Mathematical Analysis group
Calculus of variations and especially
- the aspects concerning the mechanics of elastic solids and recently fluids
- some cases of energies possessing a bulk and a surface part, with links to image reconstruction and micromagnetic materials
Initial-boundary value problem for hyperbolic equations: mainly those coming out from solid mechanics, as:
1) Kirchhoff nonlinear vibrating string 1876 (Arosio & Panizzi, Trans. Amer. Math. Soc. 348, No.1 (1996), 305-330 [MR 96f:35112] [Zbl 858:35083]
2) linear Timoshenko beam (1921)
3) nonlinear Timoshenko-Kirchhoff beam (Arosio, Chinese Ann. Math. Ser.B, 20, No.4 (1999), p. 495-506) [MR 2001a:35106] [Zbl 970.74036]
Calculus of Variations, and more specifically
- Variational problems not convex and not coercive (Newton's Problem of minimal resistance)
- The Lavrentiev phenomenon
- Regularity and simplicity of the first eigenvalue of the infinity Laplacian operator
- Shape optimization problems in the class of convex domains
Calculus of variations: semicontinuity of integral functionals, existence of minimizers for variational problems without lower semicontinuity and differential inclusions.
For several years her research focused on the Calculus of Variations, studying in particular functions of Bounded Variation, existence problems, image reconstruction and problems in fracture mechanics.
Presently she is conducting a research of medical interest, in partnership with the Allergological Department of the children section of the Parma Hospital and with several hospitals in the region Emilia Romagna, in charge of data management and mathematical-informatic-statistical treatment of data. In particular these researches address children allergies, diabetes and respiratory diseases.
Degenerate and nondegenerate elliptic and parabolic operators with unbounded coefficients in unbounded domains;
Reaction-diffusion systems and Free boundary problems with particular attention to the stability analysis.
Elliptic, hypoelliptic and parabolic differential equations. Evolution equations in Banach spaces.
Silvana Marchi (retired)
The actual research (from 2006) deals with Dirichlet forms (see Fukushima 1980). In 1995 M.Biroli and U.Mosco studied regular (bilinear) strongly local Dirichlet forms of diffusion type. In 2004 M.Biroli and P.Vernole generalized to the nonlinear case. My recent studies continue their analysis proving regularity results (as Harnack inequality, Wiener criterion, oscillation estimate) for the variational equations related to the forms. See for example: M.Biroli , S.Marchi "Harnack inequality..", Nonlinear Anal. 71 (2009), e436-e444 [MR2671852].
calculus of variations, elliptic and parabolic pde
- free-discontinuity problems: regularity, minimality conditions, variational approximation
- quasistatic evolution in plasticity problems
- variational image processing
- quasilinear elliptic equations in phase transitions problems
Calculus of Variations and Geometric Measure Theory
Calculus of Variations and PDE
Keywords: Phase transitions, Concentration, Homogenization, critical Sobolev embeddings, Nonlocal singular perturbations, Fractional operators, Dislocations, Gamma-convergence, Minimal surfaces, Measure data, Regularity in non-rearrangement spaces, Calderón-Zygmund, Yamabe problem.
Partial differential equations of hyperbolic type. In particular my papers are concerned with the following topics: Local/global in time solvability of nonlinear hyperbolic equations, particularly of Kirchhoff type.
Well-posedness of the mixed problem for second or fourth order linear/nonlinear equations in bounded domains.
Propagation of regularity for semilinear wave equations.
Variational formulations of boundary integral equations arising form wave propagation
My main research interest is the distribution of prime numbers (also in short intervals or arithmetic progressions) and additive problems similar to Goldbach's. I also studied some properties of the Riemann zeta-function.
- Workshop: Partial Differential Equations and Applications, Bologna, 22-26 May 2017
- Workshop on Calculus of Variations and its Applications, Parma April 14, 2016
- Sixth Trilateral Meeting on Nonlinear Partial Differential Equations and Applications, Parma December 14-18, 2015
- Workshop on New advances in PDE's, Inverse Problems and Control Theory, Parma July 6-10, 2015