Doubling variables: existence, relative contacts at the maximiser and localisation

Existence and finiteness of $M_{\alpha ,\rho }$. On ${\mathbb{R}}^n\times {\mathbb{R}}^n\times [0,T{]}^2$ the function $\mathrm{\Phi }$ is upper semicontinuous, being $u$ plus the upper semicontinuous $-v$ plus continuous terms by [F1]; it is bounded above by $\sup u-\inf v<\mathrm{\infty }$ because the quadratic and weight terms are nonpositive. Pick any point $p_0:=(0,0,0,0)$ and put $m_0:=\mathrm{\Phi }(p_0)\in \mathbb{R}$; the superlevel set $S_0:=\{\mathrm{\Phi }\ge m_0\}$ is nonempty, closed by upper semicontinuity, and bounded because $\rho (\mathrm{\mid }x{\mathrm{\mid }}^2+\mathrm{\mid }y{\mathrm{\mid }}^2)\le \sup u-\inf v-m_0$ on it; hence $S_0$ is compact by [F3]. On $S_0$ the restriction of $\mathrm{\Phi }$ is real-valued and upper semicontinuous, so it attains a maximum by [F2]; that maximum is a global maximum of $\mathrm{\Phi }$ because every point outside $S_0$ has value $<m_0\le {\max }_{S_0}\mathrm{\Phi }$. Hence $M_{\alpha ,\rho }$ is finite and attained.

The relative contacts and their derivatives. Let $(x_{\alpha },y_{\alpha },t_{\alpha },s_{\alpha })$ be a maximiser. Fixing $(y,s)=(y_{\alpha },s_{\alpha })$, the inequality $\mathrm{\Phi }(x,y_{\alpha },t,s_{\alpha })\le \mathrm{\Phi }(x_{\alpha },y_{\alpha },t_{\alpha },s_{\alpha })$ for all $(x,t)$ reads $u(x,t)-{\psi }_1(x,t)\le u(x_{\alpha },t_{\alpha })-{\psi }_1(x_{\alpha },t_{\alpha })$, so $u-{\psi }_1$ has a local maximum at $(x_{\alpha },t_{\alpha })$; fixing $(x,t)=(x_{\alpha },t_{\alpha })$ similarly gives $v(y,s)-{\psi }_2(y,s)\ge v(y_{\alpha },s_{\alpha })-{\psi }_2(y_{\alpha },s_{\alpha })$, a local minimum of $v-{\psi }_2$ at $(y_{\alpha },s_{\alpha })$. The displayed gradients and time derivatives are the derivatives of the two quadratic test functions: $D_x{\psi }_1=\alpha (x-y_{\alpha })+2\rho x$ evaluated at $x_{\alpha }$ gives $p_{\alpha }$, $D_y{\psi }_2=\alpha (x_{\alpha }-y)-2\rho y$ evaluated at $y_{\alpha }$ gives $q_{\alpha }$, and ${\mathrm{\partial }}_t{\psi }_1=\alpha (t-s_{\alpha })$, ${\mathrm{\partial }}_s{\psi }_2=\alpha (t_{\alpha }-s)$ both equal $p_{\alpha }^0$ at the maximiser.

