Research

Research Areas

Geometric Analysis, with particular interest in: Spectral Theory, Spectral Geometry, Analysis on manifolds, Global analysis, Partial Differential Equations (PDEs), Function Spaces, Asymptotic Analysis.

Research Topics

Spectral theory of elliptic operators of second and higher order, spectral geometry, geometric estimates for eigenvalues, spectral theory on manifolds, functional inequalities and shape optimization, eigenvalue asymptotics, Steklov-type problems, magnetic Laplacians on Riemannian surfaces, qualitative properties of PDEs on Riemannian manifolds.

Suggested online seminars:

Spectral geometry in the clouds, intergalactic seminar on spectral theory and differential geometry.
Cardiff Analysis & PDE seminar.
Electromath seminar, time harmonic Maxwell equations, eigenvalue problems and their qualitative properties.
Seminar in Analysis.

Organizaton of scientific meetings

Mini-Symposium "Spectral Geometry and related topics" at the European Congress of Mathematics, Seville (Spain), July 17-18, 2024
A Spring Day in Geometric Analys, Roma (Italy), April 11-12, 2024
Geometric Analysis and PDE's, Incontro INDAM, Cortona (Italy), June 5-9, 2023.
Recent Trends in Geometric Analysis and Spectral Theory, Roma (Italy), June 13-14, 2022.
Workshop on Spectral Geometry and Analysis of Differential Operators, Padova (Italy), September 9-11, 2019.

Scientific communications

Find out the list of my talks and seminars at international conferences and institutions in my cv.

