Research

Research field

            Mathematical Analysis 

Areas of interest

            Partial Differential Equation, Spectral theory, Theory of functions spaces, Functional Analysis, Calculus of variations, Homogenization theory.

Research topics

• • Perturbation problems for elliptic partial differential operators of second and higher order

  • Shape sensitivity analysis, spectral stability estimates and optimization 

  • Boundary homogenization for polyharmonic operators

  • Mass concentration phenomena and Steklov-type problems

  • Eigenvalue problems for quasilinear equations

  • Spectral theory for time-harmonic Maxwell's equations

  • Hardy inequalities

  • Extension theorems for Sobolev-Morrey spaces on Lipschitz domains

  • Pathological functions and pathological function spaces

PhD Students

• • Davide Buoso 2015, now Associate Professor at the University of Eastern Piedmont.

  • Luigi Provenzano 2016, now Associate Professor at the University La Sapienza in Rome.

  • Francesco Ferraresso 2018, now assistant Professor (Ricercatore di Tipo A) at the University of Sassari.

  • Michele Zaccaron 2022, now postdoc at Institut Fresnel in Marseille, France.

MS Students

• • Giovanni Franzina 2010

  • Ivan Yuri Violo 2017 

  • Damiano Greco 2020

Collaborators

• José M. Arrieta 

• Gerassimos Barbatis

• Davide Buoso

• Victor I. Burenkov

• Giuseppina di Blasio

• Giacomo Drago

• Damiano Greco 

• Maria Stella Fanciullo

• Francesco Ferraresso

• Alberto Ferrero

• Giovanni Franzina

• Massimo Lanza de Cristoforis

• Marco Perin

• Yehuda Pinchover

• Luigi Provenzano

• Giorgio Stefani

• Ioannis G. Stratis

• Paolo Toni

• Chia-Chi Tung

• Michele Zaccaron

Editorial activity

Member of the Editorial board of the Glasgow Mathematical Journal since November 2023

Member of the Editorial board of the Eurasian Mathematical Journal since 2010

Publications in scientific journals

58. Lamberti, Pier Domenico; Zaccaron, Michele; Spectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities. Mathematics in Engineering 5 (2023), no. 1, Paper No. 018, 31 pp.

57. Ferrero, Alberto; Lamberti, Pier Domenico; Spectral stability of the Steklov problem. Nonlinear Anal. 222 (2022), Paper No. 112989. 

56. Lamberti, Pier Domenico; Provenzano Luigi; On the explicit representation of the trace space H3/2 and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems. Rev. Mat. Complut. 35 (2022), no. 1, 53-88.

55. Lamberti, Pier Domenico; Paolo Luzzini, Paolo Musolino; Shape perturbation of Grushin eigenvalue. The Journal of Geometric Analysis,  31 (2021), no. 11, 10679–10717.

54. Tung, Chia-chi; Pier Domenico Lamberti; On Rellich’s lemma, The Poincaré inequality, and Friedrich extension of an operator on complex Spaces. Cubo A Mathematical Journal. 23 (2021), no. 1, 265–285

53. Lamberti, Pier Domenico; Zaccaron, Michele; Shape sensitivity analysis for electromagnetic cavities. Math. Methods Appl. Sci. 44 (2021), no. 13, 10477–10500. 

52. Lamberti, Pier Domenico; Stratis, Ioannis G.; On an Interior Calderón Operator and a Related Steklov Eigenproblem for Maxwell's Equations. SIAM J. Math. Anal. 52 (2020), no. 5, 4140–4160.

51. di Blasio, Giuseppina; Lamberti, Pier Domenico; Eigenvalues of the Finsler p-Laplacian on varying domains. Mathematika 66 (3) (2020), 765-776.

50. Lamberti, Pier Domenico; Provenzano, Luigi; On trace theorems for Sobolev spaces. Matematiche 75, n.1 (2020), 137-165.

49. Ferraresso, Francesco; Lamberti, Pier Domenico; On a Babuška Paradox for Polyharmonic Operators: Spectral Stability and Boundary Homogenization for Intermediate Problems. Integral Equations and Operator Theory 91, n.6, paper No.55, 42 p. (2019).

