Uniform subquadratic growth and the quadratic cost force all maximizers and minimizers into a common bounded spatial set. They exist there, and the maximum and minimum of a continuous function over this fixed compact set depend continuously on the parameters. The same bounded-set conclusion holds when the length varies in \((0,\ell_0]\). On that set the values of \(f\) are bounded. Comparison with the competitor \(y=x\) then gives \(|y-x|^2\le C\ell\) for an optimizer as \(\ell\downarrow0\). Uniform continuity of \(f\) on the compact set gives convergence to \(f(x,t)\) locally uniformly, as well as \(y\to x\) locally uniformly. This proves continuity at zero. The composition identities follow by optimizing first over the intermediate point and using \[\inf_y\left\{\frac{|y-x|^2}{2\ell} +\frac{|z-y|^2}{2h}\right\} =\frac{|z-x|^2}{2(\ell+h)}.\]

For the convexity assertion, let \(g(\ell)=\mathcal{P}_\ell f(x(\ell),t)\), and let a smooth function \(\psi\) touch \(g\) from above at \(\ell_0>0\). Choose an optimizer \(y_0\) at that length. After a constant adjustment, \[\Psi(y,\ell)=\psi(\ell)+\frac{|y-x(\ell)|^2}{2\ell}\] touches \((y,\ell)\mapsto f(y,t+\ell)\) from above at \((y_0,\ell_0)\). Put \(q=(y_0-x(\ell_0))/\ell_0\), which is the horizontal gradient of \(\Psi\) there. Along the direction \((q,1)\), the Hessian of the quadratic term has value \(|d|^2/\ell_0\). For example, writing \(r=y-x(\ell)\), its value along a direction with \(r\)-velocity \(q-d\) is \(\ell_0^{-1}|q-d-r/\ell_0|^2=|d|^2/\ell_0\). The viscosity inequality gives \(\psi''(\ell_0)+|d|^2/\ell_0\ge0\). Thus every upper test for \(g(\ell)+|d|^2\ell\log\ell\) has nonnegative second derivative at positive lengths. This is the one-dimensional viscosity criterion for convexity. To see it directly, a failure of convexity on \([a,b]\) lets one place a strictly concave quadratic above the chord but below the function somewhere; the positive maximum of their difference yields an upper test with negative second derivative. Continuity at zero extends the convexity to that endpoint.

Finally, the map \(\widetilde f(x,t)=-f(x,-t)\) exchanges subsolutions and supersolutions of (3), as pullback reverses the sign of \(F\). It also gives \(\mathcal{P}_\ell\widetilde f(x,t)=-\mathcal{M}_\ell f(x,-t)\). The assertion for \(\mathcal{M}\) follows.

For the last convergence assertion, fix a compact range of allowed centers and positive lengths. The common subquadratic bound and the local uniform bounds confine every optimizer for \(f_j\) and for \(f\) to one compact spatial set, by the first argument of the proof. On that set the functions inside the supremum and infimum converge uniformly. Their extrema therefore converge uniformly on the chosen parameter range. ◻

Proof. Convexity of \(\mathcal{P}_\ell f(z)\), with value \(f(z)\) at zero, gives the first monotonicity. Let \(\widetilde z_\ell=(y_\ell,t-\ell)\) be a backward optimizing point for \(\mathcal{M}_\ell f(z)\). Using \(z\) as a forward competitor from \(\widetilde z_\ell\) gives \[\mathcal{S}_\ell^-f(z)\le\mathcal{S}_\ell^+f(\widetilde z_\ell).\] The optimizer localization at length zero in Lemma 16 shows \(\widetilde z_\ell\to z\). Fix a small \(h>0\) whose forward time argument is allowed near \(z\). For all sufficiently small \(\ell<h\), monotonicity bounds the right side by \(\mathcal{S}_h^+f(\widetilde z_\ell)\). Continuity for fixed \(h\) gives an upper limit at most \(\mathcal{S}_h^+f(z)\). Now let \(h\downarrow0\) to obtain (37). This order of limits also covers the value \(-\infty\).

