latex_formula
stringlengths 6
4.11M
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\begin{equation}
\tilde{d}\circ U= U\circ d\ .
\end{equation} |
\begin{equation}
\tilde d^*=(-1)^{ni+1}*\tilde d*.
\end{equation} |
\begin{align}
\langle \sum_I(\Delta \omega_I) dx^I;\sum_I\omega_Idx^I \rangle=&\sum_{\mu\in \Z^n}\sum_I (\Delta \omega_I(\mu))
\overline{\omega_I(\mu) }\nonumber\\
=&\sum_{\mu\in \Z^n}\sum_I \sum_{\alpha=1}^n((\mathcal{D}_{\alpha}^2\omega_I)(\mu-e_\alpha))\overline{\omega_I(\mu)}\nonumber\\
=&\sum_{\mu\in \Z^n}\sum_I \sum_{\alpha=1}^n| \mathcal{D}_{\alpha}\omega_I (\mu)|^2\ .
\end{align} |
\begin{align}
\langle (\tilde d\tilde d^*)\omega, \omega \rangle &= \langle \tilde d*\omega;\tilde d*\omega\rangle\nonumber\\
&=\langle \tilde d\sum_I{\rm sign}(IJ_I) \omega_Idx^J;\tilde d\sum_I{\rm sign}(IJ_I) \omega_Idx^{J_I}\rangle\nonumber\\
&=\langle \sum_I{\rm sign}(IJ_I) \sum_{\alpha=1}^n{ \mathcal D_{\alpha}\omega_I}dx^\alpha\wedge dx^{J_I};\sum_I{\rm sign}(IJ_I) \sum_{\alpha=1}^n{ \mathcal D_{\alpha}\omega_I}dx^\alpha\wedge dx^{J_I}\rangle\nonumber\\
&=\sum_{\mu\in \Z^n}\sum_I\sum_{\substack{\alpha\neq J_I(i)\\ j+1\leq i\leq n}}| \mathcal D_{\alpha}\omega_I (\mu)|^2\nonumber\\
&=\sum_{\mu\in \Z^n}\sum_I\sum_{i=1}^j| \mathcal{D}_{I(i)}\omega_I (\mu)|^2.
\end{align} |
\begin{align}
\langle (\tilde d^*\tilde d)\omega, \omega \rangle &= \langle \tilde d\omega;\tilde d\omega\rangle\nonumber\\
&=\langle \sum_I\sum_{\alpha=1}^n{ \mathcal D_{\alpha}\omega_I}dx^\alpha\wedge dx^J;\sum_I\sum_{\alpha=1}^n{ \mathcal D_{\alpha}\omega_I}dx^\alpha\wedge dx^J\rangle\nonumber\\
&=\sum_{\mu\in \Z^n}\sum_I\sum_{\substack{\alpha\neq I(i)\\ 1\leq i\leq j}}| \mathcal D_{\alpha}\omega_I (\mu)|^2\nonumber\\
&=\sum_{\mu\in \Z^n}\sum_I\sum_{i=j+1}^n| \mathcal{D}_{J_I(i)}\omega_I (\mu)|^2.
\end{align} |
\begin{align*}
\langle \omega, \eta\rangle_{\ell^2( h\Z^n;\bigwedge^{j}(h \Z^n))}&=\frac1{h^{2j}}\sum_{\mu\in h\Z^n;I\in P^{j,n}_+}\omega_I(\mu)\overline{\eta}_I(\mu), \\
\langle f, g\rangle_{\ell^2\Big(h\Z^n;\C^{\binom{n}{j}}\Big)}&=\sum_{\mu\in h\Z^n;1\leq l\leq \binom{n}{j} }f_l(\mu)\overline{g}_l(\mu).
\end{align*} |
\begin{align*}
I(i)=&I'(i),i<l,\\
I(l)<&I'(l).
\end{align*} |
\begin{equation}
(\tilde U_{j,h} \omega)_{l}(\mu):=\frac1{h^{{j}}}\omega_{I^j_l}(\mu),1\leq l \leq \binom{n}{j}.
