Let $\Omega$ be a smooth bounded domain in ${ℝ}^N,N\>1$ and let $n\in {\mathbb{N}}^{*}$. We prove here the existence of nonnegative solutions $u_n$ in $BV\left(\Omega \right)$, to the problem $$\left(P_n\right)\left\{\begin{array}{cc}-div\sigma +2n\left({\int }_{\Omega }u-1\right){sign}^{+}\left(u\right)=0\hfill & \text{in}\Omega ,\hfill \\ \sigma ·\nabla u=\left|\nabla u\right|\hfill & \text{in}\Omega ,\hfill \\ u\text{is}\text{not}\text{identically}\text{zero},-\sigma ·\overrightarrow{n}u=u\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\overrightarrow{n}$ denotes the unit outer normal to $\partial \Omega$, and ${\mathrm{sign}}^{+}\left(u\right)$ denotes some $L^{\infty }\left(\Omega \right)$ function defined as: $${\mathrm{sign}}^{+}\left(u\right).u=u^{+},0\le {\mathrm{sign}}^{+}\left(u\right)\le 1.$$ Moreover, we prove the tight convergence of $u_n$ towards one of the first eingenfunctions for the first $1-$Laplacian Operator $-{\Delta }_1$ on $\Omega$ when $n$ goes to $+\infty$.

In this paper we consider two-dimensional quasilinear equations of the form $\text{div}\left(a\left(\left|\nabla u\right|\right)\nabla u\right)+f\left(u\right)=0$ and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of $\nabla u$(notice that $\text{arg}\left(\nabla u\right)$ is a well-defined real function since $\left|\nabla u\right|>0$ on ${\mathbb{R}}^2$) we prove that $u$ is one-dimensional, i.e., $u=u\left(\nu \cdot x\right)$ for some unit vector $\nu$. As a consequence of our result we obtain that any solution $u$ having one positive derivative is one-dimensional. This result provides...

We prove an existence result for equations of the form where the coefficients $a_{ij}(x,s)$ satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable $s$. Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients $a_{ij}(x,s)$ are supposed only Borel functions

We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system $divA(x,u,Du)+B(x,u,DU)=0$, under natural polynomial growth of the coefficient functions $A$ and $B$. We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.

Se il problema di minimo $({𝒫}_{\mathrm{\infty }})$ è il limite, in senso variazionale, di una successione di problemi di minimo con ostacoli del tipo $$\underset{u\ge {\psi }_h}{\min }{\int }_A\left[f_h(x,u,Du)+b(x,u)\right]dx,$$ allora $({𝒫}_{\mathrm{\infty }})$ può essere scritto nella forma $$\underset{u}{\min }\left\{{\int }_A\left[f_{\mathrm{\infty }}(x,u,Du)+b(x,u)\right]dx+{\int }_A{g}_{\mathrm{\infty }}(x,\tilde{u}(x))d{\lambda }_{\mathrm{\infty }}(x)\right\}$$ dove $\tilde{u}$ è un conveniente rappresentante di $u$ e ${\lambda }_{\mathrm{\infty }}$ è una misura non negativa.

We first discuss a class of inequalities of Onofri type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than $-1$. Without symmetry assumption, it holds if and only if the parameter is in the interval $(-1,0]$. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of...

We prove Harnack inequality for weak solutions to quasilinear subelliptic equation of the following kind $$\sum _{J=1}^m{X}_j^{*}{A}_j(x,u(x),Xu(x))+B(x,u(x),Xu(x))=0,$$ where $X_1,\mathrm{\dots },X_m$ are a system of non commutative locally Lipschitz vector fields. As a consequence, the weak solutions of (*) are continuous.

Si determina lo spettro di un operatore di Laplace di una «spherical space form» $(M,g)$ e si studia l’influenza di tale spettro su $(M,g)$.

Diamo condizioni sulle funzioni $f$, $g$ e sulla misura $\mu$ affinché il funzionale $$F(u)={\int }_{\mathrm{\Omega }}f(x,u,Du)dx+{\int }_{\overline{\mathrm{\Omega }}}g(x,u)d\mu$$ sia $L^1(\mathrm{\Omega })$-semicontinuo inferiormente su $W^{1,1}(\mathrm{\Omega })\cap C^0(\overline{\mathrm{\Omega }})$. Affrontiamo successivamente il problema del rilassamento.

In this paper we consider a nonlinear elliptic equation with critical growth for the operator ${\mathrm{\Delta }}^2$ in a bounded domain $\mathrm{\Omega }\subset {ℝ}^n$. We state some existence results when $n\ge 8$. Moreover, we consider $5\le n\le 7$, expecially when $\mathrm{\Omega }$ is a ball in ${ℝ}^n$.

Let $\mathrm{\Omega }$ be a bounded open convex set of class $C^2$. Let $a\left(x,H\left(u\right)\right)$ be a non linear operator satisfying the condition (A) (elliptic) with constants $\alpha$, $\gamma$, $\delta$. We prove that a number $\lambda \ge 0$ is an eigenvalue for the operator $a\left(x,H\left(u\right)\right)$ if and only if the number $\alpha \lambda$ is an eigen-value for the operator $\mathrm{\Delta }u$. If $\lambda \ge 0$ , the two systems $a\left(x,H\left(u\right)\right)=\lambda u$ and $\mathrm{\Delta }u=\alpha \lambda u$ have the same solutions. In particular, also the eventual eigen-values of the operator $a\left(x,H\left(u\right)\right)$ should all be negative. Finally, we obtain a sufficient condition for the existence of solutions $u\in H^2\cap H_0^1\left(\mathrm{\Omega }\right)$...