latex_formula
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4.11M
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\begin{equation}
\begin{aligned}
\diverg&\big(|Du|^{p-2+s }AD^2uDu\big)
-\frac{\big(\abs{Du}^{p +s-\gamma}\big)_t}{p +s-\gamma} \\
& =
\diverg\big(|Du|^{p-2+s }(D^2uDu-\Delta uDu)\big)
+u_t\diverg\big(|Du|^{p-2+s-\gamma}Du\big),
\end{aligned}
\end{equation} |
\begin{align*}
u_t=|Du|^{\gamma}(\Delta_Tu+(p-1)\ilN u), \text{and}\Delta_{p+s-\gamma}^Nu=\Delta_Tu+(p-1+s-\gamma)\ilN u
\end{align*} |
\begin{equation}
\begin{aligned}
&u_t\diverg\big(|Du|^{p-2+s-\gamma}Du\big)\\
=&|Du|^{\gamma}\big(\Delta_Tu+(p-1)\ilN u\big)\cdot|Du|^{p-2+s-\gamma}\cdot\big(\Delta_Tu+(p-1+s-\gamma)\ilN u\big) \\
=&|Du|^{p-2+s }\Big\{(\Delta_Tu)^2+(2 p-2+s-\gamma) \Delta_Tu\ilN u+(p-1)(p-1+s-\gamma)(\ilN u)^2\Big\}.
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
|Du|^{p-2+s}&
\Big\{|D^2 u|^2-(\Delta u)^2
+(p-2+s)\frac{|D^2uDu|^2}{|Du|^2}-(p-2+s )\Delta u\ilN u\Big\} \\
=&
\diverg\big(|Du|^{p-2+s}(D^2uDu-\Delta uDu)\big)
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
&|Du|^{p-2+s }
\Big\{\Big(w_2- \frac{n-2}{n-1} w_1\Big)(\Delta_T u)^2
+w_1(p +s )\abs{D_T\abs{Du}}^2 \\
& +w_2(p-1)(p-1+s-\gamma)(\ilN u)^2+\big(w_2(2p-2+s-\gamma)-w_1(p +s )\big)\Delta_Tu\ilN u\Big\} \\
&\leq
w_1\diverg\big(|Du|^{p-2+s }(D^2uDu-\Delta uDu)\big)
+w_2 u_t\diverg\big(|Du|^{p-2+s-\gamma}Du\big),
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
S
&:=
w_1\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}(D^2\ue D\ue-\Delta \ue D\ue)\big) \\
& +w_2\ue_t\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}}D\ue\big) \\
& +w_3\epsilon\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}-1}(D^2\ue D\ue-\Delta \ue D\ue)\big) \\
& +w_4\epsilon\ue_t\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}-1}D\ue\big),
\end{aligned}
\end{equation} |
\begin{align}
\theta:=\frac{|D\ue|^2}{|D\ue|^2+\epsilon}
\quad\text{and}\kappa:=1-\theta=\frac{\epsilon}{|D\ue|^2+\epsilon},
\end{align} |
\begin{equation}
\begin{aligned}
&\ez\ue_t\diverg\big((\abs{D\ue}^2+\ez)^{\frac{p-2+s-\gamma}{2}-1}D\ue\big)\\
=&\theta(\abs{D\ue}^2+\ez)^{\frac{p-2+s }{2}}\Big((\Delta_T\ue)^2+\big((2p-6+s-\gamma)\theta+2 \big)\Delta_T\ue\ilN \ue\\
&+\big((p-2)\theta+1\big)\big((p-4+s-\gamma)\theta+1 \big)(\ilN \ue)^2\Big)
\end{aligned}
\end{equation} |
\begin{align*}
\ue_t=(|D\ue|^2+\epsilon)^{\gamma/2}\Big(\Delta_T \ue+\big((p-2)\theta+1\big)\ilN\ue\Big)
\end{align*} |
\begin{equation}
\begin{aligned}
&\ez\diverg\big((\abs{D\ue}^2+\ez)^{\frac{p-2+s}{2}-1}(D^2\ue D\ue-\Delta \ue D\ue)\big)\\
=&\theta(\abs{D\ue}^2+\ez)^{\frac{p-2+s}{2}}
\Big\{|D^2 \ue|^2-(\Delta \ue)^2
+(p-4+s)\theta \abs{D\abs{D\ue}}^2\\
&-(p-4+s)\theta \Delta \ue \ilN\ue \Big\}.