\end{equation} |
\begin{equation}
(\tilde{U}_{j,h}^* f)(\mu):=h^{j}\sum_{l=1}^{\binom{n}{j}} f_l(\mu)dx^{I^j_l}
\end{equation} |
\begin{align*}
a_{h,l}(\xi):=\frac{(-1+e^{-2\pi ih\xi_l})}h,1\leq l \leq n.
\end{align*} |
\begin{align*}
(\tilde d_{j,h}\tilde U_{j,h}^* f)(\mu)&=h^{j}\sum_l^{\binom{n}{j}} (\tilde d_{0,h} f_{j,l})(\mu)\wedge dx^{I^j_l}\\
&=h^{j}\sum_l^{\binom{n}{j}} \sum_{\alpha=1}^n {(f_{j,l}(\mu+h\delta_\alpha)-f_{j,l}(\mu))}dx^\alpha\wedge dx^{I^j_l}\\
&=h^{j}\sum_{1\leq \tilde l\leq {\binom{n}{j}}} \left(\sum_{\substack{
\alpha,l \\ dx^\alpha\wedge dx^{I^j_l}=(\pm)dx^{ I^{j+1}_{\tilde l}}
}} {(f_{j,l}(\mu+h\delta_\alpha)-f_{j,l}(\mu))}\right) (\pm)dx^{ I^{j+1}_{\tilde l}}.
\end{align*} |
\begin{align}
(F H_h F^*-z)^{-1}(\xi)
&=\frac{1}{r_z(\xi)}
FH_hF^*+\frac{z}{r_z(\xi)}
\end{align} |
\begin{equation}
(\mathcal{F} H \mathcal{F}^*-z)^{-1}(\xi)=\frac{1}{R_z(\xi)}\mathcal{F} H \mathcal{F}^*+\frac{z}{R_z(\xi)}
\end{equation} |
\begin{align}
&\sum_{\mu\in\Z^n}|\hat{\varphi}(\xi+\mu)|^2=1,\xi\in\R^n,\\
&{\rm supp}(\hat{\varphi})\subset (-1,1)^n.
\end{align} |
\begin{align*}
\left(1-Q_h^*Q_h\right)g(\xi)&=g(\xi)-\sum_{\mu\in\{0,\pm1\}^n} \hat{\varphi}(h\xi)\overline{\hat{\varphi}(h\xi+\mu)}g(\xi+h^{-1}\mu)\\
&=(1-|\hat\varphi(h\xi)|^2)g(\xi)-\sum_{0\neq\mu\in\{0,\pm1\}^n} \hat{\varphi}(h\xi)\overline{\hat{\varphi}(h\xi+\mu)}g(\xi+h^{-1}\mu).
\end{align*} |
\begin{equation}
\left(Q_h^* \dfrac{a_l}{r_z} Q_h -Q_h^*Q_h\dfrac{A_l}{R_z} \right)\psi=\sum_{\mu\in\{0,\pm1\}^n} \hat{\varphi}(h\xi)\overline{\hat{\varphi}(h\xi+\mu)}\mathcal B_h(\xi+h^{-1}\mu)\psi(\xi+h^{-1}\mu),
\end{equation} |
\begin{align*}
\prescript{}{i_0}{(\hat{s}^*})(i)=&\begin{cases}
\hat{s}^*(i)&i<i_0\\
\hat{s}^*(i+1)& i_0\leq i
\end{cases}\\
=&\begin{cases}
\hat{s}(j-i+1)&i<i_0\\
\hat{s}(j-i)& i_0\leq i
\end{cases}
\end{align*} |
\begin{align*}
(\prescript{}{j-i_0+1}{\hat{s}})^*(i)=&\prescript{}{j-i_0+1}{\hat{s}}(j-i)\\
=&\begin{cases}
\hat{s}(j-i)&j-i<j-i_0+1\\
\hat{s}(j-i+1)& j-i_0+1\leq j- i
\end{cases}\\
=&\begin{cases}
\hat{s}(j-i)& i_0\leq i\\
\hat{s}(j-i+1)&i<i_0
\end{cases}
\end{align*} |
\begin{align*}
