We consider the minimizers of the energy $\|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$ with $s \in (0,1/2)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of u, and W is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer u in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.
Density estimates for a nonlocal variational model via the Sobolev inequality / O. Savin, E. Valdinoci. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 43:6(2011), pp. 2675-2687.
Density estimates for a nonlocal variational model via the Sobolev inequality
E. Valdinoci
2011
Abstract
We consider the minimizers of the energy $\|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$ with $s \in (0,1/2)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of u, and W is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer u in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.
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