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

• Jan 16th 2009, 10:23 AM
KroneckerDelta
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)
• Jan 16th 2009, 10:26 AM
Mush
Quote:

Originally Posted by KroneckerDelta
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!

If you put {math} before your mathematical statement, and then put {/math} after them, it will produce latex! But use square brackets, not curly brackets. I just used curly brackets to avoid it latexing my text!