Comparison results for solutions of elliptic problems via Steiner symmetrization. (English) Zbl 1027.35026

From the text: We consider the Dirichlet problem for a class of linear elliptic equations, whose model is \[ \begin{split} -\Delta u-\sum^n_{i= 1}\bigl(b_i(y)u \bigr)_{x_i}- \sum^n_{j=1} \bigl(\widetilde b_j(y)u \bigr)_{y_j} +\sum^n_{i=1} d_i(y)u_{x_i}\\ +\sum^m_{j=1} \widetilde d_j(y)u_{y_j}+ c(y)u= f(x,y)\quad \text{in }G,\end{split} \tag{1} \] where \(G=G'\times G''\) 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 \(p> {N\over 2}\). We prove some comparison results by using Steiner symmetrization.
More precisely, we prove the following inequality. \[ \int^s_0 u^*(\sigma,y) d\sigma\leq \int^s_0v^* (\sigma,y) d \sigma\quad \forall s\in \bigl[0,|G'|_n\bigr], \tag{2} \] where \(u\) is the weak solution of problem (1) and \(v\) is the weak solution of a problem whose data are symmetrized in the sense of Steiner, that is, \[ \begin{cases} -\Delta v-\sum^m_{j=1} \bigl( \widetilde b_j(y) v\bigr)_{y_j}+ \sum^m_{j=1}\widetilde d_j(y) v_{y_j}+ c(y)v= f^\#\text{ in }G^\#\\ v=0\text{ on }\partial G^\#. \end{cases} \] The estimate (2) allows us to get an a priori estimate of the Orlicz norm of \(u\). Moreover, we prove the result also for a more general elliptic linear operator.
The same result has been proved in A. Alvino, J. I. Diaz, P. L. Lions, and G. Trombetti [C. R. Acad. Sci., Paris, Sér I. Math. 314, 1015-1020 (1992; Zbl 0795.35022) and Commun. Pure Appl. Math. 49, 217-236 (1996; Zbl 0856.35034)] when the second-order elliptic operator does not contain lower-order terms.
Finally, we also obtain an inequality involving the gradients of \(u\) and \(v\), which gives the “energy estimate” \[ \int_G|\nabla u|^2 dx dy\leq\int_{G^\#}|\nabla v|^2 dx dy \] when \(\widetilde b_j(y)= \widetilde d_j(y)=c(y)=0\), for \(j=1,\dots,m\).


35J25 Boundary value problems for second-order elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs