##
**The constrained least gradient problem.**
*(English)*
Zbl 0668.73060

Non-classical continuum mechanics, Proc. Symp., Durham/Engl. 1986, Lond. Math. Soc. Lect. Note Ser. 122, 226-243 (1987).

[For the entire collection see Zbl 0657.00013.]

In the classical minimum principles the integrand is often quadratic and always convex. In physical terms there is a “potential well”. The problem is to minimize \(\int (W(\nabla \phi)-F\phi)dx.\) Its convexity assures that the minimum is attained (assuming that the class of admissible fields is suitably closed and convex) and the calculus of variations takes over: the Euler-Lagrange equations are elliptic and their solution is the minimizing \(\phi^*\). If W is not convex then all this is extremely likely to fail.

The paper deals with two model problems in optimal design, both initially nonconvex. They lead to a variant of the least gradient problem in \(R^ 2\), i.e.: \[ \text{Minimize }\int_{\Omega}| \nabla \phi | dx\text{ subject to }| \nabla \phi | \leq 1\text{ and }\phi |_{\partial \Omega}=g. \] Without the bound on \(| \nabla \phi |\) the solution is easy to construct. It minimizes the lengths of the level curves \(\nu_ t\) on which \(\phi =t\). The extra constraint \(| \nabla \phi | \leq 1\) requires the level curves \(\nu_ t\) and \(\nu_ s\) to be separated by at least a distance \(| t-s|\). It has been proved that each level curve still solves a minimum problem, now constrained. Two examples illustrate the simplicity of solution.

The paper has a cognizable character, but it is not far from technical applications.

In the classical minimum principles the integrand is often quadratic and always convex. In physical terms there is a “potential well”. The problem is to minimize \(\int (W(\nabla \phi)-F\phi)dx.\) Its convexity assures that the minimum is attained (assuming that the class of admissible fields is suitably closed and convex) and the calculus of variations takes over: the Euler-Lagrange equations are elliptic and their solution is the minimizing \(\phi^*\). If W is not convex then all this is extremely likely to fail.

The paper deals with two model problems in optimal design, both initially nonconvex. They lead to a variant of the least gradient problem in \(R^ 2\), i.e.: \[ \text{Minimize }\int_{\Omega}| \nabla \phi | dx\text{ subject to }| \nabla \phi | \leq 1\text{ and }\phi |_{\partial \Omega}=g. \] Without the bound on \(| \nabla \phi |\) the solution is easy to construct. It minimizes the lengths of the level curves \(\nu_ t\) on which \(\phi =t\). The extra constraint \(| \nabla \phi | \leq 1\) requires the level curves \(\nu_ t\) and \(\nu_ s\) to be separated by at least a distance \(| t-s|\). It has been proved that each level curve still solves a minimum problem, now constrained. Two examples illustrate the simplicity of solution.

The paper has a cognizable character, but it is not far from technical applications.

Reviewer: St.Jendo

### MSC:

74P99 | Optimization problems in solid mechanics |