\end{aligned}
\end{equation} |
\begin{align}
&(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{
c_1|D^2\ue|^2+c_2|D_T|D\ue||^2
+(c_3-c_1)(\Delta_T\ue)^2 \\
& +\Big((c_3+c_4)\big((p-2)\theta+1\big)-c_1\Big)(\ilN \ue)^2 \\
& +\Big(c_3\big((p-2)\theta+1\big)+(c_3+c_4)-(2c_1+c_2)\Big) \Delta_T\ue\ilN \ue
\Big\}= S,
\end{align} |
\begin{align*}
D_T|Du|:=
\begin{cases}
\la e_1,D|Du|\ra e_1
+\ldots
+\la e_{n-1},D|Du|\ra e_{n-1} &\quad\text{if }Du\neq 0, \\
0 &\quad\text{if }Du=0,
\end{cases}
\end{align*} |
\begin{align*}
\ilN u:=
\begin{cases}
\left< \frac{Du}{|Du|},D|Du|\right>=\frac{\il u}{|Du|^2} &\quad\text{if }Du\neq 0, \\
0 &\quad\text{if }Du=0.
\end{cases}
\end{align*} |
\begin{equation}
|D|Du||^2=\abs{D_T|Du|}^2+(\ilN u)^2 \quad\text{a.e.\ in space in } \Omega_T,
\end{equation} |
\begin{equation}
\Delta_T u=\Delta u-\ilN u \quad\text{a.e.\ in } \Omega_T.
\end{equation} |
\begin{equation}
\begin{aligned}
&(|Du|^2+\epsilon)^{\alpha/2}
\Big\{|D^2u|^2-(\Delta u)^2
+\alpha\frac{|D^2uDu|^2}{|Du|^2+\epsilon}-\alpha\Delta u\frac{\il u}{|Du|^2+\epsilon}\Big\}\\
=&
\diverg\big((|Du|^2+\epsilon)^{\alpha/2}(D^2uDu-\Delta uDu)\big).
\end{aligned}
\end{equation} |
\begin{align*}
&\diverg\big((|Du|^2+\ez)^{\alpha/2}(D^2uDu-\Delta uDu )\big)\\
=& \big< D^2uDu-\Delta uDu , D\big((|Du|^2+\ez)^{\alpha/2} \big)\big>+ (|Du|^2+\ez)^{\alpha/2}\diverg ( D^2uDu-\Delta uDu )\\
=&\big< D^2uDu-\Delta uDu , D\big((|Du|^2+\ez)^{\alpha/2} \big)\big>+ (|Du|^2+\ez)^{\alpha/2}\big(|D^2u|^2 -(\Delta u)^2 \big)\\
=&(|Du|^2+\ez)^{\alpha/2}\Big\{|D^2u|^2-(\Delta u)^2+\alpha\frac{|D^2uDu|^2}{|Du|^2+\ez}-\alpha\Delta u\frac{\il u}{|Du|^2+\ez}\Big\},
\end{align*} |
\begin{equation}
\begin{aligned}
&u_t(|Du|^2+\epsilon)^{\beta/2}
\Big(\Delta u +\beta\frac{\il u}{|Du|^2+\epsilon}\Big) \\
=&u_t\diverg\big((|Du|^2+\ez)^{\beta/2}Du\big) \\
=&
\begin{cases}
\displaystyle\diverg\big(u_t(|Du|^2+\epsilon)^{\beta/2} Du\big)-\Big(\frac{(|Du|^2+\epsilon)^{\frac{\beta+2}{2}}}{\beta+2}\Big)_t
&\quad\text{if }\beta\neq -2, \\
\displaystyle\diverg\big(u_t(|Du|^2+\epsilon)^{-1}Du\big)-\Big(\frac{\ln (|Du|^2+\epsilon)}{2}\Big)_t
&\quad\text{if }\beta=-2.
\end{cases}
\end{aligned}
\end{equation} |
\begin{align*}
&\diverg\big(u_t(|Du|^2+\epsilon)^{\beta/2} Du\big)-\Big(\frac{(|Du|^2+\epsilon)^{\frac{\beta+2}{2}}}{\beta+2}\Big)_t\\
=&u_t\diverg\big((|Du|^2+\epsilon)^{\beta/2} Du\big)+(|Du|^2+\epsilon)^{\beta/2} Du Du_t
-\Big(\frac{(|Du|^2+\epsilon)^{\frac{\beta+2}{2}}}{\beta+2}\Big)_t\\
=&u_t\diverg\big((|Du|^2+\epsilon)^{\beta/2} Du\big)\\
=&u_t\big<D\big((|Du|^2+\epsilon)^{\beta/2}\big), Du\big>+u_t (|Du|^2+\epsilon)^{\beta/2} \diverg(Du) \\
=&u_t(|Du|^2+\ez)^{\beta/2}\Big(\Delta u+\beta\frac{\il u}{|Du|^2+\ez}\Big).
\end{align*} |
\begin{equation}
GD_1(\alpha):=\diverg\big((|Du|^2+\epsilon)^{\alpha/2}(D^2u Du-\Delta uDu)\big)
\end{equation} |
\begin{equation}
\begin{aligned}
GD_2(\alpha):=
\begin{cases}
\displaystyle\diverg\big(u_t(|Du|^2+\epsilon)^{\frac{\alpha-\gamma}{2}} Du\big)-\Big(\frac{(|Du|^2+\epsilon)^{\frac{\alpha-\gamma+2}{2}}}{\alpha-\gamma+2}\Big)_t
&\quad\text{if }\alpha\neq \gamma-2, \\
\displaystyle\diverg\big(u_t(|Du|^2+\epsilon)^{-1} Du\big)-\Big(\frac{\ln (|Du|^2+\epsilon)}{2}\Big)_t
&\quad\text{if }\alpha=\gamma-2.
