We investigate the minima of functionals of the form$${\int }_{[a,b]}g\left(\dot{u}\left(s\right)\right)\mathrm{d}s$$where $g$ is strictly convex. The admissible functions $u:[a,b]\to ℝ$ are not necessarily convex and satisfy $u\le f$ on $[a,b]$, $u\left(a\right)=f\left(a\right)$, $u\left(b\right)=f\left(b\right)$, $f$ is a fixed function on $[a,b]$. We show that the minimum is attained by $\overline{f}$, the convex envelope of $f$.

We investigate the minima of functionals of the form $${\int }_{[a,b]}g\left(\dot{u}\left(s\right)\right)\mathrm{d}s$$ where g is strictly convex. The admissible functions $u:[a,b]⟶ℝ$ are not necessarily convex and satisfy $u\le f$ on [a,b], u(a)=f(a), u(b)=f(b), f is a fixed function on [a,b]. We show that the minimum is attained by $\overline{f}$, the convex envelope of f.

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.

Si studia il comportamento asintotico di una classe di funzionali integrali che possono dipendere da misure concentrate su strutture periodiche multidimensionali, quando tale periodo tende a 0. Il problema viene ambientato in spazi di Sobolev rispetto a misure periodiche. Si dimostra, sotto ipotesi generali, che un appropriato limite può venire definito su uno spazio di Sobolev usuale usando tecniche di $\mathrm{\Gamma }$-convergenza. Il limite viene espresso come un funzionale integrale il cui integrando è caratterizzato...

The $L^{2,\lambda }$-regularity of the gradient of local minima for nonlinear functionals is shown.

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where $\mu$ is a given positive measure on ${ℝ}^n$, and the vector-valued function $u$ belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ associated with $\mu$. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to $\mu$. More precisely, for fully general $\mu$, a notion of quasiconvexity for $f$ along the tangent bundle to $\mu$, turns out to be necessary for lower...

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where μ is a given positive measure on ${ℝ}^n$, and the vector-valued function u belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ associated with μ. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ. More precisely, for fully general μ, a notion of quasiconvexity for f along the tangent bundle to μ, turns out to be necessary for lower...

We present Z. Naniewicz method of optimization a coercive integral functional 𝒥 with integrand being a minimum of quasiconvex functions. This method is applied to the minimization of functional with integrand expressed as a minimum of two quadratic functions. This is done by approximating the original nonconvex problem by appropriate convex ones.

Studiamo le proprietà di regolarità delle mappe fra varietà di Riemann che minimizzano la $p$-energia fra quelle che soddisfano una condizione di frontiera pazialmente libera. Proviamo che tali mappe sono Hölder continue vicino alla frontiera libera fuori di un insieme singolare, e otteniamo stime ottimali per la dimensione di Hausdorff di questo insieme singolare.