In this work we deal with the numerical solution of a Hamilton-Jacobi-Bellman (HJB) equation with infinitely many solutions. To compute the maximal solution – the optimal cost of the original optimal control problem – we present a complete discrete method based on the use of some finite elements and penalization techniques.

We deal with practical aspects of an approach to the numerical realization of optimal shape design problems, which is based on a combination of the fictitious domain method with the optimal control approach. Introducing a new control variable in the right-hand side of the state problem, the original problem is transformed into a new one, where all the calculations are performed on a fixed domain. Some model examples are presented.

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

Given an open subset $\mathrm{\Omega }$ of ${ℝ}^n$ and a Borel function $f:\mathrm{\Omega }\times ℝ\times {ℝ}^n\to [0,+\mathrm{\infty }[$, conditions on $f$ are given which assure the lower semicontinuity of the functional ${\int }_{\mathrm{\Omega }}f(x,u,Du)dx$ with respect to different topologies.

Vengono presentate alcune connessioni tra gli spazi classici delle funzioni a variazione limitata ed altre classi di funzioni la cui variazione è opportunamente controllata, cioè le classi GBV introdotte da E. De Giorgi e L. Ambrosio, e le classi BBV, LBV, GBV* introdotte in questa Nota. Le dimostrazioni dei risultati enunciati, insieme con altri dettagli, appariranno in un successivo lavoro.

Sia $f=f(x,z)$ quasiconvessa in $z$, quasiperiodica in $x$ nel senso di Besicovitch e soddisfi le disuguaglianze: $${|z|}^p\le f(x,z)\le \mathrm{\Lambda }(1+{|z|}^p).$$ Allora $f$ può essere omogeneizzata: esiste una funzione $\mathrm{\Psi }$ che dipende solo da $z$ tale che i funzionali $${\int }_{\mathrm{\Omega }}f(\frac{x}{ϵ},Du(x))dx\mathit{\hspace{1em}\hspace{1em}}u\in H^{1,p}(\mathrm{\Omega };{ℝ}^m)$$ convergono, per $ϵ$ tendente a $0$ (nel senso della $\mathrm{\Gamma }$-convergenza) a $${\int }_{\mathrm{\Omega }}\mathrm{\Psi }(Du(x))dx.$$ Inoltre si può dare una formula asintotica per $\mathrm{\Psi }$.

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed...

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed...

2000 Mathematics Subject Classification: 35J40, 49J52, 49J40, 46E30By means of a suitable nonsmooth critical point theory for lower semicontinuous functionals we prove the existence of infinitely many solutions for a class of quasilinear Dirichlet problems with symmetric non-linearities having a one-sided growth condition of exponential type.The research of the authors was partially supported by the MIUR project “Variational and topological methods in the study of nonlinear phenomena” (COFIN 2001)....

We present here our most recent results ([1def]) about the definition of non-linear Weiertrass-type integrals over BV varieties, possibly discontinuous and not necessarily Sobolev's.

We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further results concern the radial symmetry of solutions as well as a precise description of their behavior near the boundary.