The following estimate holds: begin{aligned} big Vert ubig |_{Gamma _2}big Vert _{1,Gamma _2}^2+big Vert ubig |_{Gamma _3}big Vert _{1,Gamma _3}^2 le CBig (big Vert ubig |_{Gamma _4}big Vert _{1,Gamma _4}^2+big Vert u_{y_0}big |_{Gamma _4}big Vert _{0,Gamma _{0,Gamma).
Then (Tcirc P^{-1} in W') and begin{aligned} big Vert Tcirc P^{-1}big Vert _{W'} = big Vert T big Vert _{big (W^{k,p}(Delta,mu )big )'}.
(Global a priori bounds) ( displaystyle sup _{t, zeta } ; Big ( big Vert mathcal{B} t, zeta ) big Vert _{L^infty ({{mathbb R}^{N}},{{mathbb R}^{N}})} + big Vert mathcal{C} t, zeta ) big Vert _{L^infty ({{mathbb R}^{N}})} Big ) ; < ; infty ).
Then (big [varphi (A,B),Qbig ]in {varvec{S}}_1), Open image in new window and begin{aligned} big Vert big [varphi (A,B),Qbig ]big Vert _{{varvec{S}}_1} le {text {const}},Vert varphi Vert _{B_{infty,1}^1({mathbb R}^2)}big (big Vert [A,Q]big Vert _{{varvec{S}}_1}+ big Vert [B,Q]big Vert _{{varvec{S}}_1}big ).
(Global a priori bounds) ( displaystyle sup _{t, f} ; left( big Vert {mathrm{div}}_{mathbf x},mathcal{G} t, f) big Vert _{L^infty ({{mathbb R}^{N}})},+ big Vert mathcal{U} t, f) big Vert _{L^infty ({{mathbb R}^{N}})}, + big Vert mathcal{W} t, f) big Vert _{L^p({{mathbb R}^{N}})} right) ; < ; infty ).
On the basis of assumption (i) of Theorem 4, set the abbreviation begin{aligned} eta, :=, sup _{t, f} ; left( big Vert {mathrm{div}}_{mathbf x},mathcal{G} t, f) big Vert _{L^infty },+ big Vert mathcal{U} t, f) big Vert _{L^infty }, + big Vert mathcal{W} t, f) big Vert _{L^p} right) ; < ; infty.
Hence the composed functions (widetilde{{mathbf g}}), (widetilde{u}: [0,T)] (rightarrow big ( L^q, ; Vert cdot Vert _{L^q}big )) and (widetilde{w}: [0,T] rightarrow big ( L^p, ; Vert cdot Vert _{L^p}big )) are measurable.
The functions (mathcal{B}(cdot, zeta ), mathcal{C}(cdot, zeta ): [0,T] rightarrow big (L^infty, Vert cdot Vert _{L^infty } big )) are measurable for each (zeta in mathcal{M}({{mathbb R}^{N}})).
Here the constant C depends on (displaystyle rho := sup _{t,in,[0,T]} big { Vert f(t Vert _{L^p}, Vert bar{f}(t Vert _{L^p} big } < infty ) and the bounds given in hypotheses (i) and (ii) of Theorem 4.
Vertex relabeling resulting in the fewest possible labels, (min _{{lab}_{s}} vert big lbrace {lab}_{s} v) | v in mathcal {V}_{s} big rbrace vert ). .
Vertex relabeling resulting in the fewest possible labels, (min _{{lab}_{s}} vert big lbrace {lab}_{s} v) | v in mathcal {V}_{s} big rbrace vert ).
