# Giuseppe Di FazioUniversity of Catania | UNICT · Department of Mathematics and Computer Science (DMI)

Giuseppe Di Fazio

PhD Partial Differential Equations

## About

93

Publications

14,133

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

1,510

Citations

Introduction

My interest is about regularity for degenerate elliptic equations and systems.

Additional affiliations

November 2017 - November 2017

March 2017 - March 2017

September 2016 - September 2016

## Publications

Publications (93)

This paper investigates stationary mean-field games (MFGs) on the torus with Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The primary objective is to understand the existence of C1,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackag...

We prove local boundedness, Harnack's inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable matrix valued function Q(x) and two suitable non-negative weight functions. We setup an axiomatic approach in...

The aim of this paper is to study the model problem:
∂u∂t−Δpu=finQu=0on∂Ω×(0,T)u(x,0)=u0inΩ.
The main purpose of this work is to prove the existence and a limiting regularity result for the solution u of the above problem having right‐hand side f∈Lβ(0,T;LLogαL(Ω)). In particular, we will consider the cases: (i) If p≥2,β=1, and α>N−1N, then u∈Lp−1(0...

This paper deals with the existence and regularity of some unilateral problem associated to a nonlinear equation of type {-\operatorname{div}(a(x,u,\nabla u))+H(x,u,\nabla u)=f} .

We consider a class of severely degenerate elliptic operators and establish a Harnack inequality up to the boundary. As a consequence, we get smoothness of the weak solutions under very sharp assumptions regarding the lower order terms of the operator.

The aim of this paper is to give a brief account on the problem of regularity forlinear elliptic PDEs of the second order.

We study regularity for solutions of quasilinear elliptic equations of the form \mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F} in bounded domains in \mathbb{R}^n . The vector field \mathbf{A} is assumed to be continuous in u , and its growth in \nabla u is like that of the p -Laplace operator. We establish interior gradient estimates...

We prove Harnack inequality and regularity for solutions of a quasilinear doubly degenerate elliptic equation generated by Grushin vector fields. We assume the coefficients of the structure conditions to belong to suitable Stummel–Kato classes.

We study a new kind of degenerate operators related to some weighted sum operators. Although the operators are severely degenerate we show the smoothness of the weak solutions.

We study regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in bounded domains in $\R^n$. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak...

We prove that positive and bounded weak solutions of a strongly degenerate elliptic equation satisfy the Harnack inequality. The structure of the differential operator includes a nonlinear term in the gradient with quadratic growth. Moreover, the lower order terms belong to some Stummel classes defined in term of sum operators introduced in [13].

We make a further step in the study of regularizing properties of the strong solutions to the equation Open image in new windowwhere \({\Omega }\) is a bounded domain in \({\mathbb {R}}^n\) with smooth boundary. We are interested in estimating the second order derivatives of the solutions and in showing that solutions belong to the class \(C^{1,\al...

We consider quasilinear elliptic equations that are degenerate in two
ways. One kind of degeneracy is due to the particular structure
of the given vector fields. Another is a consequence of the weights
that we impose to the quadratic form of the associated differential operator.
Nonetheless we prove that positive solutions satisfy unique continuati...

In this paper we show that the shear modulus $\mu$ of an isotropic elastic body can be stably recovered by the knowledge of one single displacement field whose boundary data can be assigned independently of the unknown elasticity tensor.

We define Stummel-Kato type classes in a quasimetric homogeneous setting using sum operators introduced in [13] by Franchi, Perez and Wheeden. Then we prove a Harnack inequality for positive solutions of some linear subelliptic equations.

Our aim is to obtain L^p estimates for the second-order horizontal derivatives
of the solutions for a nondivergence form nonlinear equation in Carnot groups.

We define Stummel-Kato type classes in a quasimetric homogeneous setting using sum operators introduced in (J. Fourier Anal. Appl. 9(5), 511–540, 2003) by Franchi, Perez and Wheeden. Then we prove an embedding inequality of Fefferman–Phong type. As an application we give a unique continuation result for non negative solutions of some subelliptic eq...

Interior Lp gradient weighted estimates are proved for degenerate elliptic systems in divergence form with VMO coefficients.

Our aim is to estimate the second order horizontal derivatives of the solutions for a nondivergence form subelliptic equation
a(x,u,Xu,X2u)=f(x).a(x,u,Xu,X2u)=f(x).

We prove a Harnack inequality and regularity for solutions of a quasilinear strongly degenerate elliptic equation. We assume the coefficients of the structure conditions to belong to suitable Stummel–Kato classes.

In this note we study the global regularity in the Morrey spaces for the
second derivatives for the strong solutions of non variational elliptic
equations.

We prove interior $L^p$ gradient weighted estimates for degenerate elliptic
equations in divergence form with VMO coefficients.

We prove Harnack inequality and regularity for solutions of a linear strongly degenerate elliptic equation. We assume the
lower order terms in Stummel–Kato classes with respect to the Carnot–Carathéodory metric. We follow the pattern by Serrin
in Acta Math 111:247–302, (1964).

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong A
∞ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove C
1,α...

We establish the strong unique continuation property for positive weak solutions to degenerate quasilinear elliptic equations. The degeneracy is given by a suitable power of a strong A∞ weight (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We prove local and global regularity for the positive solutions of a quasilinear variational degenerate equation, assuming minimal hypothesis on the coefficients of the lower order terms. As an application we obtain Hölder continuity for the gradient of solutions to nonvariational quasilinear equations.

