# ## Mathematical analysis

(Marco Degiovanni - Giovanna Marchioni - Marco Marzocchi)

Our research activity focus on the study of non-linear differential equations (both ordinary and partial) by means of variational methods.

Natural phenomena are often described by functions that satisfy some equations which involves the derivatives of the of the given functions: for this reason this kind of equations are called differential equations. Differential equations are usually nonlinear and to find the solutions can be extremely difficult.

Since the XVIII was observed that the functions that describe natural phenomena often minimize (or makes stationary) appropriate functionals defined on the the functions. For one century and half, such fact was considered as an interesting property of the solutions (obtained in a different way) of some differential equations. In the first half of the XX century a change of perspective was proposed, i.e. the idea of using minimality and stationarity criteria for solving differential equations. This idea turned out to be fruitful and has been developed creating a new field in Mathematical analysis.

### Variational methods for non-regular functionals

In order to establish a connection between the functionals and the differential equation one can consider the case in which the functional itself satisfy some  regularity conditions.

For this reason (starting from the sixty for the study of the minimality and from the eighty for the study of the stationarity) a subfield dedicated to the study (by means of variational methods) of differential equations that do not satisfy standard regularity criteria was developed.

### Key words

• Non-linear differential equations
• Variational methods
• Non-regular functionals

### Collaborations

• Università degli Studi di Pisa
• Università degli Studi di Bari
• Politecnico di Torino
• Università degli Studi di Verona
• Università di Giessen (Germania)

### Ongoing projects

• PRIN07 - Metodi variazionali e topologici nello studio di fenomeni non lineari

### Main pubblications

• T. Bartsch and M. Degiovanni,
Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17, n. 1, 69-85 (2006).
• S. Cingolani and M. Degiovanni,
Nontrivial solutions for p-Laplace equations with righthand side having p-linear growth at infinity, Comm. Partial Differential Equations 30, n. 8, 1191-1203 (2005).
• M. Degiovanni and S. Lancelotti,
Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24, n. 6, 907-919 (2007).
• M. Degiovanni, A. Musesti and M. Squassina,
On the regularity of solutions in the Pucci-Serrin identity, Calc. Var. Partial Differential Equations 18, n. 3, 317-334 (2003).
• S. Lancelotti and M. Marzocchi,
Lagrangian systems with Lipschitz obstacle on manifolds, Topol. Methods Nonlinear Anal. 27, n. 2, 229-253 (2006).