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On a volume-constrained variational problem. (English) Zbl 0945.49005
Let \(W\colon {\mathbb R}^{d\times N}\to[0,+\infty[\) be a \(C^1\) quasiconvex function with \(p\) growth, \(p>1\), and such that \[ \sum_{i,j=1}^d\sum_{k=1}^N{\partial W\over\partial\xi_{ik}}(\xi)\xi_{jk}v^iv^j > 0\text{ whenever }\xi^Tv\not=0,\;\xi\in{\mathbb R}^{d\times N},\;v\in S^{d-1}, \] where \(S^{d-1}\) is the unit sphere in \({\mathbb R}^d\). Moreover, let \(\Omega\subseteq{\mathbb R}^N\) be an open, bounded, connected Lipschitz domain, \(\{z_1,\ldots,z_P\}\) be extremal points of a compact, convex set \(K\subseteq{\mathbb R}^d\) with \(P\geq 1\), and let \(\alpha_1,\ldots,\alpha_P>0\) verify \(\sum_{i=1}^P\alpha_i<{\mathcal L}^N(\Omega)\).
In the paper the authors prove an existence result for the problem \[ \min\left\{\int_\Omega W(\nabla u)dx : u\in W^{1,p}(\Omega;{\mathbb R}^d),\;{\mathcal L}^N(\{u=z_i\})=\alpha_i,\;i=1\ldots,P\right\}. \] In particular, they consider the case with \(d=1\), \(W(\xi)=|\xi|^2\), and \(P=2\), and characterize the asymptotic behaviour of the minimizers of the above problem as \(\alpha_1\to{\mathcal L}^N(\Omega)-\gamma\) and \(\alpha_2\to\gamma\) for some \(\gamma\in]0,{\mathcal L}^N(\Omega)[\). They prove that the limiting configurations satisfy the constrained least-area problem \[ \min\left\{P_\Omega(E) : E\subseteq\Omega,\;{\mathcal L}^N(E)=\gamma\right\}, \] where \(P_\Omega(E)\) denotes the perimeter of \(E\) in \(\Omega\).
Again in this case, minimizers are fully characterized when \(N=1\), and candidates for solutions are proposed for the circle and the square in the plane.
The minimization problem with \(d=1\), \(W(\xi)=|\xi|^2\), and \(P=2\) was proposed by M. E. Gurtin [Motion by mean curvature and related topics: Proceedings of the International Conference held at Trento, Italy, July 20-24, 1992. Berlin: de Gruyter. 89-97 (1994; Zbl 0809.35145)] in connection with the study of the interface between immiscible fluids. A similar problem with \(d=1\), \(W(\xi)=|\xi|^2\), and \(P=1\) (i.e., only one volume constraint), and with Dirichlet boundary conditions was studied by H. W. Alt and L. A. Caffarelli [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)], and by N. Aguilera, H. W. Alt and L. A. Caffarelli [SIAM J. Control Optimization 24, 191-198 (1986; Zbl 0588.49005)].

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49Q20 Variational problems in a geometric measure-theoretic setting
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