Ugo Gianazza's List of Papers

[79] U. Gianazza and N. Liao - Continuity of the temperature in a multi-phase transition problem. Part II - to appear in Interfaces and Free Boundaries.
Abstract: Local continuity is established for locally bounded, weak solutions to a doubly nonlinear parabolic equation that models the temperature of a material undergoing a multi-phase transition. The enthalpy, as a maximal monotone graph of the temperature, is allowed to possess several jumps and/or infinite derivatives at the transition temperatures. The effect of the $p$-Laplacian type diffusion is also considered. As an application, we demonstrate a continuity result for the saturation in the flow of two immiscible fluids through a porous medium, when irreducible saturation is present.

[78] U. Gianazza and N. Liao - Continuity of the temperature in a multi-phase transition problem. Part III - JAMA (2023), 1-24.
Abstract: We establish local continuity of locally bounded weak solutions to a doubly non-linear parabolic equation that models the temperature in multi-phase transitions. The enthalpy allows for general maximal monotone graphs of the temperature. Remarkably, moduli of continuity can be estimated without an explicit form of the enthalpy.

[77] U. Gianazza and J. Siljander - Local Bounds of the Gradient of Weak Solutions to the Porous Medium Equation - Partial Differ. Equ. Appl. 4, 8 (2023). See also this other link
Abstract: Let $u$ be a nonnegative, local, weak solution to the porous medium equation \[ \partial_t u-\Delta u^m=0 \] for $m\ge2$ in a space-time cylinder $\Omega_T$. Fix a point $(x_o,t_o)\in\Omega_T$: if the average \[ a{\buildrel\mbox{def}\over{=}}\frac1{|B_r(x_o)|}\int_{B_r(x_o)}u(x,t_o)\,dx>0, \] then the quantity $|\nabla u^{m-1}|$ is locally bounded in a proper cylinder, whose center lies at time $t_o+a^{1-m}r^2$. This implies that in the same cylinder the solution $u$ is Hölder continuous with exponent $\alpha=\frac1{m-1}$, which is known to be optimal. Moreover, $u$ presents a sort of instantaneous regularisation, which we quantify.

[76] U. Gianazza - Carleson estimates for the singular parabolic p-Laplacian in time-dependent domains - Rendiconti Lincei. Matematica ed Applicazioni 32, (2021), 669-690.
Abstract: We deal with the parabolic $p$-Laplacian in the so-called singular super-critical range $\frac{2N}{N+1}\lt p\lt 2$, and we prove Carleson estimates for non-negative solutions in suitable non-cylindrical domains $\Omega\subset{\mathbb R}^{N+1}$. The sets $\Omega$ satisfy a proper NTA condition, tailored on the parabolic $p$-Laplacian. As an intermediate step, we show that in these domains non-negative solutions which vanish at the boundary, are Hölder continuous up to the same boundary.

[75] U. Gianazza - In memoriam Emmanuele DiBenedetto (1947-2021) - Advances in Calculus of Variations (2021), 1-18.
Abstract: Emmanuele DiBenedetto passed away in May 2021, after battling cancer for fifteen months. I have had the unique privilege to collaborate and discuss Mathematics with him, almost up to his final days. Here I briefly present his life and those mathematical results of his, which I consider most familiar with.

[74] U. Gianazza and N. Liao - Continuity of the temperature in a multi-phase transition problem - Mathematische Annalen, 384, (2022), 211-245.
Abstract: Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of the $p$-Laplacian type diffusion is also considered.

[73] U. Gianazza and N. Liao - A Boundary Estimate for Singular Sub-Critical Parabolic Equations - Int. Math. Res. Notices 10, (2022), 7332-7353.
Abstract: We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-laplacian type, with $p$ in the sub-critical range $(1,\frac{2N}{N+1}]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

[72] D. Andreucci and U. Gianazza - A Short Presentation of Emmanuele's Work - in V. Vespri et al., (eds.), Harnack Inequalities and Nonlinear Operators. Springer INdAM Series, vol 46. Springer, Cham. 2021.
Abstract: As the title suggests, this is a short presentation of DiBenedetto's mathematical work. In the first part Ugo Gianazza gives a general overview, without entering too much into details of specific papers; in his contribution Daniele Andreucci focuses on DiBenedetto's accomplishments in BioMathematics.

[71] U. Gianazza and S. Salsa - On the Harnack Inequality for Non-divergence Parabolic Equations - Mathematics in Engineering, 3(3) (2020), 1-11.
Abstract: In this paper we propose an elementary proof of the Harnack inequality for linear parabolic equations in non-divergence form.

