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On a variational approximation of superlinear indefinite elliptic problems. (English) Zbl 0957.65101

The authors study a finite element discretisation of the semilinear elliptic Neumann problem \[ -\Delta u = a(x)u^p \quad \text{in }\Omega, \qquad \frac{\partial u}{\partial \nu} = 0 \quad \text{on }\partial\Omega, \qquad u > 0 \quad \text{in } \Omega,\tag{1} \] where \(a \in W^1_\infty(\Omega)\) and \(\Omega\) is a bounded regular open set of \(\mathbb R^N\) with \(1<p<(N+2)/(N-2)\) if \(N \geq 3\) or \(p>1\) if \(N=2\). It is known that there exists a solution \(u^*\) of (1) iff the function \(a\) satisfies the condition \[ \int_\Omega a(x) dx < 0 \;\text{ and} \;a(x) \;\text{ changes\;its\;sign\;in} \;\Omega. \] The existence of \(u^*\) relies basically on the solution of the problem \[ \max_{v \in S} \Biggl\{ I(v):= \int_\Omega a(x)|v(x)|^{p+1} dx \Biggr\}, \quad S:= \Biggl\{ v \in W^1_2(\Omega), \int_\Omega |\nabla v|^2 dx =1 \Biggr\}. \] This constrained maximisation problem is discretised by linear finite elements, where for \(I(v)\) a quadrature rule is used for a further discretisation. It is shown that the discretised problem has a solution \(u^*_h\). Assuming that these solutions satisfy for a sequence of triangulations with meshsize \(h\) tending to zero the a priori estimate \[ \|u^*_h \|_\infty \leq C < \infty \tag{2} \] the authors prove the weak convergence of a subsequence in \(W^1_2(\Omega)\) and the convergence of the maxima. The convergence proof relies on an abstract convergence theorem for constrained minimisation problems [see e.g. R. D. Grigorieff and R. Reemtsen, Numer. Funct. Anal. Optimization 11, No. 7/8, 701-719 (1990; Zbl 0743.65058)]. The condition (2) is shown to hold true in the case \(N=1\). For the case \(N=1\) and \(p=2\) the authors provide a numerical example.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations

Citations:

Zbl 0743.65058
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References:

[1] DOI: 10.1007/BF01206962 · Zbl 0809.35022 · doi:10.1007/BF01206962
[2] DOI: 10.1007/BF01210623 · Zbl 0840.35035 · doi:10.1007/BF01210623
[3] Brezzi F., Num. Math. 2 pp 36– (1980)
[4] DOI: 10.1016/0362-546X(93)90147-K · Zbl 0779.35032 · doi:10.1016/0362-546X(93)90147-K
[5] Ciarlet, P. 1978. ”The Finite Element Method for elliptic problems”. Amsterdam: North-Holland. · Zbl 0383.65058
[6] Daniel J. W., The approximate minimization of junctionals (1971)
[7] DOI: 10.1080/01630569008816398 · Zbl 0743.65058 · doi:10.1080/01630569008816398
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