Applied differential models
Research focus
The Peer review has evaluated this group as Average
1. Biharmonic equations. In [J7,J17] we investigate the case of Steklov boundary conditions (hinged plate) and show that positivity properties are sensitive to the parameter involved in the boundary condition. Under different boundary conditions, we consider critical growth problems in [J11,J12,J23,J24,J25]. Biharmonic Gelfand-type problems (with exponential nonlinearity) on bounded domains are considered in [J4,J10]. For the same equation in R^n, in [J3] we provided a dynamical description of the behavior of the separatrix solution in the set of entire solutions via a computer assisted proof. We applied this proof also to other different problems (the Fermi-Pasta-Ulam system, the 3-body problem), see [J1,J2,J5]. 2. Optimal shape problems. In [J19,J22] we studied with geometric tools overdetermined elliptic boundary value problems and we proved that if they admit a solution then the domain is a ball. In [J14] we partially proved an optimal shape conjecture by Polya-Szego concerning the electrostatic capacity of convex bodies. We also studied an approximation of the capacity by means of web-functions; these functions were already considered for the non-existence of an optimal shape for the torsion problem and its p-Laplacian version, see [J13]. 3. Nonlinear second order elliptic equations at critical growth. In [J6,J15,J16,J18,J26] we consider several different versions of this problem; we highlight the strong dependence of existence results on lower order perturbations and on geometric properties of the domain. 4. Blow-up for nonlinear evolution equations. Global solutions and finite time blow-up for semilinear parabolic equations in bounded domains are considered in [J21,J28]. With the same type of power-type source term, the damped hyperbolic equation is considered in [J27]. 5. Problems in hydrodynamics. A relevant problem in hydrodynamics is the determination of the wave resistance acting on a (totally or partially) submerged body in uniform motion in a heavy ideal fluid (e.g. water). This problem has been studied from analytic and numerical point of views; an exact analytic treatment is difficult, due to the nonlinear Bernoulli condition on the free surface. Therefore, most of the mathematical literature on this subject concerns the linearized version of the problem. A rigorous approach to the non linear problem was considered in the previous years. More recently, in [J20,J29,J30,J31,J32,J33,J34] we studied subcritical velocities, a typical condition for bodies moving in deep water. This is an open problem even in the linear approximation, since the existing results cannot rule out the occurrence of "singular values" of the velocity; for such values nothing is known about unique solvability. We intend to prove (for a wide class of obstacles) unique solvability of the linear problem by exploiting a special variational approach, which differs from the usual weak formulations due to the lack of coercivity. We will then use the results obtained to solve the corresponding free boundary problem by local techniques of nonlinear analysis. 6. Problems in combustion theory. The theoretical study of the combustion of solid energetic materials is a relevant subject in the applications (e.g. solid rocket propulsion) for better understanding and addressing questions such as control of burning rate, diagnostic of the 50 combustion waves, process stability. Recently, there has been an increasing interest in the implementation of experimental techniques based on radiation-driven burning (i.e., burning in presence of external radiant flux of thermal nature). In [J35] we investigated a class of mathematical models describing this situation. The models are well posed and, under the assumption of intensely irradiated boundary, there exists a global attractor for the associated evolution semigroup; furthermore, in [J36] the stabilization of all solutions towards the equilibrium solution (a travelling wave, corresponding to steady burning regimes) is derived for a special class of propellants. 7. Problems in conductivity. A relevant problem in conductivity is the determination of unknown portions of a material by overdetermined boundary measures. The aim of these nonlinear inverse problems is to get uniqueness, stability and numerical determination. In [J9] we considered, for the stationary equation, a non classical condition on the unknown line-crack inside a bidimensional body; we got uniqueness results by two additional measures at the boundary. In [J8] we also studied the case in which the unknown line is part of the boundary and is interpreted as a corroded part of the material: uniqueness and stability are obtained. Since our stability estimates are of Lipschitz type (namely "good" from a numerical point of view), our study is now devoted to find a numerical algorithm allowing to determine the corrosion.
Dipartimento di afferenza
Docenti afferenti
Full Professors
Filippo Gazzola
Carlo Pagani
Associate Professors
Gianni Arioli
Valeria Bacchelli
Dario Pierotti
Maurizio Verri