Non-Coercive Variational Problems with Constraints on the Derivatives

DOI: 10.68381/jca05-1 ISSN: 0944-6532

Cristina Marcelli

We establish a necessary and sufficient condition for the existence of the minimum of the functional ∫₀¹ f(t, v′(t))dt in the class

\mathcal{W}^p_d = \{v \in W^{1,p}([0,1]) : v(0) = 0, v(1) = d, v'(t) \geq \alpha\}

W d p = { v ∈ W 1 , p ( [ 0 , 1 ] ) : v ( 0 ) = 0 , v ( 1 ) = d , v ′ ( t ) ≥ α }

, in terms of a limitation of the slope d. Some applications to quasi-coercive and non-coercive integrands are also derived.

Outline