Luca Martinazzi

Bio: Luca Martinazzi is an academic researcher from University of Padua. The author has contributed to research in topics: Compact space & Order (ring theory). The author has an hindex of 19, co-authored 65 publications receiving 1396 citations. Previous affiliations of Luca Martinazzi include ETH Zurich & Rutgers University.

An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs

Abstract: This volume deals with the regularity theory for elliptic systems. We may find the origin of such a theory in two of the problems posed by David Hilbert in his celebrated lecture delivered on the occasion of the International Congress of Mathematicians in 1900 in Paris: 19th problem: are the solutions to regular problems in the Calculus of Variations always necessarily analytic? - 20th problem: does any variational problem have a solution, provided that certain assumptions regarding the given boundary conditions are satisfied, and provided that the notion of a solution is suitably extended? During the last century these two problems have generated a great deal of work, usually referred to as is in regularity theory, which makes this topic quite relevant in many fields and still very active for research. However, the purpose of this volume, addressed mainly to students, is much more limited. We aim to illustrate only some of the basic ideas and techniques introduced in this context, confining ourselves to important but simple situations and refraining from completeness. In fact some relevant topics are omitted. Topics covered include: harmonic functions, direct methods, Hilbert space methods and Sobolev spaces, energy estimates, Schauder and Lp-theory both with and without potential theory, including the Calderon Zygmund theorem, Harnack's and De Giorgi-Moser-Nash theorems in the scalar case and partial regularity theorems in the vector valued case; finally, harmonic maps and minimal graphs in codimension 1 and greater than 1.

Gamma-convergence of nonlocal perimeter functionals

Abstract: Given \({\Omega\subset\mathbb{R}^{n}}\) open, connected and with Lipschitz boundary, and \({s\in (0, 1)}\), we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where \({E\subset\mathbb{R}^{n}}\) is an arbitrary measurable set We prove that the functionals \({(1-s)\mathcal{J}_s(\cdot, \Omega)}\) are equi-coercive in \({L^1_{\rm loc}(\Omega)}\) as \({s\uparrow 1}\) and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, Ω) denotes the perimeter of E in Ω in the sense of De Giorgi We also prove that as \({s\uparrow 1}\) limit points of local minimizers of \({(1-s)\mathcal{J}_s(\cdot,\Omega)}\) are local minimizers of P(·, Ω)

Classification of solutions to the higher order Liouville’s equation on {\mathbb{R}^{2m}}

Abstract: We classify the solutions to the equation (−Δ) m u = (2m − 1)!e 2mu on $${\mathbb{R}^{2m}}$$ giving rise to a metric $${g=e^{2u}g_{\mathbb{R}^{2m}}}$$ with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric $${e^{2u}g_{\mathbb{R}^{2m}}}$$ at infinity, and we observe that the pull-back of this metric to S 2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.

Fractional Adams–Moser–Trudinger type inequalities

Abstract: Extending several works, we prove a general Adams–Moser–Trudinger type inequality for the embedding of Bessel-potential spaces H n p , p ( Ω ) into Orlicz spaces for an arbitrary domain Ω with finite measure. In particular we prove sup u ∈ H n p , p ( Ω ) , ‖ ( − Δ ) n 2 p u ‖ L p ( Ω ) ≤ 1 ∫ Ω e α n , p | u | p p − 1 d x ≤ c n , p | Ω | , for a positive constant α n , p whose sharpness we also prove. We further extend this result to the case of Lorentz-spaces (i.e. ( − Δ ) n 2 p u ∈ L ( p , q ) ). The proofs are simple, as they use Green functions for fractional Laplace operators and suitable cut-off procedures to reduce the fractional results to the sharp estimate on the Riesz potential proven by Adams and its generalization proven by Xiao and Zhai. We also discuss an application to the problem of prescribing the Q -curvature and some open problems.

Critical points of the Moser-Trudinger functional on a disk

Abstract: On the unit disk B-1 subset of R-2 we study the Moser-Trudinger functional E(u) = integral(B1) (e(u2) - 1)dx, is an element of H-0(1) (B-1) and its restrictions E vertical bar M-Lambda, where M-Lambda := {u is an element of H-0(1) (B-1): parallel to u parallel to(2)(H01) = Lambda} for Lambda > 0. We prove that if a sequence u(k) of positive critical points of E vertical bar(M Lambda k) ( for some Lambda(k) > 0) blows up as k -> infinity, then Lambda(k) -> 4 pi, and u(k) -> 0 weakly in H-0(1) (B-1) and strongly in C-loc(1) ((B) over bar (1) \ {0}). Using this fact we also prove that when Lambda is large enough, then E vertical bar M-Lambda has no positive critical point, complementing previous existence results by Carleson-Chang, Struwe and Lamm-Robert-Struwe.

Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

Abstract: The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.

A critical Kirchhoff type problem involving a nonlocal operator

Abstract: In this paper we show the existence of non-negative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator, that is − M ( ‖ u ‖ Z 2 ) L K u = λ f ( x , u ) + | u | 2 ∗ − 2 u in Ω , u = 0 in R n ∖ Ω where L K is an integrodifferential operator with kernel K , Ω is a bounded subset of R n , M and f are continuous functions, ‖ ⋅ ‖ Z is a functional norm and 2 ∗ is a fractional Sobolev exponent.

A critical Kirchhoff type problem involving a non-local operator

Abstract: Abstract In this paper we show the existence of non-negative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator, that is − M ( ‖ u ‖ Z 2 ) L K u = λ f ( x , u ) + | u | 2 ∗ − 2 u in Ω , u = 0 in R n ∖ Ω where L K is an integrodifferential operator with kernel K , Ω is a bounded subset of R n , M and f are continuous functions, ‖ ⋅ ‖ Z is a functional norm and 2 ∗ is a fractional Sobolev exponent.

Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies

Abstract: We obtain a sharp quantitative isoperimetric inequality for nonlocal s-perimeters, uniform with respect to s bounded away from 0. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal s-perimeter plus a non-local repulsive interaction term. In the particular case s = 1, the s-perimeter coincides with the classical perimeter, and our results improve the ones of Knuepfer and Muratov (Comm. Pure Appl. Math. 66(7):1129–1162, 2013; Comm. Pure Appl. Math., 2014) concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.