Chipot, M.; Cimatti, G. A uniqueness result for the thermistor problem. (English) Zbl 0751.35022 Eur. J. Appl. Math. 2, No. 2, 97-103 (1991). The authors establish a uniqueness result for the following nonlinear system \[ \text{div}(\sigma(u)D\varphi)=0,\quad u_ t- \text{div}(k(u)Du)=\sigma(u)| D\varphi|^ 2 \] together with Dirichlet boundary conditions on \(u\) and \(\varphi\) and initial condition on \(u\). The key assumption is a \(L^ \infty\) estimate on \(D\varphi\). Such an estimate is shown to hold true in the one- and the two- dimensional case. Reviewer: D.Blanchard (Mont-Saint-Aignan) Cited in 10 Documents MSC: 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 80A20 Heat and mass transfer, heat flow (MSC2010) Keywords:electric heating; Dirichlet boundary conditions; initial condition PDFBibTeX XMLCite \textit{M. Chipot} and \textit{G. Cimatti}, Eur. J. Appl. Math. 2, No. 2, 97--103 (1991; Zbl 0751.35022) Full Text: DOI References: [1] Bensoussan, Applications of Variational Inequalities in Stochastic control (1982) [2] Meyers, Ann. Scuola Norm. Sup. Pisa 3 pp 189– (1963) [3] Lions, Quelques m?thodes des r?solution des probl?mes aux limites non lin?aires (1969) [4] DOI: 10.1002/andp.19003060107 · JFM 31.0807.02 [5] Howison, Quart. Appl. Math. 47 pp 509– (1990) [6] DOI: 10.1093/imamat/40.1.15 · Zbl 0694.35139 [7] DOI: 10.1002/andp.19003060211 · JFM 31.0808.01 [8] Giaquinta, Multiple Integrals in the calculus of variations and nonlinear elliptic problems (1983) · Zbl 0516.49003 [9] DOI: 10.1007/BF01766151 · Zbl 0675.35039 [10] Cimatti, Quart. J. Appl. Math. 47 pp 117– (1989) [11] Gilbarg, Elliptic Partial Differential Equations of Second order (1985) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.