Strict complex convexity

Comparison Results without Strict Convexity

In this paper we establish a comparison result for solutions to the problem minimize ∫ Ω l(‖∇u(x)‖) dx or to the problem minimize ∫ Ω l(γC(∇u(x)) dx, for a special class of solutions, without assuming neither smoothness nor strict convexity of l.

Strict convexity of the singular value sequences

If A and B are compact operators on a Hilbert space, with singular values satisfying s j (A) = s j (B) = s j ((A + B)/2) Let A be a compact linear operator from a Hilbert space H into a Hilbert space K. The singular values s 1 (A) ≥ s 2 (A) ≥. .. are the eigenvalues of |A| := (A * A) 1/2. We refer to [3] for other equivalent definitions and basic properties. In this note we offer two proofs, ge...

Complex Convexity

Comparison Results and Estimates on the Gradient without Strict Convexity

In this paper we establish a comparison result for solutions to the problem minimize Z Ω f(∇u(x)) dx on {u : u − u0 ∈ W 1,1 0 (Ω)}.

Lipschitz regularity for minima without strict convexity of the Lagrangian

We give, in a non-smooth setting, some conditions under which (some of) the minimizers of ∫ Ω f (∇u(x)) dx + g(x,u(x)) dx among the functions in W1,1(Ω) that lie between two Lipschitz functions are Lipschitz. We weaken the usual strict convexity assumption in showing that, if just the faces of the epigraph of a convex function f :Rn → R are bounded and the boundary datum u0 satisfies a generali...