## 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.
Reviewer: St.Jendo

### MSC:

 74P99 Optimization problems in solid mechanics

### Keywords:

constrained optimization; optimal plastic design

Zbl 0657.00013