We consider the eigenvalue problem$$\begin{array}{c}-\mathrm{div}\left(a\right(\left|\nabla u\right|\left)\nabla u\right)=\lambda g(x,u)\text{in}\Omega \hfill \\ u=0\text{on}\partial \Omega ,\hfill \end{array}$$in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all $\lambda \>0$ are eigenvalues.

We consider the eigenvalue problem $$\begin{array}{c}{\displaystyle -\mathrm{div}\left(a\right(\left|\nabla u\right|\left)\nabla u\right)=\lambda g(x,u)\text{in}\Omega u=0\text{on}\partial \Omega ,}\hfill \end{array}$$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.

We consider minimization problems of the form ${\min }_{u\in \varphi +W_0^{1,1}\left(\Omega \right)}{\int }_{\Omega }[f\left(Du\left(x\right)\right)-u\left(x\right)]\mathrm{d}x$ where $\Omega \subseteq {ℝ}^N$ is a bounded convex open set, and the Borel function $f:{ℝ}^N\to [0,+\infty ]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of $\Omega$ and the zero level set of $f$, we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

We consider minimization problems of the form ${\min }_{u\in \varphi +W_0^{1,1}\left(\Omega \right)}{\int }_{\Omega }[f\left(Du\left(x\right)\right)-u\left(x\right)]\mathrm{d}x$ where $\Omega \subseteq {ℝ}^N$ is a bounded convex open set, and the Borel function $f:{ℝ}^N\to [0,+\infty ]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of Ω and the zero level set of f, we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

We provide a geometric rigidity estimate à la Friesecke-James-Müller for conformal matrices. Namely, we replace $\mathrm{SO}\left(n\right)$ by an arbitrary compact set of conformal matrices, bounded away from $0$ and invariant under $\mathrm{SO}\left(n\right)$, and rigid motions by Möbius transformations.

After a short introduction on micromagnetism, we will focus on a scalar micromagnetic model. The problem, which is hyperbolic, can be viewed as a problem of Hamilton-Jacobi, and, similarly to conservation laws, it admits a kinetic formulation. We will use both points of view, together with tools from geometric measure theory, to prove the rectifiability of the singular set of micromagnetic configurations.

Si prova resistenza globale della soluzione di una equazione di Riccati collegata alla sintesi di un problema di controllo ottimale. Il problema considerato rappresenta la versione astratta di alcuni problemi governati da equazioni paraboliche con il controllo sulla frontiera.