\end{cases}
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
S:=&w_1GD_1(p-2+s)+w_2GD_2(p-2+s)\\
&+\epsilon w_3GD_1(p-4+s)+\epsilon w_4GD_2(p-4+s)
\end{aligned}
\end{equation} |
\begin{align*}
\theta=\frac{|D\ue|^2}{|D\ue|^2+\epsilon}
\quad\text{and}\kappa=\frac{\epsilon}{|D\ue|^2+\epsilon},
\end{align*} |
\begin{align*}
\frac{|D^2\ue D\ue|^2}{|D\ue|^2+\epsilon}=\theta \abs{D\abs{D\ue}}^2 \quad\text{and}\frac{\il \ue}{|D\ue|^2+\epsilon}=\theta\ilN \ue.
\end{align*} |
\begin{align*}
S
=&
w_1 (|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{|D^2\ue|^2-(\Delta \ue)^2
+(p-2+s)\Big(\frac{|D^2\ue D\ue|^2}{|D\ue|^2+\epsilon}- \Delta \ue\frac{\il \ue}{|D\ue|^2+\epsilon}\Big)\Big\}\\
&+w_2\ue_t(|D\ue|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}}
\Big\{\Delta \ue +(p-2+s-\gamma)\frac{\il \ue}{|D\ue|^2+\epsilon}\Big\}\\
&+\ez w_3(|D\ue|^2+\epsilon)^{\frac{p-4+s}{2}}
\Big\{|D^2\ue|^2-(\Delta \ue)^2
+(p-4+s)\Big(\frac{|D^2\ue D\ue|^2}{|D\ue|^2+\epsilon}- \Delta \ue\frac{\il \ue}{|D\ue|^2+\epsilon}\Big)\Big\}\\
&+\ez w_4\ue_t(|D\ue|^2+\epsilon)^{\frac{p-4+s-\gamma}{2}}
\Big\{\Delta \ue +(p-4+s-\gamma)\frac{\il \ue}{|D\ue|^2+\epsilon}\Big\}.
\end{align*} |
\begin{align*}
S=&(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{
(w_1+w_3\kappa)\big(|D^2\ue|^2-(\Delta \ue)^2\big)\\
&
+\big(w_1(p-2+s)+w_3(p-4+s)\kappa\big)\theta(|D|D\ue||^2-\Delta \ue\ilN \ue) \\
&
+(w_2+w_4\kappa)(|D\ue|^2+\epsilon)^{-\gamma/2}\ue_t\Delta \ue\\
&
+\big(w_2(p-2+s-\gamma)+w_4(p-4+s-\gamma)\kappa\big)\theta(|D\ue|^2+\epsilon)^{-\gamma/2}\ue_t\ilN \ue
\Big\}\\
=&
(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{
c_1\big(|D^2\ue|^2-(\Delta \ue)^2\big)
+c_2(|D|D\ue||^2-\Delta \ue\ilN \ue) \\
& +c_3(|D\ue|^2+\epsilon)^{-\gamma/2}\ue_t\Delta \ue
+c_4(|D\ue|^2+\epsilon)^{-\gamma/2}\ue_t\ilN \ue
\Big\}
\end{align*} |
\begin{align}
\begin{cases}
c_1=w_1+w_3\kappa, &c_2=\big(w_1(p-2+s)+w_3(p-4+s)\kappa\big)\theta,\\
c_3=w_2+w_4\kappa, &c_4=\big(w_2(p-2+s-\gamma)+w_4(p-4+s-\gamma)\kappa\big)\theta.
\end{cases}
\end{align} |
\begin{equation}
\begin{aligned}
S
&=
(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{
c_1|D^2\ue|^2+c_2|D_T|D\ue||^2
-c_1(\Delta_T\ue)^2
-c_1(\ilN \ue)^2 \\
& -(2c_1+c_2)\Delta_T\ue\ilN \ue
+c_3(|D\ue|^2+\epsilon)^{-\gamma/2}\ue_t\Delta_T\ue \\
& +(c_3+c_4)(|D\ue|^2+\epsilon)^{-\gamma/2}\ue_t\ilN \ue
\Big\}
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
&(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{
c_1|D^2\ue|^2+c_2|D_T|D\ue||^2
+(c_3-c_1)(\Delta_T\ue)^2 \\
& +\big((c_3+c_4)P_\theta -c_1\big)(\ilN \ue)^2 \\
& +\big(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\big)\Delta_T\ue\ilN \ue
\Big\}=S,
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{
c_1|D^2\ue|^2+c_2|D_T|D\ue||^2 + R\Big\}=S
\end{aligned}
\end{equation} |
\begin{align*}
R
&:=
(c_3-c_1)(\Delta_T\ue)^2
+\big((c_3+c_4)P_\theta-c_1\big)(\ilN \ue)^2 \\
& +\big(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\big)\Delta_T\ue\ilN \ue
\end{align*} |
\begin{align*}
\begin{cases}
N_{11}&=c_2-c_1\\
N_{12}=N_{21}&=
\tfrac{1}{2}\big(c_3P_\theta+(c_3+c_4)
-(2c_1+c_2)\big)\\
N_{22}&=(c_3+c_4)P_\theta-c_1.
