Abstract.
We study the infimum of functionals of the form \(\int_\Omega M\nabla u\cdot\nabla u\) among all convex functions \(u\in H^1_0(\Omega)\) such that \(\int_\Omega |\nabla u|^2 =1\). (\(\Omega\) is a convex open subset of \({\mathbb R}^N\), and M is a given symmetric \(N\times N\) matrix.) We prove that this infimum is the smallest eigenvalue of M if \(\Omega\) is \(C^1\). Otherwise the picture is more complicated. We also study the case of an x-dependent matrix M.
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Received: 23 February 2000/Accepted: 4 December 2000 / Published online: 5 September 2002
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Lachand-Robert, T., Peletier, M. The minimum of quadratic functionals of the gradient on the set of convex functions. Calc Var 15, 289–297 (2002). https://doi.org/10.1007/s00526-002-0088-6
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DOI: https://doi.org/10.1007/s00526-002-0088-6