# zbMATH — the first resource for mathematics

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
Full Text: