An isoperimetric inequality for Gauss-like product measures (Q301036)

Let \(\Omega\) be a Lebesgue measurable set in \(\mathbb R^n\), let \(\varphi \in C^0(\Omega)\) be a positive function and let \(\mu\) be a finite measure on \(\Omega\) with the density \(\varphi\). Given a Borel subset \(M\) of \(\Omega\), the \(\mu\)-perimeter of \(M\) relative to \(\Omega\) is given by NEWLINE\[NEWLINE P_{\mu}(M,\Omega) = \sup \biggl\{\int_M \text{div}(\varphi v) \,dx; \;v \in C^1_0(\Omega, \mathbb R^N), |v|\leq 1 \biggr\}. NEWLINE\]NEWLINE A measurable set \(M \subset \Omega\) is called isoperimetric if it minimizes the perimeter \(P_{\mu}(M,\Omega)\) among all the sets with a fixed measure \(\mu (M)\).NEWLINENEWLINELet \(N \geq 2\) and \(-\infty \leq a_i \leq b_i \leq + \infty, \;i= 1, \cdots , N-1\). Assume that \(A_i \in C^1(a_i, b_i)\) are real functions satisfying NEWLINE\[NEWLINE A'_i(x)\geq 1 \;on \;(a_i, b_i), \quad \lim_{x\rightarrow a_i +} A_i(x)=-\infty, \quad \lim_{x\rightarrow b_i -} A_i(x)=+\infty. NEWLINE\]NEWLINENEWLINENEWLINEPut NEWLINE\[NEWLINE S'=\prod^{N-1}_{i=1} (a_i, b_i), \quad S=S'\times \mathbb R, \quad \text{and}\;\;S_{\lambda}=S'\times (\lambda, +\infty) \;\;\text{ if} \;\lambda \in \mathbb R. NEWLINE\]NEWLINENEWLINENEWLINEMoreover, let \(\mu\) be the measure on \(S\) with the density NEWLINE\[NEWLINE \varphi(x)=\exp \biggl\{-\sum^{N-1}_{i=1} \frac{A_i(x_i)^2}{2}-\frac{x_N^2}{2}\biggr\}\prod^{N-1}_{i=1} A'_i(x_i), \quad x\in S. NEWLINE\]NEWLINENEWLINENEWLINEThe main result of the paper reads as follows.NEWLINENEWLINETheorem 1. If \(\lambda \in \mathbb R\), then NEWLINE\[NEWLINEP_{\mu}(M,S) \geq P_{\mu}(S_{\lambda},S) \eqno{(1)}NEWLINE\]NEWLINE for all Lebesgue measurable subsets \(M\) of \(S\) with \(\mu(M)=\mu(S_{\lambda})\). The equality in (1) holds if and only if \(M=S_{\lambda}\).NEWLINENEWLINEAs a corollary, the authors show that the conclusion of Theorem 1 is true if the density \(\varphi\) of the measure \(\mu\) is given by NEWLINE\[NEWLINE\varphi(x)=\exp \biggl\{-\frac{|x|^2}{2}-\sum^{N-1}_{i=1} B_i(x_i)\biggr\},\eqno{(2)}NEWLINE\]NEWLINE where \(B_i \in C^2(a_i, b_i)\) with \(B''_i(x_i)\geq 0\) on \((a_i, b_i),\;i=1,\cdots, N-1\).NEWLINENEWLINENote that the particular choice \(a_i=0, \;b_i=+\infty,\;B_i(x_i)=-k_i\log x_i, \;k_i \geq 0, \;i=1,\cdots, N-1\), gives NEWLINE\[NEWLINE \varphi(x)=\exp \biggl\{-\frac{|x|^2}{2}\biggr\}\prod^{N-1}_{i=1} x^{k_i}_i NEWLINE\]NEWLINE and also note that Theorem 1 generalizes results from \textit{C. Rosales} [Anal. Geom. Metr. Spaces 2, 328--358 (2014; Zbl 1304.49096)].NEWLINENEWLINEFinally, if \(f\in L^2(G, d\mu)\), where \(G\) is an open, connected subset of \(S\), and if \(\varphi\) is given by \((2)\), then the authors apply their results to prove sharp apriory estimates for the solution of the boundary value problem NEWLINE\[NEWLINE -\text{div} (A(x)\nabla u)=\varphi(x) f(x) \;\;\text{in} \;\;G, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u=0 \quad \text{on}\;\;\partial G\cap S, NEWLINE\]NEWLINE where a measurable symmetric \((N\times N)-\)matrix \(A(x)=(a_{ij}(x))\) satisfies NEWLINE\[NEWLINE \varphi(x) |\xi |^2 \leq a_{ij}(x) \xi_i \xi_j \leq C \varphi(x) |\xi |^2, NEWLINE\]NEWLINE with some \(C\geq 1\), for almost all \(x \in G\) and all \(\xi \in \mathbb R^N\).