Moser-type inequalities for solutions of linear elliptic equations with lower-order terms
The aim of this paper is to derive Moser-type estimates for solutions \(u\in H_0^1(\Omega)\) of the Dirichlet problem for the differential operator of the second order containing lower-order terms: \[ -\sum_{i,j=1}^n\frac \partial {\partial x_j}(a_{ij} \frac {\partial}{\partial x_i}u)- \sum _{i=1}^n\frac \partial {\partial x_i} (b_iu)+ \sum _{i=1}^n d_i\frac \partial {\partial x_i}u+cu= \sum _{i=1}^n\frac \partial {\partial x_i} f_i\quad \text{in }\Omega, \] \[ u=0\quad \text{on }\partial \Omega, \] where \(\Omega \) is a bounded open subset of \(\mathbb R^n\), \(n\geq 2\), and \(a_{ij}(x)\) are bounded functions satisfying the ellipticity condition \[ \sum _{i,j=1}^na_{ij}(x)\xi _i\xi _j\geq |\xi |^2\quad \text{for a.e. \(x\in \Omega \) and all \(\xi \in \mathbb R^n\)}. \] The authors prove (under suitable assumptions on the coefficients \(a_{ij}(x)\), \(b_i(x)\), \(d_i\), and \(c(x)\)) that if \(|f|=\big (\sum _1^nf_i^2\big)^{1/2}\in L^n(\Omega)\) and \(\||f|\|_{L^n(\Omega)}\leq 1\) then there exists a constant \(C=C(n,b,|\Omega |)>0\) such that for any \(\beta \leq \tilde \beta =nC_n^{1/n}\) \[ \int _\Omega \exp \big ([ \beta |u(x)|]^{n'}\big) dx\leq C \] (\(C_n\) denotes the measure of the unit sphere in \(\mathbb R^n\), \(n'=n/(n-1)\), \(|\Omega |\) denotes the volume of \(\Omega \)). Results of this paper extend previous ones concerning elliptic equations with \(b_i, d_i, c\equiv 0\), see \textit{G. Stampacchia} [Commun. Pure Appl. Math. 16, 505-510 (1963; Zbl 0147.09202)] and \textit{E. Giarusso} and \textit{G. Trombetti} [Bull. Aust. Math. Soc. 36, 425-434 (1987; Zbl 0641.35019)]. The authors give some applications. Solutions of Hamilton-Jacobi equations are also considered.
linear elliptic equation of second order
scientific article; zbMATH DE number 1450166