Apply this conclusion to \(\widetilde f(x,t)=-f(x,-t)\) to obtain the supersolution statements. For a solution both monotone limits exist and the two inequalities identify them. Each is bounded above by the finite slope at any fixed positive allowed length, proving the range in (38). Locally in \(z\), choose a common small range of lengths. The common limit is the infimum of the continuous functions \(\mathcal{S}_\ell^+f(z)\) over that range, so it is upper semicontinuous. ◻

Proof. By Lemma 17, it suffices to bound the two slopes at length \(K\). Suppose an optimizing displacement has length \(BK\) with \(B\ge2\). Comparing its value with the competitor having no displacement, and using (27) in a box of size \((B+2)K\), gives \[\frac{B^2K}{2} \le C BK\bigl(L(K)+\log B\bigr)^2.\] This estimate holds for the supremum and for the infimum. Consequently \(B\le C(L(K)+\log B)^2\le2C L(K)^2+2C(\log B)^2\). Choose a fixed \(B_0\) so large that \(2C(\log B)^2\le B/2\) for \(B\ge B_0\). The last inequality then yields \(B\le4C L(K)^2\) for \(B\ge B_0\), and the same type of bound holds when \(B<B_0\). This proves the displacement claim. The optimizer and its time argument now lie in a box of size \(C K L(K)^2\). Applying (27) there and observing that \(L(C K L(K)^2)\le C L(K)\) bounds the absolute values of \(v\) there by \(C K L(K)^4\). Dropping the nonpositive cost in the expressions for the two slopes gives (39). ◻

Proof. For fixed \((t,s)\) put \(g(y)=f(y,t)-|y|^2/(2s)\) and \[\Gamma(A;x,t,s)=\sup_y\{g(y)-A\cdot(y-x)\}.\] The expression on the left of (43) is the supremum of \(\sum_i\theta_i g(y_i)\) over configurations of mean \(x\), because \(\sum_i\theta_i|y_i-x|^2=\sum_i\theta_i|y_i|^2-|x|^2\). It is therefore at most \(\inf_A\Gamma(A;x,t,s)\).

Fix a compact range of \((x,t,s)\) with \(s>0\). For \(A\ne0\), use \(y=x-A/|A|\) in the supremum to get \(\Gamma(A;x,t,s)\ge |A|-C\) on this range. The value at \(A=0\) is uniformly bounded above, since the negative quadratic part of \(g\) dominates the subquadratic growth of \(f\). Thus every minimizing sequence whose values are at most \(\Gamma(0)+1\) lies in a fixed bounded set of slopes. For slopes in that set, the quantity \(g(y)-A\cdot(y-x)\) tends uniformly to \(-\infty\) as \(|y|\to\infty\). Its maximizing positions therefore lie in one compact set. It follows that \(\Gamma\) is continuous on the relevant set, the minimum and maxima are attained, and their value depends continuously on \((x,t,s)\).

Let \(A_0\) be a minimizing slope, and let \(E\) be its nonempty compact set of maximizing positions. Then \(x\) belongs to the convex hull of \(E\). Otherwise strict separation gives a vector \(e\) and \(\delta>0\) with \(e\cdot(y-x)\ge\delta\) for all \(y\in E\). If \(y_h\) maximizes \(\Gamma(A_0+he;x,t,s)\) for \(h>0\), the uniform compactness of maximizers and continuity show that every limit of \(y_h\) as \(h\downarrow0\) belongs to \(E\). For small \(h\), \(e\cdot(y_h-x)\ge\delta/2\), and hence \[\Gamma(A_0+he;x,t,s) \le\Gamma(A_0;x,t,s)-h\delta/2,\] contradicting minimality. The finite-dimensional convex hull theorem now represents \(x\) as a convex combination of finitely many points of \(E\). Discard any zero weights. Their weighted values of \(g\) equal \(\Gamma(A_0;x,t,s)\), proving the reverse inequality in (43) and finite attainment in (40).

The affine majorant \(g(y)\le\Gamma(A_0;x,t,s)+A_0\cdot(y-x)\) can also be integrated against any probability measure of mean \(x\) and finite second moment. All terms are integrable by the growth hypothesis. Thus such measures cannot increase the value. The right side of (43) is an infimum of affine functions of \(x\), so it is concave in \(x\). This is (44).

For the convergence assertion, on each compact range of \((x,t,s)\) the same preceding bounds on slopes and maximizing positions hold uniformly for \(f_j\): local uniform convergence gives common bounds near the compact range, and the common subquadratic control handles large \(y\). Both extrema in (43) can therefore be restricted to fixed compact sets. On these sets the expressions converge uniformly, proving the asserted local uniform convergence. ◻

Proof. Replace \(\Phi\) by \(\Phi^\sharp=\Phi+|Y-Y_0|^4+|t-t_0|^4\). This makes the local maximum strict while preserving the derivatives through order two at the contact. Choose a small closed product neighborhood of \(((y_i^0,t_0))_{i=1}^N\), contained in the domain, such that the diagonal points \((Y,t)\) lie in this strict neighborhood. For independent time variables \(T=(t_1,\ldots,t_N)\), put \[\bar t=\frac1N\sum_i t_i,\qquad \chi(T)=\frac12\sum_i(t_i-\bar t)^2,\qquad \Psi_\lambda(Y,T)=\Phi^\sharp(Y,\bar t)+\lambda\chi(T).\] Maximize \(\sum_i\theta_i v(y_i,t_i)-\Psi_\lambda(Y,T)\) on the closed product. Comparison with the original diagonal point and boundedness on this product show \(\lambda\chi(T_\lambda)\le C\) at any maximum. Thus all time deviations tend to zero as \(\lambda\to\infty\). Every limit of such maximum points is diagonal; dropping the nonnegative penalty in the maximum inequality shows that its diagonal limit is a maximum of \(V-\Phi^\sharp\). Strictness forces this limit to be \((Y_0,t_0)\). Hence all the maximum points converge to the original point and are interior for sufficiently large \(\lambda\).