We establish a covering lemma of Besicovitch type for metric balls in the setting of Hölder quasimetric spaces of homogenous
type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions
of power decay, critical density and double ball property and with the aid of the covering theore...

We apply subelliptic Cordes conditions and Talenti–Pucci type inequalities to prove W
2,2 and C
1,α estimates for p-harmonic functions in the Grušin plane for p near 2.

In this paper we investigate the regularity of the weak solutions for degenerate elliptic equations of the following kind divA(x,u(x),∇u(x))+B(x,u(x),∇u(x))=0(1) under the structure conditions |A(x,u,ξ)|≤aω(x)|ξ| p-1 +b|u| p-1 +e,|B(x,u,ξ)|≤c|ξ| p-1 +d|u| p-1 +f, ξ·A(x,u,ξ)≥ω(x)|ξ| p -d|u| p -g· Here v is a strong A ∞ weight and ω=v 1-p/n , 1<p<n....

We prove Harnack inequality for weak solutions to quasilinear subelliptic equation of the following kind ∑ j=1 m X j * A j (x,u(x),Xu(x))+B(x,u(x),Xu(x))=0,(1) where X 1 ,⋯,X m are a system of non-commutative locally Lipschitz vector fields. As a consequence, the weak solutions of (1) are continuous.

The main result of our work [Manuscr. Math. 120, No. 4, 419–433 (2006; Zbl 1185.35037)] is the C loc 1,α regularity for subelliptic p-harmonic functions in the case of the Grušin vector fields. To this goal we prove a Calderón-Zygmund inequality and an estimate for strong solutions of a linear subelliptic equation in nondivergence form with L ∞ coe...

We establish a covering lemma of Besicovitch type for metric balls in the setting of Hölder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions of power decay, critical density and double ball property and with the aid of the covering theore...

In this note we prove the Harnack inequality and the Hölder continuity for weak solutions of quasilinear subelliptic equation of the form
where u belongs to Sobolev spaces with respect to a system of locally Lipschitz vector fields. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this note we prove a Feerman-Poincar et ype inequality in spaces with metric induced by Carnot-Carath eodory vector elds.

We consider the generation of analytic semigroups by elliptic operators with discontinuous coefficients.

The main result of our work ((2)) is the C 1;fi loc regularity for subelliptic p-harmonic functions in the case of the Grusin vector fields. To this goal we prove a Calderon-Zygmund inequality and an estimate for strong solutions of a linear subelliptic equation in nondivergence form with L 1 coecients.

We consider the generation of analytic semigroups by elliptic operators with discontinuous coefficients.

Well-posedness is proved in the space W2, p, λ(Ω)∩W1, p0(Ω) for the Dirichlet problem -- EQUATION OMITTED -- if the principal coefficients aij(x) of the uniformly elliptic operator belong to VMO∩L∞(Ω).

Strong solvability and uniqueness in the Sobolev space W 2, q (Ω), q > n , are proved for the oblique derivative problem
assuming the coefficients of the quasilinear elliptic operator to be Carathéodory functions, a ij ∈ VMO ∩ L ∞ with respect to x , and b to grow at most quadratically with respect to the gradient.

A priori estimates and strong solvability results in Sobolev space W 2,p (), 1 < p < ∞ are proved for the regular oblique derivative problem n i,j=1 a ij (x) ∂ 2 u ∂xi∂xj = f(x) a.e. ∂u ∂ℓ + σ(x)u = ϕ(x) on ∂ when the principal coefficientsa ij are VMO ∩ L ∞ functions.

In this paper we consider the Dirichlet problem for the equation $div(A(x)\nabla u) = div f in W^{1,p}_0$ and prove existence uniqueness and estimates for the gradient of the solution.
We perform the estimates via a representation formula that contains the derivatives of the solution and some harmonic analysis results.

In this paper we study the local Holder-regularity of weak solutions to is a Hörmander hypoelliptic operator and the potential V belongs to a new class of functions which is the natural extension of Morrey spaces to this situation. We improve a recent result of Citti, Garofalo, and Lanconelli.

We study the local Hölder-regularity of weak solutions to ℒu+Vu=0 where ℒ is a Hörmander hypoelliptic operator and the potential V belongs to a new class of functions which is the natural extension of Morrey spaces to this situation. We improve a recent result of Citti, Garofalo, and Lanconelli.

Let us consider the nondivergence form elliptic equation aijuxixj = ƒ In this paper we show that if the known term ƒ belongs to the Morrey space Lp, λ then the second derivatives of the W2, p-solution u belong to the same space. Next we derive a C1, αloc-regularity result that is related to a recent work by Caffarelli.

< i. < n. (For the precise statement see Section 1). The solution we consider is a very weak one introduced in [LSW] because in general the Dirichlet problem for Eq. (*) does not have a weak (variational) solution under our assumption onf. The purpose of our work is to study the regularity properties of the solu- tion as the parameter A increases f...

We give a characterization of the Hohn Nirenberg space BMO in terms of the boundedness of some commutators.
We study commutators between multiplication operators and singular integrals and commutators between multiplication operators and Riesz potentials.