[70] U. Gianazza and C. Klaus - $p$-Parabolic Approximation of Total Variation Flow Solutions - Indiana Univ. Math. J., 68 No. 5 (2019), 1519-1550.
Abstract: We show that variational solutions to the Cauchy-Dirichlet problem for the total variation flow can be built as the limit of variational solutions to the same problem for the parabolic $p$-Laplacian.

[69] F. Rossella, V. Bellani, M. Tommasini, U. Gianazza, E. Comini, C. Soldano - 3D Multi-Branched SnO₂ Semiconductor Nanostructures as Optical Waveguides - Materials, (2019), 12, 3148, 1-9.
Abstract: Nanostructures with complex geometry have gathered interest recently due to some unusual and exotic properties associated with both their shape and material. 3D multi-branched SnO₂ one-dimensional nanostructrures, characterized by a node - i.e., the location where two or more branches originate, are the ideal platform to distribute signals of different natures. In this work, we study how this particular geometrical configuration affects light propagation when a light source (i.e., laser) is focused onto it. Combining scanning electron microscopy (SEM) and optical analysis along with Raman and Rayleigh scattering upon illumination, we were able to understand, in more detail, the mechanism behind the light-coupling occurring at the node. Our experimental findings show that multi-branched semiconductor 1D structures have great potential as optically active nanostructures with waveguiding properties, thus paving the way for their application as novel building blocks for optical communication networks

[68] U. Gianazza and S. Schwarzacher - Self-improving property of the fast diffusion equation - Journal of Functional Analysis, 1-57, (2019).
Abstract: We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse Hölder inequality in suitable intrinsic cylinders. Relying on an intrinsic Calderón-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for $m\in\left(\frac{(n-2)_+}{n+2},1\right)$. Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for $m\geq 1$ (see [5] in this list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.

[67] U. Gianazza and N. Liao - A Boundary Estimate for Degenerate Parabolic Diffusion Equation - Potential Analysis, 53, (2020), 977-995. See also this other link
Abstract: We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to degenerate parabolic equations of $p$-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

[66] U. Gianazza and S. Schwarzacher - Self-improving property of degenerate parabolic equations of porous medium-type - American Journal of Mathematics, 141(2), (2019), 399-446.
Abstract: We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse Hölder inequality in suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calderón-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution $u$.

[65] U. Gianazza, N. Liao and T. Lukkari - A Boundary Estimate for Singular Parabolic Diffusion Equations - Nonlinear Differential Equations and Applications, 25(4), (2018). See also this other link as part of the Springer Nature SharedIt initiative
Abstract: We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

[64] A. Björn, J. Björn, U. Gianazza and J. Siljander - Boundary regularity for the porous medium equation - Archive for Rational Mechanics and Analysis, 230(2), (2018), 493-538. See also this other link as part of the Springer Nature SharedIt initiative.
Abstract: We study the boundary regularity of solutions to the porous medium equation $u_t=\Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values, provided that the spatial domain satisfies the elliptic Wiener criterion. This condition is known to be optimal, and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general - not necessarily cylindrical - domains in $\mathbb R^{n+1}$. One of our fundamental tools is a new strict comparison principle between sub- and superparabolic functions, which makes it essential for us to study both nonstrict and strict Perron solutions to be able to develop a fruitful boundary regularity theory. Several other comparison principles and pasting lemmas are also obtained. In the process we obtain a rather complete picture of the relation between sub/superparabolic functions and weak sub/supersolutions.

[63] E. DiBenedetto, U. Gianazza and V.Vespri - Remarks on Local Boundedness and Local Höolder Continuity of Local Weak Solutions to Anisotropic $p$-Laplacian Type Equations - J. Elliptic Parabol. Equ., 2, (2016), 157-169. The paper can also be found at this other link.
Abstract: Locally bounded, local weak solutions to a special class of quasilinear, anisotropic, $p$-Laplacian type elliptic equations, are shown to be locally Hölder continuous. Homogeneous local upper bounds are established for local weak solutions to a general class of quasilinear anisotropic equations.

[62] U. Gianazza and S. Salsa - On the Boundary Behaviour of Solutions to Parabolic Equations of $p-$Laplacian Type - Rend. Istit. Mat. Univ. Trieste, 48, (2016), 463-483.
Abstract: This is a survey, in which we describe some recent results on the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. More precisely we focus on Carleson-type estimates and boundary Harnack principles.