Fix one such \(\lambda\) and one maximum. Set \[q_i=\theta_i^{-1}\nabla_{y_i} \Phi^\sharp(Y_\lambda,\bar t_\lambda), \qquad \zeta_\lambda=((q_i,1))_{i=1}^N,\qquad A_\lambda=D^2\Psi_\lambda(Y_\lambda,T_\lambda),\] where \(A_\lambda\) is written in the separate variables \(((y_i,t_i))_i\). Apply Lemma 6 to the functions \(\theta_i v(y_i,t_i)\) and the test \(\Psi_\lambda\). For every \(\eta>0\) it supplies closed superjets with Hessians \(X_i\) and first derivatives \(D_{(y_i,t_i)}\Psi_\lambda\), with \(\operatorname{diag}(X_i)\le A_\lambda+\eta A_\lambda^2\). Dividing each jet by the positive number \(\theta_i\) gives a closed superjet of \(v\) with horizontal gradient \(q_i\). The viscosity inequalities therefore give \(X_i[(q_i,1)]\ge0\). Evaluating the matrix inequality on \(\zeta_\lambda\) yields \[0\le\sum_iX_i[(q_i,1)] \le A_\lambda[\zeta_\lambda] +\eta|A_\lambda\zeta_\lambda|^2.\] First let \(\eta\downarrow0\), keeping \(\lambda\) and its maximum fixed. The quantities \(A_\lambda\) and \(\zeta_\lambda\) do not depend on \(\eta\), so \(A_\lambda[\zeta_\lambda]\ge0\). The time component of \(\zeta_\lambda\) is \((1,\ldots,1)\), which is in the kernel of \(D^2\chi\). The linear map \((Y,T)\mapsto(Y,\bar t)\) sends \(\zeta_\lambda\) to \(((q_i)_i,1)\). We conclude that \[D^2\Phi^\sharp(Y_\lambda,\bar t_\lambda) [((q_i)_i,1)]\ge0.\] Now let \(\lambda\to\infty\). The maximum points converge to the contact, and the horizontal derivatives converge there. The derivatives of \(\Phi^\sharp\) at that point are those of \(\Phi\), giving (45). The time components of the first derivatives of \(\Psi_\lambda\) need no bound: they do not enter the direction \((\nabla_xv,1)\) in the individual viscosity inequalities. ◻

Proof. Continuity follows from Lemma 19. Fix \(c\in\mathbb{R}\), and let \(\phi\) touch \((x,t)\mapsto\mathcal{C}_t f(x,t)-ct\) from above at \((x_0,t_0)\) with \(t_0>0\). Choose an attaining finite configuration \((\theta_i,y_i^0)\) for \(\mathcal{C}_{t_0}f(x_0,t_0)\), and keep its positive weights fixed. For variable \(Y=(y_i)_i\) let \(X(Y)=\sum_i\theta_i y_i\) and define \[\Phi(Y,t)=\phi(X(Y),t)+ct +\frac{\sum_i\theta_i|y_i-X(Y)|^2}{2t}.\] The defining supremum for \(\mathcal{C}_t\) shows that \(\sum_i\theta_i f(y_i,t)-\Phi(Y,t)\) has a local maximum at \((Y_0,t_0)\): for nearby \((Y,t)\) it is at most \(\mathcal{C}_t f(X(Y),t)-ct-\phi(X(Y),t)\), and equality holds at the chosen configuration. Set \(p=\nabla_x\phi(x_0,t_0)\). The individual horizontal directions in Lemma 20 are \[q_i=\theta_i^{-1}\nabla_{y_i}\Phi =p+\frac{y_i^0-x_0}{t_0}.\] Their weighted mean is \(p\). Along the straight motion with velocities \(y_i'=q_i\) and \(t'=1\), the mean has velocity \(p\) and each offset \(y_i-X\) has velocity \((y_i^0-x_0)/t_0\). Consequently, along this motion the variance term is \[\frac{\sum_i\theta_i |(y_i^0-x_0)(1+h/t_0)|^2}{2(t_0+h)} =\frac{t_0+h}{2t_0^2} \sum_i\theta_i|y_i^0-x_0|^2,\] which is affine in \(h\). The term \(ct\) is also affine, and the Hessian of the remaining term along this motion is \(D^2\phi[(p,1)]\). Lemma 20, applied to the subsolution inequality for \(f\), therefore gives \(F[\phi](x_0,t_0)\ge0\). This proves the claim for every \(c\), including \(c=0\). ◻

