Opuscula Math. 26, no. 2 (2006), 351-359

Opuscula Mathematica

Continuous dependence of solutions of elliptic BVPs on parameters

Aleksandra Orpel

Abstract. The continuous dependence of solutions for a certain class of elliptic PDE on functional parameters is studied in this paper. The main result is as follow: the sequence \(\{x_k\}_{k\in N}\) of solutions of the Dirichlet problem discussed here (corresponding to parameters \(\{u_k\}_{k\in N}\)) converges weakly to \(x_0\) (corresponding to \(u_0\)) in \(W^{1,q}_0(\Omega,R)\), provided that \(\{u_k\}_{k\in N}\) tends to \(u_0\) a.e. in \(\Omega\). Our investigation covers both sub and superlinear cases. We apply this result to some optimal control problems.

Keywords: continuous dependence on parameters, elliptic Dirichlet problems, optimal control problem.

Mathematics Subject Classification: 49K40, 49K20, 35J20, 35J60.

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  • Aleksandra Orpel
  • University of Łódź, Faculty of Mathematics, ul. Banacha 22, 90-238 Łódź, Poland
  • Received: 2005-09-09.
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Aleksandra Orpel, Continuous dependence of solutions of elliptic BVPs on parameters, Opuscula Math. 26, no. 2 (2006), 351-359

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