\end{cases}
\end{align*} |
\begin{align*}
\|N\|_{L^\infty(\Omega_T)}
:=\sup\{|N(x,t)|:(x,t)\in \Omega_T\}
\end{align*} |
\begin{align*}
S=w_1GD_1(p-2+s)+w_2GD_2(p-2+s)+\epsilon w_3GD_1(p-4+s)+\epsilon w_4GD_2(p-4+s)
\end{align*} |
\begin{equation}
\begin{aligned}
S
&=
(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{
c_1|D^2\ue|^2+c_2|D_T|D\ue||^2+\la \bar{x},N\bar{x}\ra
\Big\}
\end{aligned}
\end{equation} |
\begin{align*}
\big(M+\lambda(N-M)\big)_{11}
=M_{11}+\lambda(N_{11}-M_{11})
\geq c-\lambda(\|N\|_{L^{\infty}(\Omega_T)}+\|M\|_{L^\infty(\Omega_T)}),
\end{align*} |
\begin{align*}
\operatorname{det}\big(M+\lambda(N-M)\big)
&=
\operatorname{det}(M)
+\lambda\big(M_{11}N_{22}+M_{22}N_{11}-2M_{12}N_{12}-2\det(M)\big) \\
& +\lambda^2\operatorname{det}(N-M) \\
&\geq
c-2\lambda\left(2\|M\|_{L^\infty(\Omega_T)}\|N\|_{L^\infty(\Omega_T)}+\|M\|_{L^\infty(\Omega_T)}^2\right)-\lambda^2\|N-M\|_{L^\infty(\Omega_T)}^2 \\
&\geq
c-4\lambda\left(\|M\|_{L^\infty(\Omega_T)}+\|N\|_{L^\infty(\Omega_T)}\right)^2.
\end{align*} |
\begin{equation}
\begin{aligned}
\int_{Q_r}&\abs{D\big((|D\ue|^2+\epsilon)^{\frac{p-2+s}{4}}D\ue\big)}^2dxdt\\
\leq&
\frac{C}{r^2}\Big(
\int_{Q_{2r}}(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}|D\ue|^2dxdt
+\int_{Q_{2r}}(|D\ue|^2+\epsilon)^{\frac{p+s-\gamma}{2}}dxdt\Big) \\
&+\epsilon\Big(\frac{C}{r^2}\int_{Q_{2r}}\big|\ln(|D\ue|^2+\epsilon)\big|dxdt
+C\int_{B_{2r}}\big|\ln(|D\ue(x,t_0)|^2+\epsilon)\big|dx\Big)
\end{aligned}
\end{equation} |
\begin{equation}
u_t-|Du|^{\gamma}\big(\Delta u+(p-2)\ilN u\big)=0,
\end{equation} |
\begin{equation}
S=w_1GD_1(p-2+s)+w_2GD_2(p-2+s)
\end{equation} |
\begin{equation}
\begin{aligned}
|Du|^{p-2+s}&
\Big\{
w_1|D^2u|^2+w_1(p-2+s)|D_T|Du||^2+(w_2-w_1)(\Delta_T u)^2 \\
& +\big(w_2(p-1)(p-1+s-\gamma)-w_1\big)(\ilN u)^2 \\
& +\big(w_2(2p-2+s-\gamma)-w_1(p+s)\big)\Delta_Tu\ilN u
\Big\}=S.
\end{aligned}
\end{equation} |
\begin{align*}
Q:
&=
\Big(w_2- \frac{n-2}{n-1} w_1\Big)(\Delta_T u)^2
+w_2(p-1)(p-1+s-\gamma)(\ilN u)^2 \\
& +\big(w_2(2p-2+s-\gamma)-w_1(p+s)\big)\Delta_Tu\ilN u.
\end{align*} |
\begin{align*}
D\varphi (x_0,t_0)=Du(x_0,t_0)\neq 0,
\phi_t(x_0,t_0)=u_t(x_0,t_0)=0
\end{align*} |
\begin{align*}
\vp_t(x_0,t_0)=C>0.
\end{align*} |
\begin{align*}
\phi(x,t)=u(x_0,t_0)+\vp_t(x_0,t_0)(t-t_0)+f(\abs x)+\sigma(t-t_0)
\end{align*} |
\begin{align}
\lim_{y\to 0,y\neq 0}F\big(Dg(y),D^2g(y)\big)=0.
