## Minimum problems over sets of concave functions and related questions.(English)Zbl 0835.49001

The aim of the paper is to provide a framework for the study of some scalar valued, non-convex problems in the calculus of variations modelled on Newton’s problem of minimal resistance in which the functional $$u\to \int_\Omega {1\over 1+ |Du|^2} dx$$ has to be minimized.
Taking inspiration from this problem, the authors first observe that, in general, it has no solutions unless some constraints on $$u$$ are imposed, and then determine some classes of functions in which it may be solved. A first existence theorem is furnished for a general functional $$F(u)= \int_\Omega f(x, u, \nabla u) dx$$, $$\Omega$$ being a convex bounded open subset of $$\mathbb{R}^n$$, on the class of admissible functions $$C_M= \{u: \Omega\to [0, M]$$, $$u$$ concave}, together with the study of some properties of the solutions and the derivation of the Euler-Lagrange equation in the Newton’s case. Later the same existence problem is treated in the classes $$E_M= \{u\in H^1_{\text{loc}}(\Omega): 0\leq u\leq M, \Delta u\leq 0$$ in $$\Omega\}$$, in which an entropy condition is taken into account, $$Q_M= \{u: \Omega\to [0, M]$$, $$u$$ quasiconcave} and existence results for the solutions of $$F$$ are proved. In proving such results interesting compactness properties of the classes $$C_M$$, $$E_M$$ and $$Q_M$$ are also proved. Finally, a physically correct class $$S_M$$ for Newton’s problem is introduced and studied.
In conclusion, boundary conditions, uniqueness and symmetry questions for the minimizers of $$F$$ in Newton’s case and relations between the admissible classes are discussed.

### MSC:

 49J10 Existence theories for free problems in two or more independent variables
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### References:

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