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Math Help - A property of the single layer potential & compatibility conditions in 2D?

  1. #1
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    A property of the single layer potential & compatibility conditions in 2D?

    So I'm reading a journal and I have a surface S of a domain D, these areas are extended with a t variable, and another surface \Gamma of a domain E which contains the domain D entirely, also extended with a time variable. I also have a dunemerable, dense set of points \{x_k,t_i\} on \Gamma\times(0,T]. F(x,t,y,\tau) represents the fundamental solution of the heat equation, f an arbitrary continuous function and the integral

    \int_{0}^{t_i} \int_{S} f(y,\tau) F(x_k,t_i,y,\tau) \,dS d\tau = 0.

    Then it introduces the potential of a simple layer

    V(x,t) = \int_{0}^{t} \int_{S} f(y,\tau) F(x,t,y,\tau) \,dS d\tau

    from the above equation we know that V(x_k,t_i)=0, and because V is continuous and is equal to zero for a denumerable, everywhere dense set of points V(x,t) = 0 for (x,t)\in\Gamma\times(0,T]. The journal then states that V(x,t)=0 in the whole external region outside of E (Basically V=0 in the infinite space E^c,) (with no explaination why) I was just wondering why this was the case that V=0 outside E, or do I need to give more details.

    Also does anyone know the compatability conditions for the heat equation in two dimensions or somewhere where I can find them on the internet, (its so I can state that these allow there to be a unique solution to the heat equation).

    Any help would be greatly appreciated. Thanks!

    (Thanks mush for the tips)
    Last edited by KroneckerDelta; January 16th 2009 at 11:35 AM.
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  2. #2
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    Quote Originally Posted by KroneckerDelta View Post
    So I'm reading a journal and I have a surface $S$ of a domain $D,$ these areas are extended with a $t$ variable, and another surface $\Gamma$ of a domain $E$ which contains the domain $D$ entirely, also extended with a time variable. I also have a dunemerable, dense set of points $\{x_k,t_i\}$ on $\Gamma\times(0,T].$ $F(x,t,y,\tau)$ represents the fundamental solution of the heat equation, $f$ an arbitrary continuous function and the integral

    $$\int_{0}^{t_i} \int_{S} f(y,\tau) F(x_k,t_i,y,\tau) \,dS d\tau = 0.$$

    Then it introduces the potential of a simple layer

    $$V(x,t) = \int_{0}^{t} \int_{S} f(y,\tau) F(x,t,y,\tau) \,dS d\tau$$

    from the above equation we know that $V(x_k,t_i)=0,$ and because $V$ is continuous and is equal to zero for a denumerable, everywhere dense set of points $V(x,t) = 0$ for $(x,t)\in\Gamma\times(0,T].$ The journal then states that $V(x,t)=0$ in the whole external region outside of $E$ (Basically $V=0$ in the infinite space $E^c,$) (with no explaination why) I was just wondering why this was the case that $V=0$ outside $E,$ or do I need to give more details.

    Also does anyone know the compatability conditions for the heat equation in two dimensions or somewhere where I can find them on the internet, (its so I can state that these allow there to be a unique solution to the heat equation).

    Any help would be greatly appreciated. Thanks!
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