\end{align} |
\begin{align}
\phi_t(x,t_0)-F\big(D\phi(x,t_0),D^2\phi(x,t_0)\big)=\vp_t(x_0,t_0)-F\big(Dg(x),D^2g(x)\big)>0.
\end{align} |
\begin{equation}
\begin{aligned}
\phi_t(x,t_0)-F\big(D\phi(x,t_0),D^2\phi(x,t_0)\big)
&=\vp_t(x_0,t_0)-F\big(Dg(x),D^2g(x)\big) \\
&=u_t(x,t_0)-F\big(Dg(x),D^2g(x)\big) \\
&\le u_t(x,t_0)-F\big(Du(x,t_0),D^2u(x,t_0)\big)=0,
\end{aligned}
\end{equation} |
\begin{align*}
\left|D(|Du|^{\frac{p-2+s}2}Du)\right|
&=\frac{1}{2}\alpha^{\frac{p+s}{2}}(\alpha-1)(p+s)|x_1|^{\frac{(\alpha-1)(p+s)}{2}-1}\\
&=C(p,s,\gamma)|x_1|^{\frac{ p+s }{2(\gamma+1)}-1}.
\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}
&(|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}
\Big\{1+(p-2+s)\theta+s(p-2)\theta^2 \\
& +(p-2-\gamma)\big((p-2)\theta+1\big)\theta\Big\}|D^2\ue|^2 \\
&=
\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2+s}{2}}(D^2\ue D\ue-\Delta\ue D\ue) \big) \\
& +\ue_t\diverg\big((|D\ue|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}}D\ue\big).
\end{aligned}
\end{equation} |
\begin{equation}
\ue_t-(|D\ue|^2+\epsilon)^{\gamma/2}\Big(\Delta\ue+(p-2)\frac{\il\ue}{|D\ue|^2+\epsilon}\Big)=0
\end{equation} |
\begin{align}
S=
w_1\diverg (D^2\ue D\ue-\Delta \ue D\ue)
+w_2\ue_t\diverg\big((|D\ue|^2+\ez)^{-\frac{\gamma}{2}}D\ue\big),
\end{align} |
\begin{equation}
\begin{aligned}
\int_{Q_r}|D^2\ue|^2dxdt
&\leq
\frac{C}{r^2}\Big(
\int_{Q_{2r}}|D\ue|^2dxdt
+\int_{Q_{2r}}(|D\ue|^2+\epsilon)^{\frac{2-\gamma}{2}}dxdt\Big) \\
& +\epsilon\Big(\frac{C}{r^2}\int_{Q_{2r}}\big|\ln(|D\ue|^2+\epsilon)\big|dxdt
+C\int_{B_{2r}}\big|\ln(|D\ue(x,t_0)|^2+\epsilon)\big|dx
\Big)
\end{aligned}
\end{equation} |
\begin{align}
S&=w_1GD_1(p-2+s)+w_2GD_2(p-2+s)+\epsilon w_3GD_1(p-4+s)+\epsilon w_4GD_2(p-4+s)\nonumber\\
&=w_1GD_1(0)+w_2GD_2(0)+w_3\epsilon GD_1(-2)+w_4\epsilon GD_2(-2),
\end{align} |
\begin{equation}
\begin{aligned}
\begin{cases}
w_1= p-\gamma -2\sqrt{(p-1)(1-\gamma)} +\eta, &w_2=2, \\
w_3=0, &w_4=0,
\end{cases}
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
\begin{cases}
w_1=p-\gamma,
&w_2=2 \\
w_3=4-p+\gamma,&w_4=2,
\end{cases}
\end{aligned}
\end{equation} |
\begin{align}
S
&=
c_1|D^2\ue|^2+c_2|D_T|D\ue||^2+(c_3-c_1)(\Delta_T \ue)^2+\big((c_3+c_4)P_\theta-c_1\big) (\ilN \ue)^2\nonumber\\
& +\big(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\big)\Delta_T \ue \ilN \ue.
\end{align} |
\begin{align}
\begin{cases}
c_1=w_1+w_3\kappa, &c_2=-2w_3\theta\kappa, \\
c_3=w_2+w_4\kappa,
&c_4=-(w_2\gamma+w_4\kappa(2+\gamma))\theta,
\end{cases}
\end{align} |
\begin{equation}
\begin{aligned}
S=&c_1\big(2|D_T|D\ue| |^2
+ (\Delta_T \ue)^2
+(\ilN \ue )^2\big)+c_2|D_T|D\ue||^2+(c_3-c_1)(\Delta_T \ue)^2\\
&+\big((c_3+c_4)P_\theta-c_1\big) (\ilN \ue)^2
+\big(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\big)\Delta_T \ue \ilN \ue\\
=&(2c_1 +c_2)|D_T|D\ue||^2+Q.