Proof. A positive finite horizontal defect. Start with a nonaffine member \(v^{(0)}\) and center it at the origin using Lemma 14. Some horizontal slice of \(v^{(0)}\) is nonaffine. To verify this, suppose instead that \(v^{(0)}(x,t)=p(t)\cdot x+c(t)\) for every \(t\). The coefficients \(p,c\) are continuous, as follows by evaluating the function at \(0\) and at the coordinate vectors. Direct optimization gives \[\mathcal{P}_\ell v^{(0)}(x,t) =p(t+\ell)\cdot x+c(t+\ell) +\frac{\ell}{2}|p(t+\ell)|^2.\] For every fixed \(x,t\) this is convex in \(\ell\). In each convexity inequality, the coefficient of \(x\) must vanish, since the inequality holds for all \(x\in\mathbb{R}^m\). Hence \(p\) satisfies the affine interpolation identity on every interval, and so is affine in \(t\). On the other hand, (27), applied at \(x=\pm Ke\) and \(|t|\le K\), implies \(|p(t)|\le C L(K)^2\). An affine function of \(t\) with this bound as \(K\to\infty\) is constant. For constant \(p\), the displayed forward formula and its backward counterpart \[\mathcal{M}_\ell v^{(0)}(x,t) =p\cdot x+c(t-\ell)-\frac{\ell}{2}|p|^2\] show that \(c\) is both convex and concave. It is therefore affine, contradicting the choice of \(v^{(0)}\).

A continuous nonaffine spatial slice has a nonzero midpoint Jensen defect, since the midpoint identity everywhere would make that slice affine. The transformation \(f(x,t)\mapsto-f(x,-t)\) preserves \(\mathcal{A}_{m,C_0}\): it exchanges the two viscosity inequalities, and it sends a fit \((p,a)\) to \((-p,a)\). It also changes the sign of a horizontal Jensen defect. Apply this transformation if needed, so that \(v^{(0)}\) has a positive defect. Define \[ \kappa=\sup_{\substack{t,\ \text{finite configurations}\\r>0}} \frac{\sum_i\theta_i v^{(0)}(y_i,t)-v^{(0)}(x,t)}{r}. \tag{50}\] Then \(\kappa>0\). It is finite as well. For any configuration in this supremum, choose a fit at its mean \((x,t)\) of radius \(r\) and use (26) with these data. At time zero the new positions are \((y_i-x)/r\), with mean zero and second moment one. The defect divided by \(r\) is unchanged, since the affine terms cancel in the weighted difference. The new function is centered. Its growth bound implies \(|\bar v(y,0)|\le C(1+|y|^2)\), with one constant for all these normalizations. Integrating this inequality over a configuration of second moment one bounds the quotient by a fixed constant. Thus \(0<\kappa<\infty\). More generally, (26) preserves every such quotient, at every time: spatial differences and their root mean square radius both scale by \(R\), and the subtracted affine terms cancel.

Attainment after compactness. Choose configurations whose quotients in (50) tend to \(\kappa\), and for each one make the normalization just described. Denote the resulting functions by \(v_j\) and the normalized finite probability measures by \(\mu_j\). Then \[v_j(0,0)=0,\qquad \int y\,d\mu_j=0,\qquad \int |y|^2\,d\mu_j=1,\qquad \int v_j(y,0)\,d\mu_j\longrightarrow\kappa.\] Every \(v_j\) is centered at the origin, and every finite configuration for it satisfies the Jensen bound with constant \(\kappa\). First use Lemma 15 to pass to a locally uniform limit \(\widehat v\in\mathcal{A}_{m,C_0}\), centered at the origin. The Jensen bound passes to this limit for each fixed finite configuration.