48. Lamberti, Pier Domenico; Pinchover, Yehuda; Lp Hardy inequality on C1,γ domains.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XIX, no. 3, (2019), 1135–1159.

47. Ferrero, Alberto; Lamberti, Pier Domenico; Spectral stability for a class of fourth order Steklov problems under domain perturbations, Calculus of Variations and Partial Differential Equations  Vol. 58  (2019), 33,  1-57.

46. Lamberti, Pier Domenico; Violo, Ivan Yuri; On Stein's extension operator preserving Sobolev-Morrey spaces. Math. Nachr. 292 (2019), 1701-1715.

45. Arrieta, José M.; Ferraresso, Francesco; Lamberti, Pier Domenico; Boundary homogenization for a triharmonic intermediate problem, Math. Methods Appl. Sci. 41 (2018), no. 3, 979–985.

44. Arrieta, José M.; Ferraresso, Francesco; Lamberti, Pier Domenico; Spectral analysis of the biharmonic operator subject to Neumann boundary conditions on dumbbell domains, Integral Equations and Operator Theory, 89 (2017), no. 3, 377–408.

43. Arrieta, José M.; Lamberti, Pier Domenico; Higher order elliptic operators on variable domains. Stability results and boundary oscillations for intermediate problems, J. Differential Equations 263 (2017), no. 7, 4222–4266.

42 Lamberti, Pier Domenico; Provenzano, Luigi; Neumann to Steklov eigenvalues: asymptotic and monotonicity results. Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), no. 2, 429–447.

41. Fanciullo, Maria Stella; Lamberti, Pier Domenico On Burenkov's extension operator preserving Sobolev-Morrey spaces on Lipschitz domains. Math. Nachr. 290 (2017), no. 1, 37–49.

40. Lamberti, Pier Domenico; Stefani, Giorgio Sobolev subspaces of nowhere bounded functions. Real Anal. Exchange 41 (2016), no. 2, 367–375.

39. Barbatis, Gerassimos; Lamberti, Pier Domenico Monotonicity, continuity and differentiability results for the Lp Hardy constant. Israel J. Math. 215 (2016), no. 2, 1011–1024.

38. Buoso, Davide; Lamberti, Pier Domenico Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system. SIAM J. Math. Anal. 47 (2015), no. 1, 407–426.

37. Barbatis, Gerassimos; Lamberti, Pier Domenico Shape sensitivity analysis of the Hardy constant. Nonlinear Anal. 103 (2014), 98–112.

36. Lamberti, Pier Domenico Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems. Complex Var. Elliptic Equ. 59 (2014), no. 3, 309–323.

35. Buoso, Davide; Lamberti, Pier Domenico Shape deformation for vibrating hinged plates. Math. Methods Appl. Sci. 37 (2014), no. 2, 237–244.

34. Lamberti, P. D.; Provenzano, L. A maximum principle in spectral optimization problems for elliptic operators subject to mass density perturbations. Eurasian Math. J. 4 (2013), no. 3, 70–83.

33. Buoso, Davide; Lamberti, Pier Domenico Eigenvalues of polyharmonic operators on variable domains. ESAIM Control Optim. Calc. Var. 19 (2013), no. 4, 1225–1235.

32. Arrieta, José M.; Lamberti, Pier Domenico Spectral stability results for higher-order operators under perturbations of the domain. C. R. Math. Acad. Sci. Paris 351 (2013), no. 19-20, 725–730.

31. Barbatis, Gerassimos; Lamberti, Pier Domenico Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients. Mathematika 58 (2012), no. 2, 324–348.

30. Burenkov, Victor I.; Lamberti, Pier Domenico Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators. Rev. Mat. Complut. 25 (2012), no. 2, 435–457.

29. Drago, Giacomo; Lamberti, Pier Domenico; Toni, Paolo A "bouquet'' of discontinuous functions for beginners in mathematical analysis. Amer. Math. Monthly 118 (2011), no. 9, 799–811.

28. Lamberti, P. D.; Perin, M. On the sharpness of a certain spectral stability estimate for the Dirichlet Laplacian. Eurasian Math. J. 1 (2010), no. 1, 111–122.

27. Franzina, Giovanni; Lamberti, Pier Domenico Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem. Electron. J. Differential Equations 2010, No. 26, 10 pp. 

