latex_formula
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4.11M
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\begin{align*}
&u_t - \cM^-_\cL u = 0 \text{ in $C_{8,3}$},\\
&\|u\|_{L^\8((-3,0] \mapsto L^1(\w_\s))} + [u]_{C^{0,1}((-3,0] \mapsto L^1(\w_\s))} \leq 1.
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\begin{align*}
K_A^\s(y) := (2-\s)\frac{\chi_A(y)}{|y|^{n+\s}},
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\begin{align*}
w_A(x) :=\varphi(x)\int (\d u(x;y) - \d u(0;y))K_A^\s(y)dy.
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\begin{align*}
P(x) := \sup_{A\ss B_1} w_A &= (2-\s)\varphi(x)\int_{B_{1/2}}\frac{(\d u(x;y)-\d u(0;y))^+}{|y|^{n+\a}}dy,\\
N(x) := \sup_{A\ss B_1} (-w_A) &= (2-\s)\varphi(x)\int_{B_{1/2}}\frac{(\d u(x;y)-\d u(0;y))^-}{|y|^{n+\a}}dy.
\end{align*} |
\begin{align*}
L(u_x-u)(0) &= \int(\d u(x;y)-\d u(0;y))K^\s(y)dy,\\
\l P(x) - \L N(x) &\leq \int_{B_1}(\d u(x;y)-\d u(0;y))K^\s(y)dy \leq \L P(x) - \l N(x).
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\begin{align*}
\int_{B^c_1} = &\int u(y)\1K^\s(y-x)\chi_{B^c_1}(y-x) - K^\s(y)\chi_{B^c_1}(y)\2dy\\
&{} + (u(x)-u(0))\int_{B^c_1} K^\s(y)dy.
\end{align*} |
\begin{align*}
\tilde w_K(x,t) = \frac{w_K(\kappa x, \kappa^\s t)}{1-\theta},
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\begin{align*}
v_A := \1\frac{1}{2} - w_A\2^+,
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\begin{align*}
G := \{w_A \geq (1/2 - s\theta)\} \cap C_{\kappa,\kappa^\s}(0,-\kappa^\s)
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\begin{align*}
0 \leq N+w_B=P-w_A \leq s\theta.
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\begin{align*}
v_B(x,t) = \1w_B(\kappa\eta x, (\kappa\eta)^\s t-\kappa^\s)+\frac{\l}{8\L}\2^+.
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\begin{align*}
\frac{\l}{8\L} = v_B(0,0) &\leq C\1 \e_1 + \eta^{-(n+\s)}s^{-\e} + \sup_{t\in[-\eta^{-\s},0]}\int_{B^c_{\eta^{-1}}} \frac{|v_B(y,t)|}{|y|^{n+\s}}dy\2.
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\begin{align*}
\int_{B^c_{\eta^{-1}}} \frac{|v_B(y,t)|}{|y|^{n+\s}}dy &= (\kappa\eta)^\s\int_{B^c_\kappa} \frac{\1w_B(y, (\kappa\eta)^\s t-\kappa^\s)+\frac{\l}{8\L}\2^+}{|y|^{n+\s}}dy,\\
&\leq C\eta^\s,
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\begin{align*}
u_t-\cM^-_\cL u=0 \text{ in viscosity in $C_{1,1}$},
\end{align*} |
\begin{align*}
&(-\D)^\s u(0)-(-\D)^\s u(x) =\\
&C\1 P(x)+N(x)+(2-\s)\int_{B^c_1}\frac{\d u(x;y)-\d u(0;y)}{|y|^{n+\s}}dy\2.
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