University of Parma
Department of Mathematics
 
Home page of Silvana Marchi
Curriculum vitae List of publications Description of the research



Curriculum vitae

Born on 1955 in Fontanellato (Parma-Italy).
After taking my degree on 1978, I availed myself of a CNR grant for the period from 1978 until 1981. I was junior lecturer from 1981 until 1988. I became professor at the University of Parma on 1988 and I worked at the School of Engineering until 1994 and at the School of Science till today.

List of publications
  1. S. Marchi: Alcune proprietà su insiemi di funzioni as-quasi periodiche , Riv. Mat. Univ. Parma 4 (1980), 65-71.
  2. S. Marchi: Soluzione quasiperiodica per disequazioni variazionali del tipo di Navier-Stokes , I : Ist. Lombardo Accad. Sci. Lett. Rend. A 113 (1979), 103-110; II : Ist. Lombardo Accad. Sci. Lett. Rend. A 114 (1980), 44-49.
  3. S. Marchi : Condizioni necessarie di equi-as-quasi periodicità , Riv. Mat. Univ. Parma 4 8 (1982), 501-508.
  4. S. Marchi , C. Risito : Generalizzazione del teorema di Markov sulla stabilità forte , Riv. Mat. Univ. Parma 4 10 (1984), 351-357.
  5. M. Biroli , S. Marchi : An estimate in homogenization problem for elliptic equations with discontinuous bounded coefficients , I.A.C. III 92 (1978).
  6. M. Biroli , S. Marchi , T. Norando : Homogenization estimates for quasi-variational inequalities , Boll. U.M.I. 5 18-A (1981) 267-274.
  7. S. Marchi , T. Norando : Homogenization estimates for variational inequalities of parabolic type , Riv. Mat. Univ. Parma 4 9 (1983) 473-484.
  8. S. Marchi , T. Norando : Homogenization estimates for quasi-variational inequalities of parabolic type , Riv. Mat. Univ. Parma 4 11 (1985), 15-24.
  9. M. Biroli , S. Marchi : Regular points for degenerate elliptic equations , Le Matematiche XL I-II (1985), 145-153.
  10. M. Biroli , S. Marchi : Wiener estimates at boundary points for degenerate elliptic equations , Boll. U.M.I. 6 5-B (1986), 689-706 ; Boll. U.M.I. 7 2-B (1988), 713.
  11. M. Biroli , S. Marchi : Remark on "Wiener estimates at boundary points for degenerate elliptic equations" , Boll. U.M.I. , 6 V-C (1986) , 257-267.
  12. M. Biroli , S. Marchi : Wiener estimates for degenerate elliptic equations II , Differential and Integral Equations , 2 4 (1989), 511-523.
  13. S. Marchi : Remark on "Wiener estimates at boundary points for degenerate quasi-linear elliptic equations" , Ist. Lombardo Accad. Sci. Lett. Rend. A 120 (1986) 17-33.
  14. S. Marchi : Boundary regularity for weighted quasi-minima ,Riv. Mat. Univ. Parma 4 15* (1989) 63-74.
  15. S. Marchi : Boundary regularity for weighted quasi-minima II , Nonlinear Anal. 20 5 (1993) 461-467.
  16. S. Marchi : A Wiener type criterion for weighted quasiminima , Rend. Mat. Acc. Lincei 9 2 (1991) 25-28.
  17. S. Marchi : Boundary regularity for parabolic quasiminima , Ann. Mat. Pura Appl. 4 166 (1994) 17-26.
  18. S. Marchi : Holder continuity and Harnack inequality for De Giorgi classes related to Hormander vector fields , Ann. Mat. Pura Appl. 4 168 (1995) 171-188.
  19. S. Marchi : Note on a paper of J.M. Rakotoson , Ist. Lombardo Accad. Sci. Lett. Rend. A 127 (1993) 25-31.
  20. S. Marchi : Capacities for Dirichlet forms , Riv. Mat. Univ. Parma 5 2 (1993) 103-114.
  21. U. Gianazza , S. Marchi : Interior regularity for solutions to some degenerate quasilinear obstacle problems , Nonliner Anal. 36 7 (1999) Ser: A : Theory Methods ,
  22. S. Marchi : Influence of the nonlocal term on the regularity of equations involving Dirichlet forms , Ist. Lombardo Accad. Sci. Lett. Rend. A 131 (1997) N°1-2, 189-199..
  23. S. Marchi : $C^[2,\beta]_[loc]$ regularity for degenerate elliptic equations , Riv. Mat. Univ. Parma 5 6 (1997) 123-128.
  24. S. Marchi : $W^[2,p]$ and $W^[2,p]_[loc]$ regularity for degenerate elliptic equations , Ist. Lombardo Accad. Sci. Lett. Rend: A 133 (1999) 87-101.
  25. S. Marchi : Existence of nontrivial solutions to a nonlinear Dirichlet problem for the Q-Laplacian to Hormander vector fields , Riv. Mat. Univ. Parma 6 3 (2000) 87-100.
  26. S. Marchi : $C^1_[\alpha]$ local regularity for the solutions of the p-Laplacian on the Heisenberg group for $2 \le p < 1+ \sqrt[5]$ , Z. Anal. Anwendungen 20 3 (2001) 617-636; 22 2 (2003) 471-472.
  27. S. Marchi : Regularity for the solutions of double obstacle problems involving nonlinear elliptic operators on the Heisenberg group , Le Matematiche XVI (2001) 109-127.
  28. S. Marchi : $C^1_[\alpha]$ local regularity for the solutions of the p_Laplacian on the Heisenberg group. The case $1+ \frac[1][\sqrt[5]] < p \le 2$ , Comment. Math. Univ. Carolinae 44 1 (2003) 33-56.
  29. S. Marchi : $L^p$ regularity for the derivative in the second commutator direction for nonlinear elliptic equations on the Heisenberg group , Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 5 25 (2002) 1-15.
  30. M. Biroli , S. Marchi : Oscillation estimates relative to p-homogeneous forms and Kato measures data , Le Matematiche LXI (2006) 335-361.
  31. M. Biroli , S. Marchi : Harnack inequality for the Schroedinger problem relative to strongly local Riemannian p-homogeneous forms with a potential in the Kato class , Boundary Value Problems I (2007) 1-19.
  32. M. Biroli , S. Marchi : Wiener criterion at the boundary related to p-homogeneous strongly local Dirichlet forms , Le Matematiche LXII (2007) 37-52.
  33. M. Biroli , F. Dal Fabbro , S. Marchi : Wiener criterion for the relaxed Dirichlet problem relative to p-homogeneous Riemannian Dirichlet forms , Ukranian Math. Bull. V (2008) 1-15.
  34. M. Biroli , S. Marchi : Harnack inequality for harmonic functions relative to a nonlinear p-homogeneous Riemannian Dirichlet form , Nonlinear Anal. 71 (2009) e436-e444.