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
Q=c_3 (\Delta_T \ue )^2+ (c_3+c_4)P_\theta (\ilN \ue)^2 +\big(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\big) \Delta_T \ue \ilN \ue,
\end{aligned}
\end{equation} |
\begin{align}
2c_1+c_2=2(w_1+w_3\kappa)-2w_3\theta\kappa\geq c
\end{align} |
\begin{align}
c_3=w_2+w_4\kappa\geq c,
\end{align} |
\begin{align}
\det(M)=c_3(c_3+c_4)P_\theta-\frac{\big(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\big)^2}{4}\geq c
\end{align} |
\begin{align*}
\begin{cases}
c_1=w_1, &c_2=0,\\
c_3=w_2,
&c_4=-w_2\gamma\theta.
\end{cases}
\end{align*} |
\begin{equation}
\begin{aligned}
S=2w_1|D_T|D\ue||^2+ w_2(\Delta_T \ue )^2+w_2 P_\theta R_\theta(\ilN \ue)^2+\big(w_2(P_\theta+R_\theta)-2w_1\big)\Delta_T \ue \ilN \ue,
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
S=&2w_1|D_T|D\ue||^2+ 2\Big((\Delta_T \ue )^2+ P_\theta R_\theta(\ilN \ue)^2+( P_\theta+R_\theta -w_1)\Delta_T \ue \ilN \ue\Big)\\
=& 2w_1|D_T|D\ue||^2+Q,
\end{aligned}
\end{equation} |
\begin{align*}
X_1^\prime(\theta)
=&p-2-\gamma-\frac{(p-2) R_\theta-\gamma P_\theta }{\sqrt{P_\theta R_\theta}}\\
=&\frac{(\sqrt{P_\theta}-\sqrt{R_\theta})\big((p-2)\sqrt{R_\theta}+\gamma\sqrt{P_\theta}\big)}{ \sqrt{P_\theta R_\theta }}.
\end{align*} |
\begin{align*}
\sup_\theta X_1 (\theta)&
=\max\{X_1 (0), X_1 (1)\}\\
&=\max\Big\{0, \big(\sqrt{p-1}-\sqrt{1-\gamma}\big)^2 \Big\}\\
&= \big(\sqrt{p-1}-\sqrt{1-\gamma}\big)^2.
\end{align*} |
\begin{align*}
\inf_\theta X_2(\theta)&
=\min\Big\{X_2(0),X_2\Big(\frac{p-2-\gamma}{(p-2)\gamma}\Big), X_2(1)\Big\}\\
&=\min\Big\{4, \frac{p-2}{\gamma}+\frac{\gamma}{p-2}+2,\big(\sqrt{p-1}+\sqrt{1-\gamma}\big)^2 \Big\}\\
&=\min\Big\{4, \big(\sqrt{p-1}+\sqrt{1-\gamma}\big)^2 \Big\}.
\end{align*} |
\begin{align*}
w_1=\big(\sqrt{p-1}-\sqrt{1-\gamma}\big)^2+\eta.
\end{align*} |
\begin{align}
\begin{cases}
2c_1+c_2&=2(w_1+w_3\kappa)-2w_3\theta\kappa\geq c,\\
c_3&=w_2+w_4\kappa\geq c,\\
\det(M)&=c_3(c_3+c_4)P_\theta-\frac{\left(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\right)^2}{4}\geq c
\end{cases}
\end{align} |
\begin{align*}
Q=c_3 (\Delta_T \ue )^2+ (c_3+c_4)P_\theta (\ilN \ue)^2 +\big(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\big) \Delta_T \ue \ilN \ue.
\end{align*} |
\begin{align}
\begin{cases}
w_1 =p-\gamma,
&w_2=2, \\
w_3 =4-p+\gamma,
&w_4=2,
\end{cases}
\end{align} |
\begin{align}
\begin{cases}
2c_1+c_2&=2(w_1+w_3\kappa)-2w_3\theta\kappa\geq c,\\
c_3&=w_2+w_4\kappa\geq c,\\
c_3+c_4&=w_2+w_4\kappa-\big(w_2\gamma+w_4\kappa(2+\gamma)\big)\theta\geq c
\end{cases}
\end{align} |
\begin{align*}
2c_1+c_2
=&2(p-\gamma)+2(4-p+\gamma)\kappa^2.
\end{align*} |
\begin{equation}
\begin{aligned}
2c_1+c_2\geq\min\{2(p-\gamma), 8\}>0.
\end{aligned}
\end{equation} |
\begin{align}
c_3=2+2\kappa\geq2>0.
\end{align} |
\begin{align*}
c_3+c_4
=&2(1-\gamma)-2\kappa+2(2+\gamma)\kappa^2.
\end{align*} |
\begin{align}
c_3+c_4=2(1-\gamma)-\frac{1}{2(2+\gamma)}>0
\end{align} |
\begin{align}
u_t-|Du|^{\gamma}\big(\Delta u+(p-2)\Delta_\infty^Nu\big)=0.
