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A feak notion of convergence in capacity with applications to thin obstacle problems. (English) Zbl 0959.49014

Ioffe, Alexander (ed.) et al., Calculus of variations and differential equations. Technion 1998. Selected papers of the international conference on the calculus of variations and related topics held at Technion - Israel Institute of Technology, Haifa, Israel, March 25-31, 1998. Vol. I. Boca Raton, FL: Chapman & Hall. Pitman Res. Notes Math. Ser. 410, 56-64 (2000).
In this paper the authors give a new general notion of weak convergence in \(C_p\)–capacity and prove, as a consequence of the main result, the lower semicontinuity of the functionals \[ u\mapsto\int_\Omega f(x,u) d\mu \] with respect to the weak topology of \(H^{1,p}_0(\Omega,\mathbb R^m)\), when \(\Omega\) is a bounded open set in \(\mathbb R^n\), \(1<p<\infty\), \(f:\Omega\times\mathbb R^m \to [0, \infty]\) is a Borel function such that \(f(x,\cdot)\) is lower semicontinuous on \(\mathbb R^m\) for each \(x\in\Omega\), and \(\mu\geq 0\) is a Borel measure on \(\Omega\), which is zero on all Borel sets of \(p\)-capacity zero. A standard application of the direct method of the calculus of variations leads to the existence of a solution of the minimum problem \[ \min_{u\in H^{1,p}_0(\Omega,\mathbb R^m)} \left\{\int_\Omega|Du|^p dx+ \int_\Omega f(x,u) d\mu\right\}, \] provided that there is some \(u\in H^{1,p}_0(\Omega,\mathbb R^m)\) with \[ \int_\Omega|Du|^p dx+ \int_\Omega f(x,u) d\mu<\infty. \] Also existence of solutions for other related minimum vector valued problems is obtained, as well as a new proof of an unpublished interesting result due to Ennio De Giorgi.
For the entire collection see [Zbl 0921.00022].
Reviewer: P.Pucci (Perugia)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
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