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An isoperimetric inequality for Gauss-like product measures. (English. French summary) Zbl 1356.49072

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 \[ P_{\mu}(M,\Omega) = \sup \biggl\{\int_M \text{div}(\varphi v) \,dx; \;v \in C^1_0(\Omega, \mathbb R^N), |v|\leq 1 \biggr\}. \] 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 \(\mu (M)\).
Let \(N \geq 2\) 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 \[ A'_i(x)\geq 1 \;on \;(a_i, b_i), \quad \lim_{x\rightarrow a_i +} A_i(x)=-\infty, \quad \lim_{x\rightarrow b_i -} A_i(x)=+\infty. \]
Put \[ S'=\prod^{N-1}_{i=1} (a_i, b_i), \quad S=S'\times \mathbb R, \quad \text{and}\;\;S_{\lambda}=S'\times (\lambda, +\infty) \;\;\text{ if} \;\lambda \in \mathbb R. \]
Moreover, let \(\mu\) be the measure on \(S\) with the density \[ \varphi(x)=\exp \biggl\{-\sum^{N-1}_{i=1} \frac{A_i(x_i)^2}{2}-\frac{x_N^2}{2}\biggr\}\prod^{N-1}_{i=1} A'_i(x_i), \quad x\in S. \]
The main result of the paper reads as follows.
Theorem 1. If \(\lambda \in \mathbb R\), then \[ P_{\mu}(M,S) \geq P_{\mu}(S_{\lambda},S) \eqno{(1)} \] for all Lebesgue measurable subsets \(M\) of \(S\) with \(\mu(M)=\mu(S_{\lambda})\). The equality in (1) holds if and only if \(M=S_{\lambda}\).
As a corollary, the authors show that the conclusion of Theorem 1 is true if the density \(\varphi\) of the measure \(\mu\) is given by \[ \varphi(x)=\exp \biggl\{-\frac{|x|^2}{2}-\sum^{N-1}_{i=1} B_i(x_i)\biggr\},\eqno{(2)} \] where \(B_i \in C^2(a_i, b_i)\) with \(B''_i(x_i)\geq 0\) on \((a_i, b_i),\;i=1,\cdots, N-1\).
Note 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 \[ \varphi(x)=\exp \biggl\{-\frac{|x|^2}{2}\biggr\}\prod^{N-1}_{i=1} x^{k_i}_i \] and also note that Theorem 1 generalizes results from C. Rosales [Anal. Geom. Metr. Spaces 2, 328–358 (2014; Zbl 1304.49096)].
Finally, 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 \[ -\text{div} (A(x)\nabla u)=\varphi(x) f(x) \;\;\text{in} \;\;G, \]
\[ u=0 \quad \text{on}\;\;\partial G\cap S, \] where a measurable symmetric \((N\times N)-\)matrix \(A(x)=(a_{ij}(x))\) satisfies \[ \varphi(x) |\xi |^2 \leq a_{ij}(x) \xi_i \xi_j \leq C \varphi(x) |\xi |^2, \] with some \(C\geq 1\), for almost all \(x \in G\) and all \(\xi \in \mathbb R^N\).

MSC:

49Q05 Minimal surfaces and optimization
35J70 Degenerate elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Citations:

Zbl 1304.49096
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References:

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