Differential and Integral Equations

Comparison results for solutions of elliptic problems via Steiner symmetrization

F. Chiacchio and V. M. Monetti

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We consider the Dirichlet problem for a class of linear elliptic equations, whose model is \begin{align*} & -\Delta u-\sum _{i=1}^{n}\left( b_{i}(y)u\right) _{x_{i}}-\sum _{j=1}^{m}\left( \widetilde{b}_{j}(y)u\right) _{y_{j}}+ \sum _{i=1}^{n}d_{i}(y)u_{x_{i}} \\ & +\sum\limits_{j=1}^{m}\widetilde{d} _{j}(y)u_{y_{j}}+c(y)u=f(x,y) \ \ \text{ in }G, \end{align*} where $G=G^{\prime }\times G^{\prime \prime }$ is an open, bounded and connected subset of ${\mathbb R}^{N}={\mathbb R}^{n}\times {\mathbb R}^{m}$, the coefficients $b_{i}(y),$ $\widetilde{b}_{j}(y),$ $d_{i}(y)$, $\widetilde{d} _{j}(y)$ and $c(y)$ are in $L^{\infty }(G)$ and the datum $f(x,y)$ belongs to $ L^{p}(G)$ with $p>\frac{N}{2}$. We prove some comparison results by using Steiner symmetrization.

Article information

Differential Integral Equations, Volume 14, Number 11 (2001), 1351-1366.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35A30: Geometric theory, characteristics, transformations [See also 58J70, 58J72] 35B05: Oscillation, zeros of solutions, mean value theorems, etc.


Chiacchio, F.; Monetti, V. M. Comparison results for solutions of elliptic problems via Steiner symmetrization. Differential Integral Equations 14 (2001), no. 11, 1351--1366. https://projecteuclid.org/euclid.die/1356123028

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