For a bounded and sufficiently smooth domain $\mathrm{\Omega }$ in ${\mathbb{R}}^N$, $N\ge 2$, let ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$ and ${\left({\phi }_k\right)}_{k=1}^{\mathrm{\infty }}$ be respectively the eigenvalues and the corresponding eigenfunctions of the problem (with Neumann boundary conditions) $$-\text{div}\left(a\left(x\right)\nabla {\phi }_k\right)+q\left(x\right){\phi }_k={\lambda }_k\varrho \left(x\right){\phi }_k\text{ in }\mathrm{\Omega },a\frac{\partial }{\partial \mathbf{n}}{\phi }_k=0\text{ su }\partial \mathrm{\Omega }.$$ We prove that knowledge of the Dirichlet boundary spectral data ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$, ${\left({\phi }_{k|\partial \mathrm{\Omega }}\right)}_{k=1}^{\mathrm{\infty }}$ determines uniquely the Neumann-to-Dirichlet (or the Steklov- Poincaré) map $\gamma$ for a related elliptic problem. Under suitable hypothesis on the coefficients $a,q,\varrho$ their identifiability is then proved. We prove also analogous results for Dirichlet...

We present a revisited form of a result proved in [Boccardo, Murat and Puel, Portugaliae Math.41 (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 deal with the existence of solutions of the Dirichlet problem for sublinear and superlinear partial differential inclusions considered as generalizations of the Euler-Lagrange equation for a certain integral functional without convexity assumption. We develop a duality theory and variational principles for this problem. As a consequence of the duality theory we give a numerical version of the variational principles which enables approximation of the solution for our problem.

We prove existence of a positive, radial solution for a semilinear elliptic problem with a discontinuous nonlinearity. We use an approximating argument which requires no monotonicity assumptions on the nonlinearity.

In this paper we study a quasilinear resonant problem with discontinuous right hand side. To develop an existence theory we pass to a multivalued version of the problem, by filling in the gaps at the discontinuity points. We prove the existence of a nontrivial solution using a variational approach based on the critical point theory of nonsmooth locally Lipschitz functionals.

The convergence of discrete approximations of generalized reflected backward stochastic differential equations with random terminal time in a general convex domain is studied. Applications to investigation obstacle elliptic problem with Neumann boundary condition for partial differential equations are given.