Linear Elliptic Equations and Gauss Measure

Volume 4, Issue 5, 2003

Article 106

LINEAR ELLIPTIC EQUATIONS AND GAUSS MEASURE

G. DI BLASIO

DIPARTIMENTO DI MATEMATICA E APPLICAZIONI ``R. CACCIOPPOLI''
UNIVERSITÀ DEGLI STUDI DI NAPOLI ``FEDERICO II''
COMPLESSO MONTE S. ANGELO - VIA CINTIA.\ NAPOLI-ITALY
E-Mail: giuseppina.diblasio@dma.unina.it

Received 20 February, 2003; Accepted 30 June, 2003.
Communicated by: A. Fiorenza

ABSTRACT. In this paper we study a Dirichlet problem relative to a linear elliptic equation with lower-order terms, whose ellipticity condition is given in terms of the function $\varphi (x)=(2\pi )^{-\frac{n}{2}}$ $\exp\left( -\left\vert x\right\vert ^{2}/2\right)$ , the density in the Gaussian measure. Using the notion of rearrangement with respect to the Gauss measure, we prove a comparison result with a problem of the same type defined in a half space, with data depending only on the first variable.

Key words:
Linear elliptic equation, Comparison theorem, Rearrangements of functions, Gauss measure

2000 Mathematics Subject Classification:
35B05, 35J70.

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