26. Burenkov, Victor I.; Lamberti, Pier Domenico Spectral stability of the p-Laplacian. Nonlinear Anal. 71 (2009), no. 5-6, 2227–2235.

25. Lamberti, Pier Domenico Absence of critical mass densities for a vibrating membrane. Appl. Math. Optim. 59 (2009), no. 3, 319–327.

24. Burenkov, Victor I.; Lamberti, Pier Domenico Spectral stability of Dirichlet second order uniformly elliptic operators. J. Differential Equations 244 (2008), no. 7, 1712–1740.

24. Burenkov, V. I.; Lamberti, P. D.; Lantsa de Kristoforis, M. Spectral stability of nonnegative selfadjoint operators. (Russian) Sovrem. Mat. Fundam. Napravl. 15 (2006), 76--111; translation in J. Math. Sci. (N.Y.) 149 (2008), no. 4, 1417–1452

22. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo A real analyticity result for symmetric functions of the eigenvalues of a domain-dependent Neumann problem for the Laplace operator. Mediterr. J. Math. 4 (2007), no. 4, 435–449.

21. Burenkov, Victor I.; Lamberti, Pier Domenico Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators. J. Differential Equations 233 (2007), no. 2, 345–379.

20. Lamberti, Pier Domenico; Lanza De Cristoforis, Massimo Persistence of eigenvalues and multiplicity in the Dirichlet problem for the Laplace operator on nonsmooth domains. Math. Phys. Anal. Geom. 9 (2006), no. 1, 65–94.

19. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems. J. Math. Soc. Japan 58 (2006), no. 1, 231–245.

18. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo Persistence of eigenvalues and multiplicity in the Neumann problem for the Laplace operator on nonsmooth domains. Rend. Circ. Mat. Palermo (2) Suppl. No. 76 (2005), 413–427.

17. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo A global Lipschitz continuity result for a domain dependent Dirichlet eigenvalue problem for the Laplace operator. Z. Anal. Anwendungen 24 (2005), no. 2, 277–304.

16. Lamberti, Pier Domenico Representation of the eigenvalues of the p-Laplacian by means of a surface integral. Math. Nachr. 278 (2005), no. 11, 1282–1284.

15. Burenkov, V. I.; Lamberti, P. D. Spectral stability of nonnegative selfadjoint operators. (Russian) Dokl. Akad. Nauk 403 (2005), no. 2, 159–164.

14. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo A global Lipschitz continuity result for a domain-dependent Neumann eigenvalue problem for the Laplace operator. J. Differential Equations 216 (2005), no. 1, 109–133.

13. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator. J. Nonlinear Convex Anal. 5 (2004), no. 1, 19–42.

12. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo An analyticity result for the dependence of multiple eigenvalues and eigenspaces of the Laplace operator upon perturbation of the domain. Glasg. Math. J. 44 (2002), no. 1, 29–43.

Book chapters

11. Barbatis, Gerassimos; Burenkov, Victor I.; Lamberti, Pier Domenico Stability estimates for resolvents, eigenvalues, and eigenfunctions of elliptic operators on variable domains. Around the research of Vladimir Maz'ya. II, 23–60, Int. Math. Ser. (N. Y.), 12, Springer, New York, 2010.

10. Burenkov, Victor; Lamberti, Pier Domenico Spectral stability of higher order uniformly elliptic operators. Sobolev spaces in mathematics. II, 69–102, Int. Math. Ser. (N. Y.), 9, Springer, New York, 2009.

9. Lamberti, Pier Domenico A differentiability result for the first eigenvalue of the p-Laplacian upon domain perturbation. Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vol. 1, 2, 741–754, Kluwer Acad. Publ., Dordrecht, 2003.

Conference proceedings

8. Lamberti, Pier Domenico; Provenzano, Luigi Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues. Current trends in analysis and its applications, 171–178, Trends Math., Birkhäuser/Springer, Cham, 2015.

7. Buoso, Davide; Lamberti, Pier Domenico On a classical spectral optimization problem in linear elasticity. New trends in shape optimization, 43–55, Internat. Ser. Numer. Math., 166, Birkhäuser/Springer, Cham, 2015.