The moment bound gives \(\mu_j(\{|y|>M\})\le M^{-2}\), so the measures are tight. After a further subsequence they converge weakly to a probability measure \(\mu\). Lower semicontinuity of the second moment gives \(\int|y|^2\,d\mu\le1\). The mean remains zero: the bound \(\int_{|y|>M}|y|\,d\mu_j\le M^{-1}\) makes the first moments uniformly integrable, and the same bound holds for the limit. To retain the extremal defect, we also need convergence of the value integrals despite the varying functions and measures. For a compactly supported continuous cutoff \(\chi_M\), local uniform convergence and weak convergence give \[\int\chi_M(y)v_j(y,0)\,d\mu_j \longrightarrow \int\chi_M(y)\widehat v(y,0)\,d\mu.\] Choose \(\chi_M=1\) on \(\{|y|\le M\}\) and zero outside \(\{|y|<2M\}\). By (31), \[\int_{|y|>M}|v_j(y,0)|\,d\mu_j \le \sup_{\substack{j\\|y|>M}}\frac{|v_j(y,0)|}{|y|^2} \int|y|^2\,d\mu_j \longrightarrow0\] uniformly in \(j\); the analogous tail bound holds for \(\widehat v\) and \(\mu\). Letting first \(j\to\infty\) and then \(M\to\infty\) proves \[ \int y\,d\mu=0,\qquad \int|y|^2\,d\mu\le1,\qquad \int\widehat v(y,0)\,d\mu-\widehat v(0,0)=\kappa. \tag{51}\]

Apply (42) to \(\widehat v\). By Lemma 19, the measure in (51) is admissible for the value of \(\mathcal{C}_{1/\kappa}\widehat v(0,0)\). It gives \[\mathcal{C}_{1/\kappa}\widehat v(0,0)-\widehat v(0,0) \ge\kappa-\frac{\kappa}{2}\int|y|^2\,d\mu \ge\frac{\kappa}{2}.\] This is equality by (42). In particular, loss of second moment in the weak limit causes no loss of contact; the equality also forces \(\int|y|^2\,d\mu=1\).

Set \(R=1/\kappa\), and choose a fit \((p_0,a_0)\) for \(\widehat v\) at \((0,0)\) of radius \(R\). Define the final function by the single change of frame and time translation \[v(y,t)= \frac{\widehat v(R(y+p_0(t-1)),R(t-1))-\widehat v(0,0)}{R} -p_0\cdot y -\left(a_0+\frac{|p_0|^2}{2}\right)(t-1).\] It is centered at \((0,1)\) and still has the global Jensen bound with constant \(\kappa\). Under the change of frame (26), at corresponding points \(Z\) and \((y,s)\) the envelope gaps obey \[\mathcal{C}_\sigma\bar v(y,s)-\bar v(y,s) =\frac{\mathcal{C}_{R\sigma}\widehat v(Z)-\widehat v(Z)}{R}.\] This follows by scaling the spatial variance by \(R^2\) in (40); the affine terms again cancel. The preceding equality at parameter \(1/\kappa=R\) therefore becomes \[ \mathcal{C}_1v(0,1)-v(0,1)=\frac{\kappa^2}{2}. \tag{52}\]

The subsolution below \(v\). Define \(W\) by (46). Equations (42) and (52) give (47). At any contact, (46) also gives (48). Lemma 19 gives continuity and the stated spatial semiconcavity. The sandwich and (31) give subquadratic spatial growth on every compact positive-time interval. Since \(v\) is a solution with the growth in (31), Corollary 21, with \(f=v\) and \(c=\kappa^2/2\), gives the subsolution property of \(W\).

A finite slope at the contact. Lemma 17 already bounds \(H_v\) above at every point. We first bound it below at the finitely many points supporting the extremal configuration, using one common forward paraboloid, and then transfer this bound to their barycenter through equality in the Jensen bound. Choose a finite attaining configuration in (52), with mean \(x=0\), radius \(r\), and positive weights. If \(D=\sum_i\theta_i v(y_i,1)-v(0,1)\), its attainment and the Jensen bound imply \[\frac{\kappa^2}{2}=D-\frac{r^2}{2} \le\kappa r-\frac{r^2}{2} =\frac{\kappa^2}{2}-\frac{(r-\kappa)^2}{2}.\] All inequalities are equalities, so \[ r=\kappa,\qquad \sum_i\theta_i v(y_i,1)-v(0,1)=\kappa r. \tag{53}\] Let \(A\) be a minimizing slope in (43) for \((x,t,s)=(0,1,1)\). The weighted attaining value equals its affine majorant. Since all weights are positive, each \(y_i\) must attain that majorant. The identity \[v(y,1)-\frac{|y|^2}{2}-A\cdot y =v(y,1)-\frac{|y+A|^2}{2}+\frac{|A|^2}{2}\] therefore shows that all \(y_i\) are optimizers for \(\mathcal{P}_1v(b,0)\) with \(b=-A\).