\prescript{}{i_1}{(\prescript{}{i_0}{\hat{s}})}(i)=&\begin{cases}
\prescript{}{i_0}{\hat{s}}(i)&i<i_1\\
\prescript{}{i_0}{\hat{s}}(i+1)& i_1\leq i
\end{cases}\\
=&\begin{cases}
\hat{s}(i)&i<i_1\\
\hat{s}(i+1)&i_1\leq i\leq i_0-2\\
\hat{s}(i+2)&i_0-1\leq i
\end{cases}
\end{align*} |
\begin{align*}
\prescript{}{i_0-1}{(\prescript{}{i_1}{\hat{s}})}(i)
=&\begin{cases}
\prescript{}{i_1}{\hat{s}}(i)&i<i_0-1\\
\prescript{}{i_1}{\hat{s}}(i+1)& i_0-1\leq i
\end{cases}
\end{align*} |
\begin{align*}
\partial (\overline{s}) =&\cup_{i=1}^j \{(-1)^{j-i} (\lfloor \overline{s} \rfloor;\prescript{}{i}{(\hat{s}^*)}) \}\bigcup\cup_{i=1}^j\{(-1)^{i} (\lceil \overline{s} \rceil;(\prescript{}{i}{(\hat{s}^*)})^*) \}\\
=&\cup_{i=1}^j \{(-1)^{j-i} (\lceil s \rceil\lfloor ;(\prescript{}{j-i+1}{\hat{s}})^*) \}\bigcup\cup_{i=1}^j\{(-1)^{i} (\lfloor s \rfloor;\prescript{}{j-i+1}{\hat{s}}) \}\\
=&\cup_{m=1}^j \{(-1)^{m-1} (\lceil s \rceil\lfloor ;(\prescript{}{m}{\hat{s}})^*) \}\bigcup\cup_{m=1}^j\{(-1)^{j-m+1} (\lfloor s \rfloor;\prescript{}{m}{\hat{s}}) \}\\
=&\cup_{m=1}^j \overline{\{(-1)^{m} (\lceil s \rceil\lfloor ;(\prescript{}{m}{\hat{s}})^*) \}}\bigcup\cup_{m=1}^j\overline{\{(-1)^{j-m} (\lfloor s \rfloor;\prescript{}{m}{\hat{s}}) \}}=\overline{\partial(s)}\ .
\end{align*} |
\begin{align*}
B_l(A_i)(s)=B_l((-1)^{j-i}(\lfloor s \rfloor ; \prescript{}{i}{\hat{s}})=&B_l((\lceil s \rceil-\delta_i ; (\prescript{}{i}{\hat{s}})^*)\\
=&(-1)^{l}(\lfloor s \rfloor ; (\prescript{}{l}{((\prescript{}{i}{\hat{s}})^*}))^*)\\
=&(\lfloor s \rfloor ; \prescript{}{j-l}{(\prescript{}{i}{\hat{s}})})=
\begin{cases}(\lfloor s \rfloor ; \prescript{}{j-l}{(\prescript{}{i}{\hat{s}})}) &\text{ if }j-l<i\\
(\lfloor s \rfloor ; \prescript{}{i}{(\prescript{}{j-l+1}{\hat{s}})}) &\text{ if }i\leq j-l
\end{cases}\ .
\end{align*} |
\begin{equation}
(\om+\ddc\f)^n=\mu\tag{MA}
\end{equation} |
\[
\MA_\om(\f):=\tfrac1{[\om]^n}(\om+\ddc\f)^n.
\] |
\begin{equation}
\MA_\om(\f_w)=\d_w,\f_w(w)=0\tag{$\star$}
\end{equation} |
\[
\f_\Sigma:=\sup\{\f\in\CPSH(Y,\theta)\mid \f|_\Sigma\le 0\}.
\] |
\[
\MA_\om(\pi^\star\f)=\sigma_\star\MA_\theta(\f).
