On the best constant in {G}affney inequality

We discuss the value of the best constant in Gaffney inequality namely \lVert \nabla \omega \rVert_{L^{2}}^{2}\leq C\left( \lVert d\omega\rVert_{L^{2}}^{2}+\lVert \delta\omega\rVert_{L^{2}% }^{2}+\lVert \omega\rVert_{L^{2}}^{2}\right) when either $\nu\wedge\omega=0$ or $\nu\,\lrcorner\,\omega=0$ on $\partial\Omega.

On convexity properties of homogeneous functions of degree one

We provide an explicit example of a function that is homogeneous of degree one, rank-one convex, but not conve

Existence of solutions for some implicit partial differential equations and applications to variational integrals involving quasi-affine functions

We discuss some existence theorems for partial differential inclusions, subject to Dirichlet boundary conditions, of the form where Φ is a quasi-affine function and so, in particular, for Φ(Du) = det Du. We then apply it to minimization problems of the for

On the equation det ∇u=f{{\rm det}\,\nabla{u}=f} with no sign hypothesis

We prove existence of $${u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}$$ satisfying $$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$$ where k≥1 is an integer, $${\Omega}$$ is a bounded smooth domain and $${f\in C^{k}(\overline{\Omega}) }$$ satisfies $$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$$ with no sign hypothesis on