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Daniele Tampieri
Daniele Tampieri
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this post is concerned with functionals defined in measures. Consider the following functional

$$\mathcal{W}[\mu]=-\int_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$

were we define $-\log\vert0\vert=+\infty$ and $\mu$ is a finite, non negative Radon measure in $\mathbb{R}$. It is clear that if $\mu$ does given mass to points of $\mathbb{R}$, then the functional above is $\infty$, also is easy to build absolutely continuous measures with respect to the Lebesgue measure were this functional is finit. The question is: There are singular measures (obviously without atoms) such that the functional is finit? I have tried of build an example using the Cantor set and a suitable Hausdorff measure retrictrestricted to the Cantor set, but I can not limit such integral, therefore I think that such measure exist. Best regads.

this post is concerned with functionals defined in measures. Consider the following functional

$$\mathcal{W}[\mu]=-\int_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$

were we define $-\log\vert0\vert=+\infty$ and $\mu$ is a finite, non negative Radon measure in $\mathbb{R}$. It is clear that if $\mu$ does given mass to points of $\mathbb{R}$, then the functional above is $\infty$, also is easy to build absolutely continuous measures with respect to the Lebesgue measure were this functional is finit. The question is: There are singular measures (obviously without atoms) such that the functional is finit? I have tried of build an example using the Cantor set and a suitable Hausdorff measure retrict to the Cantor set, but I can not limit such integral, therefore I think that such measure exist. Best regads.

this post is concerned with functionals defined in measures. Consider the following functional

$$\mathcal{W}[\mu]=-\int_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$

were we define $-\log\vert0\vert=+\infty$ and $\mu$ is a finite, non negative Radon measure in $\mathbb{R}$. It is clear that if $\mu$ does given mass to points of $\mathbb{R}$, then the functional above is $\infty$, also is easy to build absolutely continuous measures with respect to the Lebesgue measure were this functional is finit. The question is: There are singular measures (obviously without atoms) such that the functional is finit? I have tried of build an example using the Cantor set and a suitable Hausdorff measure restricted to the Cantor set, but I can not limit such integral, therefore I think that such measure exist. Best regads.

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Julio Valencia
Julio Valencia
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On the domain of functionals in measure with singular kernels

this post is concerned with functionals defined in measures. Consider the following functional

$$\mathcal{W}[\mu]=-\int_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$

were we define $-\log\vert0\vert=+\infty$ and $\mu$ is a finite, non negative Radon measure in $\mathbb{R}$. It is clear that if $\mu$ does given mass to points of $\mathbb{R}$, then the functional above is $\infty$, also is easy to build absolutely continuous measures with respect to the Lebesgue measure were this functional is finit. The question is: There are singular measures (obviously without atoms) such that the functional is finit? I have tried of build an example using the Cantor set and a suitable Hausdorff measure retrict to the Cantor set, but I can not limit such integral, therefore I think that such measure exist. Best regads.