Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients

Abstract

In this paper we prove existence of radially symmetric minimizers u_A (x)=U_A (|x| ), having U_A (∙) AC monotone & l^(**) (U_A (∙),0) increasing, for the convex scalar multiple integral

∫_(B_R) l^(**)(u(x),|∇u(x)|ρ_1(|x| ) ).ρ_2 (|x| ) dx (*)

Among those u(∙) in the Sobolev space A+W_0^1,1∩C^0 (¯B_R ). Here |∇u(x) | is the Euclidian norm of the gradient vector and B_R is the ball { x∈R^d ∶ |x|<R }.

Besides being e.g. superlinear (but no growth needed if (*) is known to have minimum), our lagrangian l^(**):R×R→[0,∞] is just convex lsc & ∃min l^(**) (R,0) & l^(**) (s,∙) is even ∀s; while ρ_1 (∙) & ρ_2 (∙) are Borel bounded away from 0 & ∞.

Remarkably, (*) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state & gradient pointwise constraints are, in a sense, built-in, since l^(**) (s,v)=∞ is freely allowed.