The weight bound (3). At a maximiser, $\mathrm{\Phi }(x_{\alpha },y_{\alpha },t_{\alpha },s_{\alpha })=M_{\alpha ,\rho }$, that is $u(x_{\alpha },t_{\alpha })-v(y_{\alpha },s_{\alpha })-\frac{\alpha }{2}{d}_{\alpha }^2-\rho (\mathrm{\mid }{x}_{\alpha }{\mathrm{\mid }}^2+\mathrm{\mid }{y}_{\alpha }{\mathrm{\mid }}^2)=M_{\alpha ,\rho }$; since $u(x_{\alpha },t_{\alpha })\le \sup u$ and $-v(y_{\alpha },s_{\alpha })\le -\inf v$, the sum of the two penalty terms is at most $\sup u-\inf v-M_{\alpha ,\rho }$, which is (3). If in addition $M_{\alpha ,\rho }\ge m$ then $\rho (\mathrm{\mid }{x}_{\alpha }{\mathrm{\mid }}^2+\mathrm{\mid }{y}_{\alpha }{\mathrm{\mid }}^2)\le \sup u-\inf v-m$, and $\mathrm{\mid }{q}_{\alpha }-p_{\alpha }\mathrm{\mid }\le 2\rho (\mathrm{\mid }{x}_{\alpha }\mathrm{\mid }+\mathrm{\mid }{y}_{\alpha }\mathrm{\mid })\le 2\rho \sqrt{2(\mathrm{\mid }{x}_{\alpha }{\mathrm{\mid }}^2+\mathrm{\mid }{y}_{\alpha }{\mathrm{\mid }}^2)}\le 2(2\rho (\sup u-\inf v-m){)}^{1\mathrm{/}2}$.

Localisation as $\alpha \to \mathrm{\infty }$. Fix any sequence ${\alpha }_j\to \mathrm{\infty }$ and any corresponding sequence of maximisers $(x_j,y_j,t_j,s_j)$; these are exactly the sequences quantified in part (2). For fixed $\rho$, $\alpha \mapsto M_{\alpha ,\rho }$ is nonincreasing and bounded below by $\mathrm{\Phi }(0,0,0,0)$, so it converges. Put $d_j^2:=\mathrm{\mid }{x}_j-y_j{\mathrm{\mid }}^2+\mathrm{\mid }{t}_j-s_j{\mathrm{\mid }}^2$. Evaluating the ${\alpha }_j\mathrm{/}2$-function at this same maximiser gives $M_{{\alpha }_j\mathrm{/}2,\rho }\ge M_{{\alpha }_j,\rho }+\frac{{\alpha }_j}{4}{d}_j^2$, hence ${\alpha }_j{d}_j^2\le 4(M_{{\alpha }_j\mathrm{/}2,\rho }-M_{{\alpha }_j,\rho })\to 0$. In particular $\mathrm{\mid }{x}_j-y_j\mathrm{\mid }+\mathrm{\mid }{t}_j-s_j\mathrm{\mid }\to 0$. By step 3.1, $\mathrm{\mid }{x}_j\mathrm{\mid }$ is uniformly bounded for fixed $\rho$. With $p_j={\alpha }_j(x_j-y_j)+2\rho x_j$, the bounds ${\alpha }_j\mathrm{\mid }{x}_j-y_j\mathrm{\mid }(\mathrm{\mid }{x}_j-y_j\mathrm{\mid }+\mathrm{\mid }{t}_j-s_j\mathrm{\mid })\le 2{\alpha }_j{d}_j^2\to 0$ and $2\rho \mathrm{\mid }{x}_j\mathrm{\mid }(\mathrm{\mid }{x}_j-y_j\mathrm{\mid }+\mathrm{\mid }{t}_j-s_j\mathrm{\mid })\to 0$ give $(1+\mathrm{\mid }{p}_j\mathrm{\mid })(\mathrm{\mid }{x}_j-y_j\mathrm{\mid }+\mathrm{\mid }{t}_j-s_j\mathrm{\mid })\to 0$. Finally $q_j-p_j=-2\rho (x_j+y_j)$ and step 3.1 gives the asserted bound on $\mathrm{\mid }{q}_j-p_j\mathrm{\mid }$, which need not vanish for fixed $\rho$. The argument applies to every given sequence of maximisers and selects none.

Conclusion. Part (1) is steps 1.1 and 2.1, part (2) is step 4.1, and part (3) is step 3.1; the maximiser is obtained from the compactness of a closed bounded superlevel set and the extreme-value property for upper semicontinuous functions, and no sequence, point or index is selected in the construction.