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Entropy solutions for nonlinear elliptic equations in \(L^1\). (English) Zbl 1155.35352

Summary: Results on existence and continuous dependence are proved for the nonlinear Dirichlet problem \((*) -\text{div}\, (a(x,u,\nabla u))=f-\text{div}\, (F)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), with \(\Omega\subset \mathbb R^{N}\), \(N\geq 2\), being a bounded domain. It is assumed that \(a: \Omega\times\mathbb R\times\mathbb R^{N}\to\mathbb R^{N}\) is a Carathéodory function such that for every \(s\in \mathbb R\), \(\xi,\eta\in\mathbb R^{N}\) \((\xi\neq \eta)\) and for almost all \(x\in\Omega\) one has \(a(x,s,\xi)\cdot\xi\geq \alpha|\xi|^{p}\), \(|a(x,s,\xi)|\leq b(|s|)(k(x)+|\xi|^{p-1})\), \((a(x,s,\xi)-a(x,s,\eta))\cdot(\xi-\eta)>0\), where \(p>1,\) \(\alpha>0\) is a constant, \(k(x)\in L^{p'}(\Omega)\) \((1/p+1/p'=1)\) and \(b:[0,+\infty)\to(0,+\infty)\) is a continuous function.
Since no growth conditions on \(b\) are required, the term \(-\text{div} (a(x,u,\nabla u))\) does not make sense even as a distribution, i.e., \(a(x,u,\nabla u)\in (L^{1}(\Omega))^{N}\). This difficulty is overcome by the authors by considering a weaker formulation of (*) and introducing the notion of entropy solution.

MSC:

35J60 Nonlinear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
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