Put \(h(s)=\mathcal{P}_s v(b,0)\) for \(0\le s\le1\). It is finite and convex by Lemma 16. For \(0<\ell<1\), use the point \(y_{i,\ell}^-=b+(1-\ell)(y_i-b)\) at time \(1-\ell\) as a competitor for \(\mathcal{M}_\ell v(y_i,1)\). Its value is bounded above using \(\mathcal{P}_{1-\ell}v(b,0)\). The two quadratic costs on this linear path add, giving explicitly \[\mathcal{M}_\ell v(y_i,1) \le h(1-\ell)+\frac{|y_i-b|^2}{2}, \qquad v(y_i,1)=h(1)+\frac{|y_i-b|^2}{2}.\] It follows that \[\mathcal{S}_\ell^-v(y_i,1) \ge\frac{h(1)-h(1-\ell)}{\ell} \ge h(1)-h(0).\] The last inequality is the chord inequality for the convex function \(h\). Taking the slope limit shows \(H_v(y_i,1)\ge h(1)-h(0)>-\infty\). These finitely many values are also finite above by Lemma 17.

For each \(\ell>0\), choose an optimizer \(\widehat y_i\) for \(\mathcal{P}_\ell v(y_i,1)\) and let \(\widehat x=\sum_i\theta_i\widehat y_i\). Write \(\xi_i=y_i-x\), \(\widehat\xi_i=\widehat y_i-\widehat x\), and \(r'=(\sum_i\theta_i|\widehat\xi_i|^2)^{1/2}\). The exact orthogonal decomposition and the reverse triangle inequality in the weighted Euclidean space give \[\sum_i\theta_i|\widehat y_i-y_i|^2 =|\widehat x-x|^2+ \sum_i\theta_i|\widehat\xi_i-\xi_i|^2 \ge|\widehat x-x|^2+(r'-r)^2.\] Use the Jensen bound at time \(1+\ell\), (53) at time \(1\), and \(\widehat x\) as a forward competitor from \(x\). The decomposition yields \[\begin{align*} \ell\sum_i\theta_i\mathcal{S}_\ell^+v(y_i,1) &\le \ell\mathcal{S}_\ell^+v(x,1) +\kappa(r'-r)-\frac{(r'-r)^2}{2\ell}\\ &\le \ell\mathcal{S}_\ell^+v(x,1)+\frac{\kappa^2\ell}{2}. \end{align*}\] Divide by \(\ell\) and take \(\ell\downarrow0\). There are only finitely many endpoints, all with finite limits, so \[H_v(z_*)\ge \sum_i\theta_i H_v(y_i,1)-\frac{\kappa^2}{2} \ge h(1)-h(0)-\frac{\kappa^2}{2}>-\infty.\] Its upper finiteness is part of Lemma 17; hence \(H_v(z_*)\in\mathbb{R}\).

It remains to identify the forward slope of \(W\). At the contact, the sandwich gives for every \(\ell>0\) \(\mathcal{S}_\ell^+W(z_*)\le\mathcal{S}_\ell^+v(z_*)\). For \(0<\ell<1\) it also gives \(\mathcal{S}_\ell^-W(z_*)\ge\mathcal{S}_\ell^-v(z_*)\), since the backward time is positive. The subsolution property and growth of \(W\) allow the one-sided inequality (37) to be applied to it. Consequently \[H_v(z_*)\le \limsup_{\ell\downarrow0}\mathcal{S}_\ell^-W(z_*) \le\lim_{\ell\downarrow0}\mathcal{S}_\ell^+W(z_*) \le H_v(z_*).\] This proves (49). ◻

Proof. Put \(s=t+\ell\). The function \(f(\cdot,s)-|\cdot|^2/(2s)\) is concave, because the same is true of \(\mathcal{C}_s v(\cdot,s)-|\cdot|^2/(2s)\). Thus \[y\longmapsto f(y,s)-\frac{|y-x|^2}{2\ell}\] is strictly concave: after subtracting the concave function \(f(y,s)-|y|^2/(2s)\), its quadratic part has Hessian \((s^{-1}-\ell^{-1})I=-t(\ell s)^{-1}I\). It has a maximizer by the subquadratic growth and the quadratic cost, and strict concavity makes the maximizer unique. Keeping this maximizer fixed when \(x\) is replaced by \(x+h\) gives \[ f^\ell(x+h,t)\ge f^\ell(x,t)+\frac{y-x}{\ell}\cdot h -\frac{|h|^2}{2\ell}. \tag{60}\]

For the upper bound, the identity \[\frac{|y|^2}{2s}-\frac{|y-x|^2}{2\ell}-\frac{|x|^2}{2t} =-\frac{t}{2\ell s}\left|y-\frac{s}{t}x\right|^2\] shows that \[f^\ell(x,t)-\frac{|x|^2}{2t} =\sup_y\left\{f(y,s)-\frac{|y|^2}{2s} -\frac{t}{2\ell s}\left|y-\frac{s}{t}x\right|^2\right\}\] is concave in \(x\). Indeed, the expression in braces is jointly concave in \((x,y)\); applying that concavity to a pair of maximizing points proves concavity of its partial supremum. A finite concave function on \(\mathbb{R}^m\) has an upper supporting affine function at each point. Adding back the quadratic therefore gives a vector \(q\) such that \[f^\ell(x+h,t)\le f^\ell(x,t)+q\cdot h+\frac{|h|^2}{2t} \quad\text{for every }h.\] Comparison with (60), first for \(h=\rho e\) and then for \(h=-\rho e\), and passage to \(\rho\downarrow0\), shows that \(q=(y-x)/\ell\). The two quadratic remainders now prove differentiability and (58)–(59). ◻