\end{align} |
\begin{equation}
\begin{cases}
\begin{aligned}
\ue_t-(|D\ue|^2+\epsilon)^{\gamma/2}\Big(\Delta\ue+(p-2)\frac{\il\ue}{|D\ue|^2+\epsilon}\Big)=0
\quad&\text{in } U_{t_1,t_2};\\
\ue=u
\quad&\text{on } \partial_pU_{t_1,t_2},
\end{aligned}
\end{cases}
\end{equation} |
\begin{equation}
\begin{aligned}
\int_{Q_r}|D^2\ue|^2dxdt
&\leq
\frac{C}{r^2}\Big(
\int_{Q_{2r}}|D\ue|^2dxdt
+\int_{Q_{2r}}(|D\ue|^2+\epsilon)^{\frac{2-\gamma}{2}}dxdt\Big) \\
& +\epsilon\Big(\frac{C}{r^2}\int_{Q_{2r}}\big|\ln(|D\ue|^2+\epsilon)\big|dxdt \\
& +C\int_{B_{2r}}\big|\ln\big(|D\ue(x,t_0)|^2+\epsilon\big)\big|dx
\Big)
\end{aligned}
\end{equation} |
\begin{equation}
\begin{cases}
\begin{aligned}
\bar{u}_t-|D\bar{u}|^\gamma\big(\Delta\bar{u}+(p-2)\ilN\bar{u}\big)=0
\quad&\text{in } U_{t_1,t_2};\\
\bar{u}=u
\quad&\text{on } \partial_pU_{t_1,t_2}.
\end{aligned}
\end{cases}
\end{equation} |
\begin{align*}
\int_{Q_{r}}|D^2u|^2dxdt
&\leq \liminf_{\epsilon\to 0}\int_{Q_{r}}|D^2\ue|^2dxdt \\
&\leq \liminf_{\epsilon\to 0} \Bigg(\frac{C}{r^2}\Big(\int_{Q_{2r}} |D\ue|^2 dxdt
+\int_{Q_{2r}}(|D\ue|^2+\epsilon)^{\frac{2-\gamma}{2}}dxdt\Big) \\
& +\epsilon\Big(\frac{C}{r^2}\int_{Q_{2r}}\big|\ln(|D\ue|^2+\epsilon)\big|dxdt
+C\int_{B_{2r}}\big|\ln\big(|D\ue(x,t_0)|^2+\epsilon\big)\big|dx
\Big)\Bigg) \\
&= \frac{C}{r^2}\Big(\int_{Q_{2r}}|Du|^2 dxdt
+\int_{Q_{2r}}|Du|^{2-\gamma} dxdt\Big),
\end{align*} |
\begin{align*}
\abs{\ue_t}=& \abs{ (|D\ue|^2+\epsilon)^{\gamma/2}\Big(\Delta\ue+(p-2)\frac{\il\ue}{|D\ue|^2+\epsilon}\Big)}\nonumber\\
\leq& (|D\ue|^2+\epsilon)^{\gamma/2}(\abs{\Delta\ue}+\abs{p-2}\abs{D^2\ue} )\nonumber\\
\leq & (p+2)(|D\ue|^2+\epsilon)^{\gamma/2}\abs{D^2\ue}.
\end{align*} |
\begin{equation}
\begin{aligned}
\int_{Q_r}&\abs{D\big((|Du|^2+\epsilon)^{\frac{p-2+s}{4}}Du\big)}^2dxdt\\
\leq&
\frac{C}{r^2}\Big(
\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s}{2}}|Du|^2dxdt
+\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p+s-\gamma}{2}}dxdt\Big) \\
&+\epsilon\Big(\frac{C}{r^2}\int_{Q_{2r}}\abs{\ln(|Du|^2+\epsilon)}dxdt
+C\int_{B_{2r}}\abs{\ln\big(|Du(x,t_0)|^2+\epsilon\big)}dx
\Big)
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
\lambda(|Du|^2+&\epsilon)^{\frac{p-2+s}{2}}|D^2u|^2\\
&\leq
w_1\diverg\big((|Du|^2+\epsilon)^{\frac{p-2+s}{2}}(D^2uDu-\Delta uDu)\big) \\
& +w_2
\diverg\big(u_t(|Du|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}} Du\big)
-w_2\Big(\frac{(|Du|^2+\epsilon)^{\frac{p+s-\gamma}{2}}}{p+s-\gamma}\Big)_t \\
& +\epsilon w_3\diverg\big((|Du|^2+\epsilon)^{\frac{p-4+s}{2}}(D^2uDu-\Delta uDu)\big) \\
& +\epsilon w_4
\diverg\big(u_t(|Du|^2+\epsilon)^{\frac{p-4+s-\gamma}{2}} Du\big)
-\epsilon w_4\Big(\frac{(|Du|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}}}{p-2+s-\gamma}\Big)_t.