\] |
\[
Z_Y(\f):=c_Y\{\f<\sup\f\},
\] |
\[
\te(\Sigma):=\sup\f_\Sigma
\] |
\begin{equation}
\B_-(\theta)=\bigcup_{\e\in\Q_{>0}}\B_+(\theta+\e\om)
\end{equation} |
\begin{align*}
\theta\in\Psef(X) & \Leftrightarrow \B_-(\theta)\ne X;\\
\theta\in\Nef(X) & \Leftrightarrow \B_-(\theta)=\emptyset;\\
\theta\in\Mov(X) &\Leftrightarrow \codim\B_-(\theta)\ge 2.
\end{align*} |
\begin{equation}
\PSH(\pi^\star\theta)=\pi^\star\PSH(\theta);
\end{equation} |
\begin{equation}
\hf^\la:=\inf_{t>0}\{t\cdot\f-t\la\}
\end{equation} |
\begin{equation}
\f=\sup_{\la\le\sup\f}\{\hf^\la+\la\}.
\end{equation} |
\begin{equation}
\hf^{\max}(v)=\lim_{t\to 0_+}\frac{\f(tv)-\f(v_\triv)}{t}
\end{equation} |
\begin{equation}
\f=\sup_{\la\le\la_{\max}}\{\p_{B_\la}+\la\},\p_{B_\la}=\hf^\la.
\end{equation} |
\begin{equation}
Z_X(S):=\bigcup_{v\in S} Z_X(v).
\end{equation} |
\[
Z_X(\f):=Z_X(\{\f<\sup\f\}).
\] |
\begin{equation}
V_\theta:=\sup\left\{\f\in\PSH(\theta)\mid\f\le 0\right\}.
\end{equation} |
\begin{equation}
v(\theta):=-V_\theta(v)=\inf\{-\f(v)\mid\f\in\PSH(\theta),\,\f\le 0\}\in\R_{\ge 0}.
\end{equation} |
\begin{equation}
v(\theta)=\sup_{\e>0} v(\theta+\e\om)
\end{equation} |
\begin{align*}
v(\|L\|) : & =\lim_{m\to\infty}\tfrac{1}{m}\min\left\{v(s)\mid s\in\Hnot(X,mL)\setminus\{0\}\right\}\\
& =\inf\left\{v(D)\mid D\sim_\Q L\text{ effective }\Q{-divisor}\right\}.
\end{align*} |
\[
\Nz(\pi^\star\theta)=\Nz(\theta)
\quad\text{and} \env(\pi^\star\theta)=\env(\theta)
\] |
\begin{equation}
\Nz_X(\theta)=\sum_{E\subset X} \ord_E(\theta) E
\end{equation} |
\begin{equation}
\Nz_X(D):=\Nz_X([D])\le D.
\end{equation} |
\begin{equation}
\f_\Sigma=\f_{\om,\Sigma}:=\sup\{\f\in\PSH(\om)\mid \f|_{\Sigma}\le 0\}.
\end{equation} |
\begin{equation}
\|\mu\|=\sup_{\f\in\cE^1(\om)}\left(\en(\f)-\int\f\,\mu\right)\in [0,+\infty],
\end{equation} |
\begin{equation}
\pi^\star\hf^\la_\Sigma
=\sup\{\tau\in\PSH_\hom(\pi^\star\om-\la D)\}-\la\p_D=V_{\pi^\star\om-\la D}-\la\p_D.
\end{equation} |
\[
Z_X(\f_\Sigma)
=Z_X(\hf_\Sigma^{\max})
=\pi(Z_Y(\pi^*\hf_\Sigma^{\max})).
\] |
\[
\pi^\star\hf^{\max}_\Sigma=V_{\pi^\star\om-\la_\psef D}-\la_\psef\p_D.
\] |
\[
X^{\mathrm{qm}}:=\bigcup_\cX\D_\cX\subset X^\an
\] |
\begin{equation}
\PL(X)=\bigcup_\cX p_\cX^\star\PL(\D_\cX).