Proof. Convexity in the length of \(\mathcal{P}_\ell f(z)\) gives \(D\ge0\), and concavity in the length of \(\mathcal{M}_\ell v(z)\) gives \(\widetilde D\ge0\). The point \(z\) is a forward competitor from \(z^-\) and a backward competitor from \(z^+\), which gives \(e\ge0\) and \(\widetilde e\ge0\). The identities in (61) and the definitions of the competitor gaps yield \[\begin{align*} f^\epsilon(z^+)+f^\epsilon(z^-)-2f^\epsilon(z) &=D+e+\frac{\epsilon}{2}\bigl(|Q_+|^2-|Q_-|^2\bigr),\\ g_\epsilon(z^+)+g_\epsilon(z^-)-2g_\epsilon(z) &=-\widetilde D-\widetilde e +\frac{\epsilon}{2}\bigl(|Q_+|^2-|Q_-|^2\bigr). \end{align*}\] The same difference of the two half-step costs occurs in both lines, so it cancels on subtraction. Dividing by \(4\epsilon\) proves the identity and its first inequality.

For the final inequality let \(b,d\in\mathbb{R}^m\) be arbitrary. Apply the path-convexity property of the forward envelope to \[x(\ell)=x+d+\frac{\ell}{2\epsilon}(b-d),\qquad 0\le\ell\le2\epsilon.\] For this path the function \(\mathcal{P}_\ell f(x(\ell),t)+|b-d|^2\ell\log\ell/(4\epsilon^2)\) is convex, with \(\ell\log\ell\) assigned its continuous value zero at \(\ell=0\). The midpoint inequality, and \(2\epsilon\log(2\epsilon)-2\epsilon\log\epsilon=2\epsilon\log2\), give \[ f(x+d,t)+f^{2\epsilon}(x+b,t) -2f^\epsilon\left(x+\frac{b+d}{2},t\right) \ge-\frac{\log2}{2\epsilon}|b-d|^2. \tag{64}\] Using \(x+d\) as a forward competitor from \(z^-\) and expanding its cost gives \[f(x+d,t)\le f(z)+e+Q_-\cdot d+\frac{|d|^2}{2\epsilon}.\] Use the upper supporting inequality in (59) for \(f^{2\epsilon}(x+b,t)\), and the lower one for \(f^\epsilon(x+(b+d)/2,t)\). The left side of (64) is then at most \[D+e+(Q_+-Q_0)\cdot b+(Q_--Q_0)\cdot d +\frac{|b|^2}{2t}+\frac{|d|^2}{2\epsilon} +\frac{|b+d|^2}{4\epsilon}.\] Combining this upper bound with (64), using \(t>\epsilon\) and \(|b\pm d|^2\le2(|b|^2+|d|^2)\), proves \[D+e\ge (Q_0-Q_+)\cdot b+(Q_0-Q_-)\cdot d -\frac{1+\log2}{\epsilon}(|b|^2+|d|^2).\] Choose \[b=\frac{\epsilon(Q_0-Q_+)}{2(1+\log2)},\qquad d=\frac{\epsilon(Q_0-Q_-)}{2(1+\log2)}.\] The result is \(D+e\ge\epsilon(|Q_+-Q_0|^2+|Q_--Q_0|^2)/(4(1+\log2))\), as required. ◻

Proof. Lemma 14, followed by the time translation that puts the fitted point at \((0,1)\), proves the class and centering assertions. At each fixed time the spatial offsets are multiplied by \(r\), and values are multiplied by \(r\) up to an affine function. The affine terms cancel from a barycentric defect, so its quotient by the root mean square radius is unchanged. This proves the Jensen assertion.

For the remaining calculations put \[A(y,s)=\frac{v(x_0,r)}{r}+p_0\cdot y +\left(a_0+\frac{|p_0|^2}{2}\right)(s-1), \qquad \frac{v(x,t)}{r}=\bar v(y,s)+A(y,s).\] Scaling the positions of a barycentric configuration and cancelling its weighted affine terms gives \[\frac{\mathcal{C}_{r\sigma}v(x,t)}{r} =\mathcal{C}_\sigma\bar v(y,s)+A(y,s).\] Subtracting the displayed formula for \(v/r\) proves the first identity in (66). For \(s>0\), take \(\sigma=s\). The old time and the old barycentric parameter in \(W\) are both \(t=rs\), while the correction \(\kappa^2t/(2r)\) is \(\kappa^2s/2\). Hence \[\frac{W(x,t)}{r}=\bar W(y,s)+A(y,s),\] which proves the second gap identity. This is why the frame uses \(s=t/r\): it preserves the time-zero boundary in the definition of \(W\).

