Existence and uniqueness for elliptic equations with lower-order terms. (English) Zbl 1232.35051

The author considers the Dirichlet problem
\[ \left\{\begin{matrix} \text{div}(\mathcal A(x)\nabla u+E(x)u)=\text{div}F & \text{in}\;\;\Omega,\\ u=0 & \text{on}\;\;\partial\Omega,\end{matrix} \right.\tag{1} \]
where \(\Omega\) is a bounded regular domain in \(\mathbb R^N\), \(N>2\). Here \(\mathcal A(x)=(a_{ij}(x))\) is a symmetric positive definite \(N\times N\) matrix and \(x\to E(x):\Omega\to\mathbb R^N\) is a vector field. It is also assumed that the entries of \(\mathcal A(x)\) satisfy \[ \begin{aligned} &\alpha|\xi|^2\leq\sum_{i,j=1}^Na_{ij}(x)\xi_i\xi_j,\\ &\|\mathcal A(x)\|\leq\beta, \end{aligned} \] with \(0<\alpha<\beta\), for every \(\xi\in\mathbb R^N\) and for \(x\in\Omega\) a.e. Under various assumptions on \(E(x)\), uniqueness or/and existence (of some weak solutions) in Sobolev spaces \(W_0^{1,p}(\Omega)\), \(p>1\), is/are proved for problem (1). The corresponding spaces for \(F\) are described for different cases.


35J25 Boundary value problems for second-order elliptic equations
35J50 Variational methods for elliptic systems
35D30 Weak solutions to PDEs
Full Text: DOI


[1] Alvino A., Boll. Un. Mat. It. A 14 (5) pp 148– (1977)
[2] Alvino A., (Napoli Settembre pp 269– (1982)
[3] Boccardo L., Boll. Un. Mat. It. 2 (9) pp 285– (2009)
[4] DOI: 10.1023/A:1015709329011 · Zbl 1161.35362
[5] Greco L., Diff. Int. Eq. 6 pp 5– (1993)
[6] DOI: 10.1007/BF02678192 · Zbl 0869.35037
[7] DOI: 10.1515/crll.1994.454.143 · Zbl 0802.35016
[8] Iwaniec T., I. H. Poincaré AN 18 pp 5– (2001)
[9] Meyers N., Ann. Sc. Norm. Sup. Pisa 17 pp 189– (1963)
[10] DOI: 10.1016/0022-247X(68)90233-3 · Zbl 0155.27805
[11] DOI: 10.5802/aif.204 · Zbl 0151.15401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.