\end{equation} |
\begin{equation}
\RPL(X)=\bigcup_\cX p_\cX^\star\RPL(\D_\cX),
\end{equation} |
\begin{equation}
\sup_X\f=\max_{\Ga_\cX}\f
\end{equation} |
\[
\f_w:=\f_{\om,w}:=\sup\{\f\in\PSH(\om)\mid \f(w)\le 0\}.
\] |
\begin{equation}
\env(\f\circ p_\cX)=\sup\left\{\p\in\PSH(\om)\mid\p\le\f\text{ on }\D_\cX\right\}.
\end{equation} |
\[
\MA_{\pi^\star\om}(\f_w)=\d_w\quad\text{and} \f_w(w)=0.
\] |
\[
\f_w=\pi^\star\f_v.
\] |
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\begin{align}
\begin{cases}
\mu \nabla \times \nabla \times \u = \lambda \u & \hbox{in}~\Omega, \\
\nabla \cdot \u =0 & \hbox{in}~\Omega, \\
{\bm n} \times \u= \bm {0} & \hbox{on}~\partial\Omega,
\end{cases}
\end{align} |
\begin{align}
\begin{cases}
\mu \nabla \times \nabla \times \u + \nabla p = \lambda \u & \hbox{in}~\Omega, \\
\nabla \cdot \u =0 & \hbox{in}~\Omega, \\
{\bm n} \times \u= \bm {0} ~\hbox{and}~ p=0& \hbox{on}~\partial\Omega.
\end{cases}
\end{align} |
\begin{align}
\mu (\nabla \times \u, \nabla \times \vt) = \lambda (\u,\vt) , \forall \vt \in H_0({\textbf{curl}},\Omega),
\end{align} |
\begin{align}
\mu (\nabla \times \u_h, \nabla \times \vt_h) = \lambda (\u_h,\vt_h) , \forall \vt_h \in {\cal V}_h .
\end{align} |
\begin{align}
B_{ }([\u,p],[\vt,q]) = \lambda (\u,\vt) , \forall [\vt,q] \in {\cal X},
\end{align} |
\begin{align*}
B_{ }([\u,p],[\vt,q]) = \mu (\nabla \times \u, \nabla \times \vt) + (\nabla p, \vt ) - (\nabla q, \u ) .
\end{align*} |
\begin{align}
B_{ }([\u_h,p_h],[\vt_h,q_h]) = \lambda_h (\u_h,\vt_h) , \forall [\vt_h,q_h] \in {\cal X}_h.
\end{align} |
\begin{align}
B_{\text{S}}([\u_h,p_h],[\vt_h,q_h]) = \lambda_h (\u_h,\vt_h) , \forall [\vt_h,q_h] \in {\cal X}_h,
\end{align} |
\begin{align*}
B_{\text{S}}([\u_h,p_h],[\vt_h,q_h]) &= B([\u_h,p_h],[\vt_h,q_h]) \\
&+ \sum_{K} \tau_{p} (\tilde{P}(\nabla p_h), \tilde{P}(\nabla q_h))_K \\ &+ \sum_{K} \tau_{\u}(\tilde{P}(\nabla \cdot \u_h), \tilde{P}(\nabla \cdot \vt_h))_K.
\end{align*} |
\begin{align*}
\mu \nabla \times \nabla \times \u +\nabla p = \lambda \u & \hbox{in}~\Omega, \\
-\frac{\ell}{\mu} \Delta p+ \nabla \cdot \u =0 & \hbox{in}~\Omega, \\
{\bm n} \times \u= \bm {0} & \hbox{on}~\partial\Omega,
\end{align*} |
\begin{align}
B_{ }([\u,p],[\vt,q]) = (\f,\vt) , \forall [\vt,q] \in {\cal X}.