Publications

Publications in scientific journals

M. Grossi, L. Provenzano, On the critical points of semi-stable solutions on convex domains of Riemannian surfaces. Math. Ann., (2023). (published, preprint)
B. Colbois, C. Léna, L. Provenzano, A. Savo, Geometric bounds for the magnetic Neumann eigenvalues in the plane. J. Math. Pures Appl. (9)179 (2023), 454–497. (published, preprint)
D. Buoso, P. Luzzini, L. Provenzano, J. Stubbe, Semiclassical estimates for eigenvalue means of Laplacians on spheres. J. Geom. Anal. 33 (2023), no. 9, 280. (published, preprint)
P. Luzzini, L. Provenzano, J. Stubbe, The first Grushin eigenvalue on cartesian product domains. Adv. Calc. Var.17(2024), no.2, 353–371. (published, preprint)
F. Ferraresso, L. Provenzano, On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set. Bull. Lond. Math. Soc.55 (2023), no.3, 1154–1177. (published, preprint)
B. Colbois, L. Provenzano, A. Savo, Isoperimetric inequalities for the magnetic Neumann and Steklov problems with Aharonov-Bohm magnetic potential. J. Geometric Analysis 32, article n. 285, 2022 (published, preprint)
D. Buoso, L. Provenzano, J. Stubbe, Semiclassical bounds for spectra of biharmonic operators. Rendiconti di Matematica e delle sue Applicazioni, (7) 43 (2022), 267-314 (published, preprint).
B. Colbois, L. Provenzano, Neumann eigenvalues of the biharmonic operator on domains: geometric bounds and related results. J. Geometric Analysis, vol. 32, article n. 218 (2022) (published, preprint).
L. Provenzano, A. Savo, Isoparametric foliations and the Pompeiu property. Mathematics in Engineering 5 (2022), no. 2, 1-27 (published, preprint).
L. Provenzano, Upper bounds for the Steklov eigenvalues of the p-Laplacian. Mathematika, 68: 148-162, 2021 (published, preprint).
D. Buoso, P. Luzzini, L. Provenzano, J. Stubbe, On the spectral asymptotics for the buckling problem. J. Math. Phys. 62 (2021), no. 12, Paper No. 121501 (published, preprint).
B. Colbois, L.Provenzano, Conformal upper bounds for the eigenvalues of the p-Laplacian. J. Lond. Math. Soc. (2)104(2021), no.5, 2128–2147. (published, preprint).
P.D. Lamberti, L. Provenzano, 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. (published, preprint) .
P.D. Lamberti, L. Provenzano, On Trace Theorems for Sobolev Spaces. Le Matematiche, Volume 75, Issue 1 (2020), 137–175 (published, preprint).
E. Harrell II, L. Provenzano, J. Stubbe, Complementary asymptotically sharp estimates for eigenvalue means of Laplacians , Int. Math. Res. Not. IMRN (2021), no.11, 8405–8450 (online version, preprint).
L. Provenzano, Inequalities between Dirichlet and Neumann eigenvalues of the polyharmonic operators, Proceedings of the American Mathematical Society, Volume 147, Issue 11 (2019), 4813–4821 (published, preprint).
L. Provenzano, J. Stubbe Weyl-type bounds for Steklov eigenvalues, Journal of Spectral Theory, Volume 9, Issue 2, (2019). (published, preprint)
D. Buoso, L.M. Chasman, L. Provenzano, On the stability of some isoperimetric inequalities for the fundamental tones of free plates, Journal of Spectral Theory Volume 8, Issue 3 (2018), 2928-2967. (published, preprint)
M. Dalla Riva, L. Provenzano, On vibrating membranes with mass concentrated near the boundary: an asymptotic analysis", SIAM Journal on Mathematical Analysis, Volume 50, Issue 3, (2018), to appear. (published, preprint)
B. Colbois, L. Provenzano, Eigenvalues of elliptic operators with density, Calculus of Variations and Partial Differential Equations (2018) 57:36. (published, preprint)
L. Provenzano, A note on the Neumann eigenvalues of the biharmonic operator, Mathematical Methods in the Applied Sciences, 41 (2018), no. 3, 1005–1012. (published, preprint.)
P.D. Lamberti, L. Provenzano, Neumann to Steklov eigenvalues: asymptotic and monotonicity results, Proceedings of the Royal Society of Edinburgh: A, Volume 147, Issue 2 (2017), 429-447. (published, preprint)
L. Andreis, D. Barbato, F. Collet, M. Formentin, Luigi Provenzano, Strong existence and uniqueness of the stationary distribution for a stochastic inviscid dyadic model, Nonlinearity, Volume 29, Issue 3 (2016) , 1156-1169. (published, preprint)
D. Buoso, L. Provenzano, A few shape optimization results for a biharmonic Steklov problem, Journal of Differential Equations Volume 259, Issue 5 (2015), 1778–1818. (published, preprint)
P.D. Lamberti, L. Provenzano, A maximum principle in spectral optimization problems for elliptic operators subject to mass density perturbations Eurasian Mathematical Journal,ISSN 2077-9879 Volume 4, Issue 3 (2013), 70 – 83. (published, preprint)

Publications in books and proceedings

P.D. Lamberti, L. Provenzano, Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues, Current Trends in Analysis and Its Applications, Proceedings of the 9th ISAAC Congress, Kraków 2013 171-178, Birkhäuser, Basel (2015). (published, preprint version)
D. Buoso, L. Provenzano, On the eigenvalues of a biharmonic Steklov problem, Integral Methods in Science and Engineering: Theoretical and Computational Advances, ch.7, Springer (2015), 81-89. (published, preprint version)

Submitted

A. Enciso, A. Pistoia, L. Provenzano, Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator (preprint version).
L. Provenzano, A. Savo, Geometry of the magnetic Steklov problem on Riemannian annuli (preprint version).
L. Battaglia, A. Pistoia, L. Provenzano, On the critical points of Steklov eigenfunctions (preprint version)
B. Colbois, C. Léna, L. Provenzano, A. Savo, A reverse Faber-Krahn inequality for the magnetic Laplacian (preprint version)

Photo: mountains of Haute-Savoie from the dock of St. Sulpice (VD), Switzerland.