Description of the research

Actual research
The actual research deals with Dirichlet forms (see the book of Fukushima 1980). They are non-negative definite, symmetric, bilinear forms a(u,v) defined on a linear subspace of the Hilbert space H=L^2(X,m), where X is a given separable measurable space and m is a \sigma-finite positive measure on X. Moreover a(u,v) are closed and Markovian.
In virtue of the Beuerling-Deny formula 1958, 1959 any regular (if it possesses a core) Dirichlet form can be represented as the sum of three parts, the "diffusion part" (it is the integral of the "energy measure" \mu (u,v), a Radon-measure-valued positive-semidefinite symmetric bilinear form) the global or "nonlocal part" and the "killing part".
This context offered to me the occasion to write the papers [20] and [22].
On 1995 Biroli and Mosco concentrated their study on the regular, strongly local Dirichlet forms expressed as integral of the energy measure \mu, which may be used in describing the variational behavior of possibly highly nonhomogeneous and nonisotropic bodies. One of the most important properties of \mu is its local character: the restriction of the measure \mu (u,v) to any open subset A of X only depends on the restrictions of u and v to A. This property entitles us to interprets \mu as a measure-valued description of the phisical characteristic of the body X. The structure of differential manifold could not be required to X.
The local minimizer u of the "energy functional" E(u)=a(u,u)/2 is the local solution of the equation a(u,v)=0. They studied the regularity of the local solutions.
On 2004 Biroli and Vernole extended to the nonlinear case defining a Dirichlet form expressed as integral of the energy measure \mu which is Riemannian, strongly local and p-homegeneous (the energy measure \mu(u,v) is homogeneous of degree p-1 in u and linear in v) and they studied the regularity of the local solutions.
The papers [30],...,[34] continue their analysis.
In [31] we proved Harnack's inequality for the positive solutions of an homogeneous Schroedinger problem defined by the Biroli-Vernole form, with a potential in the Kato class.
On this aim we utilized the result previously proved in [30], concerning the estimate of the oscillation of the solutions of Dirichlet problems with measure data. In [32] we established a Wiener criterion at the boundary for the homogeneous equations. In [33] and [34] we established a Wiener criterion in the interior for relaxed Dirichlet problems with data in the Kato class.

Preprints
  1. M. Biroli , S. Marchi: Asymptotic behavior of relaxed Dirichlet problems related to p-homogeneous strongly local forms.
  2. F. Dal Fabbro , S. Marchi: \Gamma-convergence of strongly local Dirichlet functionals.
In [1] we prove the weak convergence of the solutions of a sequence of relaxed Dirichlet problems related to a sequence of Borel measures.
In [2] we give a variational motivation of the above result proving the convergence of the minima of the functionals associated to the above relaxed problems.

Previous research
The papers [1],...,[4] deal with the properties of quasi-periodic functions.
The papers [5],...,[8] concern homogenization estimates for elliptic or parabolic quasi-variational inequalities.
The following papers concern the regularity (as Wiener estimate, Harnack inequality, Holder inequality) of the solutions of degenerate elliptic equations.
The degeneracy is due to the lack of coercivity.
The parabolic case is sometimes also considered.
The papers [14],...,[18] deal in particular with quasiminima.
back to top