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
\lambda\int_{Q_{2r}}(|Du|^2+&\epsilon)^{\frac{p-2+s}{2}}|D^2u|^2\phi^2dxdt \\
&\leq
-2w_1\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s}{2}}\la D^2uDu-\Delta uDu,D\phi\ra \phi dxdt \\
& -2w_2\int_{Q_{2r}} u_t(|Du|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}} \la Du,D\phi\ra\phi dxdt \\
& +\frac{2w_2}{p+s-\gamma}\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p+s-\gamma}{2}}\phi_t\phi dxdt \\
& -2\epsilon w_3\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-4+s}{2}}\la D^2uDu-\Delta uDu,D\phi\ra \phi dxdt \\
& -2\epsilon w_4
\int_{Q_{2r}}u_t(|Du|^2+\epsilon)^{\frac{p-4+s-\gamma}{2}}\la Du,D\phi\ra\phi dxdt \\
& +\frac{2\epsilon w_4}{p-2+s-\gamma}\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}}\phi_t\phi dxdt.
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
&\lambda\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s}{2}}|D^2u|^2\phi^2dxdt \\
&\leq
C\Big(\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s}{2}}|D^2u||Du||D\phi|\phi dxdt \\
& +\int_{Q_{2r}} |u_t|(|Du|^2+\epsilon)^{\frac{p-2+s-\gamma}{2}}|Du||D\phi|\phi dxdt \\
& +\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p+s-\gamma}{2}}|\phi_t|\phi dxdt\Big),
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
&(\lambda-2\eta)\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s}{2}}|D^2u|^2\phi^2dxdt \\
&\leq
\frac{C}{\eta}\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p-2+s}{2}}|Du|^2|D\phi|^2dxdt
+C\int_{Q_{2r}}(|Du|^2+\epsilon)^{\frac{p+s-\gamma}{2}}|\phi_t|\phi dxdt,
\end{aligned}
\end{equation} |
\begin{equation}
(|Du|^2+\epsilon)^{-\gamma/2}|u_t|
=\abs{\Delta u+(p-2)\frac{\il u}{|Du|^2+\epsilon}}\leq C|D^2u|
\end{equation} |
\begin{equation}
\begin{aligned}
&\lambda\int_{Q_{2r}}(|Du|^2+\epsilon)^{\gamma/2}|D^2u|^2\phi^2dxdt \\
&\leq
C\Big(\int_{Q_{2r}}(|Du|^2+\epsilon)^{\gamma/2}|D^2u||Du||D\phi|\phi dxdt
+\int_{Q_{2r}} |u_t||Du||D\phi|\phi dxdt \\
& +\int_{Q_{2r}}(|Du|^2+\epsilon)|\phi_t|\phi dxdt
+\epsilon\int_{Q_{2r}}\big|\ln(|Du|^2+\epsilon)\big||\phi_t|\phi dxdt \\
& +\epsilon\int_{B_{2r}}\abs{\ln\big(|Du(x,t_0)|^2+\epsilon\big)}\phi^2(x,t_0)dx
\Big),
\end{aligned}
\end{equation} |
\begin{equation}
\begin{aligned}
&(\lambda-2\eta)\int_{Q_{2r}}(|Du|^2+\epsilon)^{\gamma/2}|D^2u|^2\phi^2dxdt \\
&\leq
\frac{C}{\eta}\int_{Q_{2r}}(|Du|^2+\epsilon)^{\gamma/2}|Du|^2|D\phi|^2dxdt
+C\Big(\int_{Q_{2r}}(|Du|^2+\epsilon)|\phi_t|\phi dxdt \\
& +\epsilon\int_{Q_{2r}}\big|\ln(|Du|^2+\epsilon)\big||\phi_t|\phi dxdt
+\epsilon\int_{B_{2r}}\big|\ln\big(|Du(x,t_0)|^2+\epsilon\big)\big|\phi^2(x,t_0)dx
\Big)
\end{aligned}
\end{equation} |
\begin{align*}
Q
&=
\Big(w_2-\frac{n-2}{n-1}w_1\Big)(\Delta_T u)^2
+w_2(p-1)(p-1+s-\gamma)(\ilN u)^2 \\
& +\big(w_2(2p-2+s-\gamma)-w_1(p+s)\big)\Delta_Tu\ilN u.
\end{align*} |
\begin{equation}
P:=p-1\quad\text{and}K:=\gamma+1,
\end{equation} |
\begin{equation}
G:=p-1+s-\gamma\quad\text{and}E:=s+1+\frac{p-1}{n-1}.
\end{equation} |
\begin{align*}
\operatorname{det}(M)
=a\Big(w_2-\frac{n-2}{n-1}\Big)^2+b\Big(w_2-\frac{n-2}{n-1}\Big)+c
\end{align*} |
\begin{align}
a=-\frac{1}{4}(G-P)^2, b=P\cdot E+\frac{1}{2}(G-P)\Big(\frac{G}{n-1}+K-\frac{(n-2)P}{n-1}\Big) \nonumber
\end{align} |
\begin{align*}
b^2-4ac=G\cdot P\cdot E\Big(\frac{G}{n-1}+K\Big).
\end{align*} |