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A comparison result related to Gauss measure. (English. Abridged French version) Zbl 1004.35044

Summary: We prove a comparison result for weak solutions to linear elliptic problems of the type \[ -\bigl(a_{ij} (x)u_{x_i}\bigr)_{x_j} =f(x)\varphi (x)\text{ in }\Omega, \quad u=0\text{ on }\partial\Omega, \tag{1} \] where \(\Omega\) is an open set of \(\mathbb{R}^n\) \((n\geq 2)\), \(\varphi(x)=(2\pi)^{-n/2}\exp (-|x|^2/2)\), \(a_{ij}(x)\) are measurable functions such that \(a_{ij}(x) \xi_i\xi_j \geq\varphi (x)|\xi |^2\) a.e. \(x\in\Omega\), \(\xi\in \mathbb{R}^n\) and \(f(x)\) is a measurable function taken in order to guarantee the existence of a solution \(u\in H_0^1 (\varphi,\Omega)\) of (1). We use the notion of rearrangement related to Gauss measure to compare \(u(x)\) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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References:

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