6. Burenkov, Victor I.; Lamberti, Pier Domenico Spectral stability of elliptic selfadjoint differential operators with Dirichlet and Neumann boundary conditions. Differential & difference equations and applications, 237–245, Hindawi Publ. Corp., New York, 2006.

5. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo Lipschitz type inequalities for a domain dependent Neumann eigenvalue problem for the Laplace operator. Advances in analysis, 227–233, World Sci. Publ., Hackensack, NJ, 2005.

4. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo Analyticity results for the eigenvalues of compact selfadjoint operators. Int. J. Differ. Equ. Appl. 3 (2001), no. 4, 427–438.

3. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo A spectral perturbation result for the Laplace operator. Tenth International Colloquium on Differential Equations (Plovdiv, 1999). Int. J. Differ. Equ. Appl. 1A (2000), no. 1, 107–115.

Books

2. Constanda, C. (ed.); Dalla Riva, M. (ed.); Lamberti, P. D. (ed.); Musolino, P. (ed.) Integral methods in science and engineering. Volume 1. Theoretical techniques (to appear). Basel: Birkhäuser/Springer (ISBN 978-3-319-59383-8/hbk; 978-3-319-59384-5/ebook). x, 350 p. (2017).Constanda, C. (ed.); Dalla Riva, M. (ed.); Lamberti, P. D. (ed.); Musolino, P. (ed.)

1. Constanda, C. (ed.); Dalla Riva, M. (ed.); Lamberti, P. D. (ed.); Musolino, P. (ed.)  Integral methods in science and engineering, Volume 2. Practical applications (to appear).  Basel: Birkhäuser/Springer (ISBN 978-3-319-59386-9/hbk; 978-3-319-59387-6/ebook). x, 350 p. (2017).

Main results

• • Sharp spectral stability estimates for elliptic partial differential operators of second and higher order via Lebesgue measure of  the symmetric difference of the domains. [31], [30], [23], [21], [11].

  • Notion of atlas distance between domains and related spectral stability estimates for elliptic partial differential operators of second and higher order subject to Dirichlet and Neumann boundary conditions on domains with arbitrarily strong boundary degenerations. [10],  [26].

  • Notion of transition operator and formulation of a general spectral stability theorem giving necessary and sufficient conditions for the validity of  spectral stability estimates for self-adjoint operators with compact resolvents. [21]

  • Unified variational approach to the analysis of  the spectral stability of elliptic partial differential operators of second and higher order subject to homogeneous boundary conditions of intermediate type. [43], [32]

  • Identification of the strange curvature in the Babuska Paradox:  a periodic perturbation of a macroscopically flat boundary produces a microscopic curvature in a biharmonic problem related to the study of a hinged plate. [43], [32]

  • Formulation of a natural Steklov problem  for the biharmonic operator, see the paper of my students.

  • Analytic dependence of the symmetric functions of polyharmonic operators upon domain perturbations and criticality of balls in isovolumetric domain perturbations. [35], [33], [19], [13]

  • Stability results for the Lp Hardy constant on non-convex domains upon perturbation of the domain and of p. [39], [37]

  • Existence and non-existence results for the minimizers in the Hardy inequality on C1,γ domains. [46]

  • Formulation of a nonlinear eigenvalue problem involving a normalizing nonlocal factor and proof of related existence and uniqueness results. [27], see also the related breakthrough result of my former student.

  • Preservation of Sobolev-Morrey norms for extension operators on Lipschitz domains.  [41]

  • Existence of closed  infinite dimensional Sobolev subspaces of nowhere bounded functions. [40]

  • Definition of a number of pathological functions. [29]

Editorial activity

Member of the editorial board of the international journal Eurasian Mathematical Journal which is indexed in Scopus, Mathematical Reviews, MathSciNet (American Mathematical Society, USA), Zentralblatt MATH (ZMATH, GERMANY), Referativnyi Zhurnal-Matematika (RUSSIA)

Organization of scientific meetings

• Member of the organizing committee of the online seminar Online seminar on Mathematical Methods in the Theory of Electromagnetism  

• Member of the organizing committee of the Mini-courses in Mathematical Analysis since 2006.  

• Member of the organizing committee of the Workshop on Spectral Geometry and Analysis of Differential Operators

• Member of the organizing committee of the international conference IMSE 2016 - 14th International Conference on Intergram Methods in Science and Engineering