\end{align} |
\begin{align}
B_{\text{S}}([\u_h,p_h],[\vt_h,q_h]) = (\f,\vt_h) , \forall [\vt_h,q_h] \in {\cal X}_h,
\end{align} |
\begin{align}
\Vert [\vt,q] \Vert^2_{\rm AG} := \mu \Vert\nabla\times \vt \Vert^2_{L^2(\Omega)} + \frac{\mu}{\ell^2} \Vert \vt \Vert^2_{L^2(\Omega)}
+ \frac{\ell^2}{\mu} \Vert \nabla p \Vert^2_{L^2(\Omega)},
\end{align} |
\begin{align}
\Vert [\vt,q] \Vert^2_{\rm OSGS} := \mu \Vert\nabla\times \vt \Vert^2_{L^2(\Omega)} + \frac{\mu}{\ell^2} \Vert \vt \Vert^2_{L^2(\Omega)}
+ \frac{\ell^2}{\mu} \Vert P_h^\bot(\nabla p) \Vert^2_{L^2(\Omega)}
+ \frac{h^2}{\mu} \Vert P_h(\nabla p) \Vert^2_{L^2(\Omega)}.
\end{align} |
\[
\|\u-\u_h\|_{L^2(\Omega)}=\|T\f-T_h\f\|_{L^2(\Omega)}\lesssim \varphi(h) \|\f\|_{L^2(\Omega)},
\] |
\[
\aligned
&|\lambda-\lambda_h^i|=O(h^{2r}),&&i=1,\dots,m,\\
&\hat\delta(E,E_h)=O(h^r),
\endaligned
\] |
\[
\vertiii{\u-\u_h}=O(h^r),
\] |
\[
P1ml,9iAjLMWUS&quh,sQmHS9\d<+ER*0oV'OAjSo&=F[D_g^ADf2X!:,p4TG?N0>@5@QbN*g6\c.op-9^Lr,J;AZkT*YIiVUQNn'c996i5YggPV+
(ap.`Xki/."+J3Y+WFG#)X525]/aY;'IkD0?Gn'tiumY,>]'>DB[I0($Q>N,#IJ'El*4]/p/=DVi-4?j
?;X!9KolSG1.)A.XYcXS]<qGlah&@:^2",eKqqcP3qq]K^qEsCfl1jL=J,90=^HY&N*dHUo?i8n"_.u,Xh;5PkZGFlQ77X-9sg!rd=33IBD[nWSsRM5"[&4F.]/'cR[W(dDV;M]"hn4O`.hPp*2,_nqaVDe)bQSR^*d
W`_F[h)Wr.m)(.cgfhIq9W8jLD:kFnB4Eup#^I(]fBH_\_AsD:>p=bc'i*BKTU5_A,IM=!:cERng5rm\
#?K'c]I4r.@Y$d1:J^;M&EaS[[FOB[jZr`k2Q/c:BTplaDn+a1LF2EC\*l!Mnh24(VXV1mp8:H)Mt&hm5n"UE[sCg:BXK(>[@JBI$`:tRg:8i?8O_'^"$:(l/W4_C7P[1fd=QPV#sF@`$0)]M3Rcg)Y`s4:RW19f$!QI98J*BV;1p-U"I`=>jt4-<EQV_>OD=^7YT2EId3MqnN1.pV-C8bj5*mk&hQB$2[8>F6uUOCi/hAJRhl3MXm>,J&2aOo!dS)@\.`)r-Jj=$T;Nm^7.5`
'0i10PP&#f"f)fV0fg)IF)ljr6KY/GF.E^NEO#hWSq=e!U=T(n[R3S`<#f#+bh<4)E>j'o)@.'h2#OiM3gh<^KcA)BA13Spt0eX;mbZ/(03T^EnrkAY^Kq
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`Iu.RQmA`]A!ms6']E.KJ9E@1,)-\L-RMWWPBeKk9Q\ICa5u7!WSQKLG1KEtCNX:@0ssYGVU2/"Us8Jg
*$mRdD#)eLqbDj98V3L*iGUc?9O$$R]1^XA,bE,A/b?&;8?buF^EG:!-fe!;-l?_=D"S"@c!tl6aiM,t
Dh-F[=hj@gl0iNH-24$Wh]\.C:an4YA*N1"!jEqaX#>i10"NR0me6:XeDI_:[ajVk&6oSlZXg!k-/QlFSXqcFY"^'t.6Dld]Be&M`5V!d7AZ0_$d@3k'GHk/F.0FoU'R8rjbV#&K[C*[j(l]n0";5Nk^8k/+-]jB
)cs<G"`s@X^^g8Ugc!tl6ZQSkN^6*\H`bN@S6/HD[@3\KKqlm[b/<P4G:^a6K14a?ZXo83&IssP*Jl^Mi>=-3Dabt+r4>B<YDp`!V+W`^a"s-"d/'U"Z+>l3Oa16iaLj-82,TU8`7$B/eOf&[J><Lc(/d)?EmCcoV3",5<_2$p
