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.


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