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edit approved Oct 16, 2014 at 21:55
Margaret Friedland
edit approved Oct 16, 2014 at 21:55
Margaret Friedland
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Analytic extension of the exterior Newtonian protientialpotential into the domain

I just want to know for a soomth boundary domain whetherwith smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domiandomain.

Definition of Newtonian potential of domain $\Omega$ at x is defined to be $u(x)=\int_{R^{n}}\Gamma(|x-y|)\chi_{\Omega}(y)dy$. When $x\in \Omega$ we say it is the interior Newtonian potential, otherwise we call it exterior Newtonian potential. Here $\Gamma(x)=\frac{1}{|x|^{n-2}}$ is the fundamental solution of Laplace, $\chi_{\Omega}$ is the characterization fucntion of $\Omega$

For example, the exterior Newtonian potential of the ball $B(a,R)$ is just $\frac{R^{n-2}}{n(n-2)}\frac{1}{|x-a|^{n-2}}$ when $n>2$. Even though it is Newtonian potential of the ball out of it, we can regard it as a singular analytic function on $R^{n}$. So we can say the exterior newtonian potential of the ball have singular analytic extension in it.

Or in particular,: what is the exterior newtonianNewtonian potential of the domain which is formed by moving the center of a ball along a arbitrary small smooth curve.? Does this case allow singular analytic extension in the domain.?

Analytic extension of the exterior Newtonian protiential into the domain

I just want to know for a soomth boundary domain whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domian.

Definition of Newtonian potential of domain $\Omega$ at x is defined to be $u(x)=\int_{R^{n}}\Gamma(|x-y|)\chi_{\Omega}(y)dy$. When $x\in \Omega$ we say it is the interior Newtonian potential, otherwise we call it exterior Newtonian potential. Here $\Gamma(x)=\frac{1}{|x|^{n-2}}$ is the fundamental solution of Laplace, $\chi_{\Omega}$ is the characterization fucntion of $\Omega$

For example, the exterior Newtonian potential of the ball $B(a,R)$ is just $\frac{R^{n-2}}{n(n-2)}\frac{1}{|x-a|^{n-2}}$ when $n>2$. Even though it is Newtonian potential of the ball out of it, we can regard it as a singular analytic function on $R^{n}$. So we can say the exterior newtonian potential of the ball have singular analytic extension in it.

Or in particular, what is the exterior newtonian potential of the domain which is formed by moving the center of a ball along a arbitrary small smooth curve. Does this case allow singular analytic extension in the domain.

Analytic extension of the exterior Newtonian potential into the domain

I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain.

Definition of Newtonian potential of domain $\Omega$ at x is defined to be $u(x)=\int_{R^{n}}\Gamma(|x-y|)\chi_{\Omega}(y)dy$. When $x\in \Omega$ we say it is the interior Newtonian potential, otherwise we call it exterior Newtonian potential. Here $\Gamma(x)=\frac{1}{|x|^{n-2}}$ is the fundamental solution of Laplace, $\chi_{\Omega}$ is the characterization fucntion of $\Omega$

For example, the exterior Newtonian potential of the ball $B(a,R)$ is just $\frac{R^{n-2}}{n(n-2)}\frac{1}{|x-a|^{n-2}}$ when $n>2$. Even though it is Newtonian potential of the ball out of it, we can regard it as a singular analytic function on $R^{n}$. So we can say the exterior newtonian potential of the ball have singular analytic extension in it.

Or in particular: what is the exterior Newtonian potential of the domain which is formed by moving the center of a ball along a arbitrary small smooth curve? Does this case allow singular analytic extension in the domain?

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Analytic extension of the exterior Newtonian protiential into the domain

I just want to know for a soomth boundary domain whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domian.

Definition of Newtonian potential of domain $\Omega$ at x is defined to be $u(x)=\int_{R^{n}}\Gamma(|x-y|)\chi_{\Omega}(y)dy$. When $x\in \Omega$ we say it is the interior Newtonian potential, otherwise we call it exterior Newtonian potential. Here $\Gamma(x)=\frac{1}{|x|^{n-2}}$ is the fundamental solution of Laplace, $\chi_{\Omega}$ is the characterization fucntion of $\Omega$

For example, the exterior Newtonian potential of the ball $B(a,R)$ is just $\frac{R^{n-2}}{n(n-2)}\frac{1}{|x-a|^{n-2}}$ when $n>2$. Even though it is Newtonian potential of the ball out of it, we can regard it as a singular analytic function on $R^{n}$. So we can say the exterior newtonian potential of the ball have singular analytic extension in it.

Or in particular, what is the exterior newtonian potential of the domain which is formed by moving the center of a ball along a arbitrary small smooth curve. Does this case allow singular analytic extension in the domain.