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  • An eigenvalue Dirichlet problem with weight and ▫$L^1$▫ data = Marian Bocea, Vicenţiu Rǎdulescu
    Bocea, Marian ; Rǎdulescu, Vicenţiu, 1958-
    Let ▫$\Omega$▫ be a smooth bounded domain in ▫${\bold R}^N, \; N \ge 2$▫, and let ▫$\lambda$▫ be a real number. The authors study the Dirichlet problem with weight (1) ▫$-\Delta u = \lambda au + f$▫ ... in ▫$\Omega,\; u = 0$▫ on ▫$\partial \Omega$▫, with ▫$a \in L^{\infty}(\Omega), \; a(x) \ge 0$▫ for a.e. ▫$x \in \Omega, \; f \in L^1(\Omega)$▫. Since for ▫$f \not\in L^2(\Omega)$▫ the variational method does not work for finding solutions in ▫$H_0^1(\Omega)$▫, the authors use an approximation method in the following sense: ▫$f_n \to f$▫ strongly in ▫$L^1 (\Omega)$▫ and ▫$f_n \in L^2(\Omega)$▫. Let ▫$u_n$▫ be the variational solution of (1) corresponding to ▫$f_n$▫. Then they prove that the problem (1) has a solution by approximation and the solution is unique. Further, they prove that the solution obtained by approximation coincides with that found by the duality method of Stampacchia. In the last part of the paper they prove that for ▫$N \le 3$▫ the solution by approximation coincides with the unique solutionof the renormalized problem, which was introduced by P.-L. Lions and F. Murat. The case ▫$a \equiv 1$▫ was first proved by L. Orsina [Rend. Sem. Mat. Univ. Padova 90 (1993), 207--238].
    Source: Mathematische Nachrichten. - ISSN 0025-584X (Vol. 198, 1999, str. 5-17)
    Type of material - article, component part
    Publish date - 1999
    Language - english
    COBISS.SI-ID - 15279705

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