The quadratic envelopes obey the following value identities for \(\ell>0\), with \(s>0\) in the identity involving \(W\): \[ \begin{split} \frac{\mathcal{P}_{r\ell}v(x,t)}{r} &=\mathcal{P}_\ell\bar v(y,s)+A(y,s)+a_0\ell,\\ \frac{\mathcal{M}_{r\ell}v(x,t)}{r} &=\mathcal{M}_\ell\bar v(y,s)+A(y,s)-a_0\ell,\\ \frac{\mathcal{P}_{r\ell}W(x,t)}{r} &=\mathcal{P}_\ell\bar W(y,s)+A(y,s)+a_0\ell. \end{split} \tag{69}\] To check the first identity, parameterize an old forward competitor by a new position \(y'\) at time \(s+\ell\). Its old displacement divided by \(r\) is \(y'-y+p_0\ell\). The cost divided by \(r\) is \[\frac{|y'-y+p_0\ell|^2}{2\ell} =\frac{|y'-y|^2}{2\ell}+p_0\cdot(y'-y) +\frac{|p_0|^2\ell}{2}.\] The affine change in \(A\) from \((y,s)\) to \((y',s+\ell)\) is \(p_0\cdot(y'-y)+(a_0+|p_0|^2/2)\ell\), leaving \(a_0\ell\) after the cost is subtracted. For a backward competitor the old displacement divided by \(r\) is \(y'-y-p_0\ell\); the same expansion with the cost added leaves \(-a_0\ell\). These prove the first two identities. The last follows by the same forward calculation using \(W/r=\bar W+A\) at every positive time. Subtract the formula for \(v/r\) from the forward identity, and subtract the backward identity from that formula. Dividing each result by \(\ell\) gives (67).

The competitor calculations also show that old optimizers map to optimizers for the transformed envelopes. The affine coordinate map sends their straight intermediate points to the new straight intermediate points. Thus the mapped steps are precisely the steps of (61) for the transformed pair. The first two velocity equalities in (68) follow from (65) and \(t^\pm-t=\pm\epsilon\). For the reference velocity, differentiate the last identity of (69) in \(y\) with \(r\ell=\epsilon\). The derivative of the left side is \(\nabla_x\mathcal{P}_\epsilon W=Q_0\), since \(D_yx=rI\), and the derivative of \(A\) is \(p_0\). This proves the equality for \(Q'_0\) and the invariance of the velocity discrepancies, with no additional scale factor. ◻

Proof. Let \(s=t+\epsilon\), so \(2\epsilon<s\le K+1\). At the point \(x\), choose any upper supporting slope \(p\) for the semiconcave function \(f(\cdot,s)\): \[ f(x+h,s)\le f(x,s)+p\cdot h+\frac{|h|^2}{2s} \quad(h\in\mathbb{R}^m). \tag{71}\] Such a slope exists because \(f(\cdot,s)-|\cdot|^2/(2s)\) is finite and concave on all of \(\mathbb{R}^m\). If \(p\ne0\), insert \(h=-sp/|p|\) to obtain \[f(x+h,s)-f(x,s)\le-s|p|+\frac{s}{2}.\] The sandwich bound for \(W\) gives, in the other direction, \[f(x+h,s)-f(x,s) \ge v(x+h,s)-v(x,s)-\frac{\kappa^2s}{2}.\] The spatial increment estimate from the fits, applied with increment length \(s\) at the center \((x,s)\) in the box of size \(2K\), bounds the right side below by \[-Cs\bigl(L(2K)+|\log s|\bigr)-\frac{\kappa^2s}{2}.\] The resulting estimate for \(p\) also holds when \(p=0\). Since \(1<s/t<2\), we have \(|\log s|\le|\log t|+\log2\), and hence \[ |p|\le C\bigl(L(K)+|\log t|\bigr). \tag{72}\]

Let \(h\) be the optimizing displacement for \(f^\epsilon(x,t)\). Its value is at least the value of the competitor \(h=0\). In conjunction with (71), this gives \[\frac{|h|^2}{2\epsilon}\le p\cdot h+\frac{|h|^2}{2s}.\] For \(h\ne0\), using \(s>2\epsilon\) yields \(|h|/\epsilon\le4|p|\); the same inequality is immediate when \(h=0\). By (58), \(Q_0=h/\epsilon\), so (72) proves the lemma. ◻