3oY*sh0lT--fck\T$dMLsiZp.fT[:F)cQf,?<)ZRQaaQ6\qTCfpc/=I9rGjVs4gZ6&`*r)2,-i,#JfF/[8B
-TBoEZ+>nIOa16i!3c:%p"Ree5I,R$gjeI/>7C3[p\&(o-f%S9^sbmjki;mX-=6!Z\] |
\begin{align}
u_t-|Du|^{\gamma}\big(\Delta u+(p-2)\Delta_\infty^Nu\big)=0
\end{align} |
\begin{align*}
\ilN u:=\abs{Du}^{-2}\sum_{i,j=1}^nu_{x_i}u_{x_j}u_{x_ix_j}=\abs{Du}^{-2}\la Du,D^2uDu\ra =\abs{Du}^{-2}\il u
\end{align*} |
\begin{equation}
u_t-\Delta_p u=0
\end{equation} |
\begin{equation}
u_t-\plN u=0
\end{equation} |
\begin{align*}
\Delta u:=\sum_{i=1}^nu_{x_ix_i}
\end{align*} |
\begin{align*}
\il u:=\sum_{i,j=1}^nu_{x_i}u_{x_j}u_{x_ix_j}=\la Du,D^2uDu\ra
\end{align*} |
\begin{align*}
\ilN u:=\frac{\il u}{|Du|^2}.
\end{align*} |
\begin{align}
u_t-|Du|^{\gamma}\big(\Delta u+(p-2)\ilN u\big)=0\quad\text{in }\Omega_T,
\end{align} |
\begin{equation}
F(Du,D^2 u):=\abs{Du}^{\gamma}\big(\Delta u+(p-2)\Delta_{\infty}^{N} u\big)
\end{equation} |
\[
\begin{split}
f(0)=f'(0)=f''(0)=0,\ f''(r)>0 \text{ for all }r>0,
\end{split}
\] |
\[
\begin{split}
\lim_{x\to 0,x\neq 0}F(Dg(x),D^2g(x))=0.
\end{split}
\] |
\[
\begin{split}
\Sigma=\{ \sigma \in C^{1}(\R) \,:\, \sigma \text{ is even},\, \sigma(0)=\sigma'(0)=0,\text{ and }\sigma(r)>0 \text{ for all } r\neq 0 \}.
\end{split}
\] |
\[
\begin{split}
\begin{cases}
\vp_t(x_0,t_0)-F(D\vp(x_0,t_0),D^2\vp(x_0,t_0))\ge 0 & \text{if }D\vp(x_0,t_0)\neq 0,\\
\vp_t(x_0,t_0)\ge 0 & \text{if }D\vp(x_0,t_0)= 0.
\end{cases}
\end{split}
\] |
\begin{equation}
\ue_t-(\abs{D\ue}^2+\epsilon)^{\gamma/2}\Big(\Delta\ue+(p-2)\frac{\il\ue}{\abs{D\ue}^2+\epsilon}\Big)=0
\end{equation} |
\begin{equation}
\begin{aligned}
&(|D\ue|^2+\epsilon)^{\frac{p-2-\gamma}{2}}(\ue_{x_k})_t
-\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2}{2}}AD\ue_{x_k}\big) \\
& +(p-2-\gamma)(|D\ue|^2+\epsilon)^{\frac{p-4-\gamma}{2}}\ue_t\la D\ue,D\ue_{x_k}\ra=0
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
&(|D\ue|^2+\epsilon)^{\frac{p-2-\gamma+s}{2}}\ue_{x_k}(\ue_{x_k})_t
-(|D\ue|^2+\epsilon)^{s/2}\ue_{x_k}\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2}{2}}AD\ue_{x_k}\big) \\
& +(p-2-\gamma)(|D\ue|^2+\epsilon)^{\frac{p-4-\gamma+s}{2}}\ue_t\la D\ue,D\ue_{x_k}\ra \ue_{x_k}=0.
