Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

Abstract

In this paper, we prove existence of radially symmetric minimizers u A ( x ) = U A ( | x | ) , having U A ( ⋅ ) AC monotone and ℓ ∗ ∗ ( U A ( ⋅ ) , 0 ) increasing, for the convex scalar multiple integral (∗ ) ∫ B R ℓ ∗ ∗ ( u ( x ) , | ∇ u ( x ) | ρ 1 ( | x | ) ) ⋅ ρ 2 ( | x | ) d x among those u ( ⋅ ) in the Sobolev space A + W 0 1 , 1 ( B R ) . Here, | ∇ u ( x ) | is the Euclidean norm of the gradient vector and B R is the b a l l { x ∈ R d : | x | < R } ; while A is the boundary data. 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 and ∃ min ℓ ∗ ∗ ( R , 0 ) and ℓ ∗ ∗ ( s , ⋅ ) is even ∀ s ; while ρ 1 ( ⋅ ) and ρ 2 ( ⋅ ) are Borel bounded away from 0 and ∞ . Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ℓ ∗ ∗ ( s , v ) = ∞ is freely allowed.