In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies$$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$where $\Omega$ is an open set of ${ℝ}^2$ and $E$ is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on $\varphi$.

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies $$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$ where Ω is an open set of ${ℝ}^2$ and E is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on φ.

In questo lavoro si considera un’equazione alle derivate parziali del primo ordine con una condizione sulla frontiera di tipo integrale. Si studia resistenza, l'unicità e il comportamento asintotico delle soluzioni.

We discuss the existence of almost everywhere solutions of nonlinear PDE’s of first (in the scalar and vectorial cases) and second order.

The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton − Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of some approximation procedures are derived.

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The $x$-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The x-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

On s’intéresse à des systèmes symétriques hyperboliques multidimensionnels en présence d’une semilinéarité. Il est bien connu que ces systèmes admettent des solutions discontinues, régulières de part et d’autre d’une hypersurface lisse caractéristique de multiplicité constante. Une telle solution $u^0$ étant donnée, on montre que $u^0$ est limite quand $\epsilon \to 0$ de solutions ${\left(u^{\epsilon }\right)}_{\epsilon \in ]0,1]}$ du système perturbé par une viscosité de taille $\epsilon$. La preuve utilise un problème mixte parabolique et des développements de couches limites....