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
&\frac{\big((|D\ue |^2+\epsilon)^{\frac{p+s-\gamma}{2}}\big)_t}{p+s-\gamma}
-(|D\ue|^2+\epsilon)^{s/2}\sum_{k=1}^n\ue_{x_k}\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2}{2}}AD\ue_{x_k}\big) \\
& +(p-2-\gamma)(|D\ue|^2+\epsilon)^{\frac{p-2-\gamma+s}{2}}\ue_t\frac{\il \ue}{|D\ue|^2+\epsilon}=0.
\end{aligned}
\end{equation} |
\begin{align*}
&\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}AD^2\ue D\ue \big)\\
=&\sum_{k=1}^n\diverg\Big(\big((|D\ue|^2+\epsilon)^{s/2} \ue_{x_k} \big)\big((|D\ue|^2+\epsilon)^{\frac{p-2}{2}}A D\ue_{x_k}\big)\Big)\\
=&(|D\ue|^2+\epsilon)^{s/2}\sum_{k=1}^n\ue_{x_k}\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2}{2}}AD\ue_{x_k}\big)\\
&+(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{|D^2\ue|^2+(p-2+s)\frac{|D^2\ue D\ue|^2}{|D\ue|^2+\epsilon}
+s(p-2)\frac{(\il \ue)^2}{(|D\ue|^2+\epsilon)^2}
\Big\},
\end{align*} |
\begin{equation}
\begin{aligned}
&(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{|D^2\ue|^2+(p-2+s)\frac{|D^2\ue D\ue|^2}{|D\ue|^2+\epsilon}
+s(p-2)\frac{(\il \ue)^2}{(|D\ue|^2+\epsilon)^2} \\
& +(p-2-\gamma)(|D\ue|^2+\epsilon)^{-\gamma/2}\ue_t\frac{\il \ue}{|D\ue|^2+\epsilon} \Big\} \\
&=
\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}AD^2\ue D\ue \big)
-\frac{\big((|D\ue |^2+\epsilon)^{\frac{p+s-\gamma}{2}}\big)_t}{p+s-\gamma}.
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
&|Du|^{p-2+s}
\Big\{\frac{1}{n-1}(\Delta_Tu)^2+(p+s)|D_T|Du||^2+(p-1)(s+1)(\ilN u)^2 \\
& +(p-2-\gamma)\abs{Du}^{-\gamma}u_t\ilN u\Big\} \\
&\leq
\diverg\big(|Du|^{p-2+s}AD^2uDu\big)
-\frac{\big(\abs{Du}^{p+s-\gamma}\big)_t}{p+s-\gamma},
\end{aligned}
\end{equation} |
\begin{equation}
\abs{D_T\abs{Du}}^2:=\frac{|D^2uDu|^2}{|Du|^2}-(\ilN u)^2
\quad\text{and} \Delta_T u:=\Delta u-\ilN u.
\end{equation} |
\begin{equation}
\begin{aligned}
Q:
&=\frac{1}{n-1}(\Delta_Tu)^2+(p-1)(p-1+s-\gamma)(\ilN u)^2+(p-2-\gamma)\Delta_Tu\ilN u \\
&=:\la \bar{x},M\bar{x}\ra,
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
u_t |Du|^{q-2}\Delta_q^N u=u_t\diverg\big(|Du|^{q-2}Du\big)
&=
\diverg(u_t|Du|^{q-2}Du)-\frac{(|Du|^{q})_t}{q}
\end{aligned}
\end{equation} |