Comparison results for solutions of elliptic problems via symmetrization (Q1284417)

Weak solutions of the homogeneous Dirichlet problem \[ Au=f \quad \text{in }\Omega , \qquad u=0 \quad \text{on }\partial\Omega\tag{1} \] are studied, where \(\Omega \) is an open bounded subset of \(\mathbb{R}^{N}\). It is important that in the linear elliptic differential operator \(A\) the first-order terms are included, that is to say \[ Au=-\sum_{i,j=1}^{N}(a_{ij}(x)u_{x_{i}})_{x_{j}}+ \sum_{i=1}^{N}b_{i}(x)u_{x_{i}}+\sum_{i=1}^{N}(d_{i}(x)u)_{x_{i}}+c(x)u,\tag{2} \] where the coefficients are assumed to be measurable and bounded. One of the main goals is to estimate the concentration of \(u\) by the concentration of \(v\), \[ \int_{0}^{s}u^{\ast }(\sigma)d\sigma \leq \int_{0}^{s}v^{\ast }(\sigma)d\sigma \quad \forall s\in [0,| \Omega | ],\tag{3} \] where \(\varphi ^{\ast }\) denotes the decreasing rearrangement of \(\varphi \), the function \(u\in H_{0}^{1}(\Omega)\) is a solution of \((1)\) and the function \(v\) is a weak solution of a suitable spherically symmetric problem. However in \textit{A. Alvino}, \textit{G. Trombetti} and \textit{P.-L. Lions} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 2, 37-65 (1990; Zbl 0703.35007)] was proved that if \(v\) is a solution of a standard symmetrized problem then \((3)\) is not satisfied. In order to prove \((3)\), a new symmetrized problem is given in which the zero-order term is improved. In the sequel, these results are shown in the case of variational inequalities.