Advanced Search

The aim of this note is to indicate how inequalities concerning the integral of ${\left|\nabla u\right|}^2$ on the subsets where |u(x)| is greater than k ($k\in I{R}^{+}$) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of ${\left|\nabla u\right|}^2$ on the subsets where |u(x)| is less than k ($k\in I{R}^{+}$) or...

Inequalities concerning the integral of |∇u| on the subsets where |u(x)| is greater than k can be used in order to prove regularity properties of the function u. This method was introduced by Ennio De Giorgi e Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems.

In the paper [5] in collaboration with Italo Capuzzo Dolcetta, the use of the Lewy-Stampacchia inequality was the main tool for the study of the G-convergence in unilateral problems with linear differential operators. In this paper we prove a Lewy-Stampacchia inequality for unilateral problems with more general differential operators (quasilinear operators with lower order term having quadratic growth with respect to the gradient) in order to study the G-convergence in unilateral problems with such...

This paper dedicated to the memory of Enrico Magenes, concerning a nonlinear Dirichlet problem, follows the previous one ([1]) dedicated to the memory of Guido Stampacchia, concerning a similar linear problem (see [14]).

This paper, dedicated to the memory of Guido Stampacchia in the thirtieth anniversary of his death, starts from his lectures on Dirichlet problems of forty years ago. As Sergei Prokofiev named his first symphony the "Classical", since it was written in the style that Joseph Haydn would have used if he had been alive at the time, this paper strongly follows the one by Guido Stampacchia about elliptic equations with discontinuous coefficients ([8]).

We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is $$\left\{\begin{array}{cc}-\text{div}\left(\left(1+{\left|u\right|}^r\right)\nabla u\right)+{\left|u\right|}^{m-2}u{\left|\nabla u\right|}^2=f\hfill & \text{in}\mathrm{\Omega }\hfill \\ u=0\hfill & \text{su}\partial \mathrm{\Omega }.\hfill \end{array}\right.$$

We present a revisited form of a result proved in [Boccardo, Murat and Puel, (1982) 507–534] and then we adapt the new proof in order to show the existence for solutions of quasilinear elliptic problems also if the lower order term has quadratic dependence on the gradient and singular dependence on the solution.

We study degenerate elliptic problems of the type $$\left\{\begin{array}{cc}-\text{div}\left(\frac{\nabla u}{{(1+|u\left|\right)}^{\theta }}\right)=f\hfill & \text{in}\Omega \hfill \\ u=0\hfill & \text{on}\Omega .\hfill \end{array}\right.$$

In this paper we deal with the existence of critical points for functionals defined on the Sobolev space $W_0^{1,2}\left(\mathrm{\Omega }\right)$ by $J\left(v\right)={\int }_{\mathrm{\Omega }}\mathfrak{I}\left(x,v,Dv\right)dx$, $v\in W_0^{1,2}\left(\mathrm{\Omega }\right)$, where $\mathrm{\Omega }$ is a bounded, open subset of ${\mathbb{R}}^N$. Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of Calculus of Variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we...

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

In this note we study the summability properties of the minima of some non differentiable functionals of Calculus of the Variations.

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