[61] V. Bögelein, F. Duzaar and U. Gianazza - Sharp boundedness and continuity results for the singular porous medium equation - Israel Journal of Mathematics, 214, (2016), 259-314. See also this other link as part of the Springer Nature SharedIt initiative.
Abstract: We consider non-homogeneous, singular ($m\in(0,1)$) parabolic equations of porous medium type of the form $$ u_t-\operatorname{div} \mathbf A(x,t,u,Du)=\mu\qquad\mbox{in $E_T$,} $$ where $E_T$ is a space time cylinder, and $\mu$ is a Radon-measure having finite total mass $\mu(E_T)$. In the range $m\in(\frac{(N-2)_+}{N},1)$ we establish sufficient conditions for the boundedness and the continuity of $u$ in terms of a natural Riesz potential of the right-hand side measure $\mu$.

[60] E. DiBenedetto and U. Gianazza - Some Properties of DeGiorgi Classes - Rend. Istit. Mat. Univ. Trieste, 48, (2016), 245-260.
Abstract: The DeGiorgi classes $[DG]_p(E;\gamma)$, defined in (1.1)$_\pm$ of the manuscript encompass, solutions of quasilinear elliptic equations with measurable coefficients as well as minima and $Q$-minima of variational integrals. For these classes we present some new results (§ 2 and § 3.1), and some known facts scattered in the literature (§ 3-§ 5), and formulate some open issues (§ 6).

[59] A. Björn, J. Björn and U. Gianazza - The Petrovskiĭ criterion and barriers for degenerate and singular p-parabolic equations - Math. Ann., 368, (2017), 885-904. See also this other link as part of the Springer Nature SharedIt initiative.
Abstract: In this paper we obtain sharp Petrovskiĭ criteria for the $p$-parabolic equation, both in the degenerate case $p$>2 and the singular case 1<$p$<2. We also give an example of an irregular boundary point at which there is a barrier, thus showing that regularity cannot be characterized by the existence of just one barrier.

[58] F.G. Düzgün, U. Gianazza and V. Vespri - 1-Dimensional Harnack Estimates - Discrete and Continuous Dynamical Systems, Series S, 9(3), (2016), 675-685.
Abstract: Let $u$ be a non-negative super-solution to a 1-dimensional singular parabolic equation of $p$-Laplacian type, $p\in(1,2)$. If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay of order $\frac p{2-p}$ with respect to the space variable $x$ in ${\mathbb R}\times[T/2,T]$. This fact, stated quantitatively in Proposition 1.1, is a sidewise spreading of positivity of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.

[57] B. Avelin, U. Gianazza and S. Salsa - Boundary Estimates for Certain Degenerate and Singular Parabolic Equations - J. Eur. Math. Soc. 18, (2016), 381-424.
Abstract: We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations, of porous medium type.

[56] E. DiBenedetto, U. Gianazza and C. Klaus - A Necessary and Sufficient Condition for the Continuity of Local Minima of Parabolic Variational Integrals with Linear Growth - Adv. Calc. Var. 10(3), (2017), 209-221.
Abstract: For proper minimizers of parabolic variational integrals with linear growth with respect to $|Du|$, we establish a necessary and sufficient condition for $u$ to be continuous at a point $(x_o,t_o)$, in terms of a sufficient fast decay of the total variation of $u$ about $(x_o,t_o)$. These minimizers arise also as proper solutions to the parabolic $1$-Laplacian equation. Hence, the continuity condition continues to hold for such solutions.

[55] E. DiBenedetto and U. Gianazza - A Wiener-Type Condition for Boundary Continuity of Quasi-Minima of Variational Integrals - manuscripta math. 149, (2016), 339-346. Unfortunately, the online version does not take into account all the corrections the authors asked for in the galley proofs; interested readers can refer to preprint.
Abstract: A Wiener-type condition for the continuity at the boundary points of Q-minima, is established, in terms of the divergence of a suitable Wiener integral.

[54] V. Bögelein, F. Duzaar and U. Gianazza - Very weak solutions of singular porous medium equations with measure data - Communications on Pure and Applied Analysis, 14(1), (2015), 23-49.
Abstract: We consider non-homogeneous, singular ($m\in(0,1)$) porous medium type equations with a non-negative Radon-measure $\mu$ having finite total mass $\mu(E_T)$ on the right-hand side. We deal with a Cauchy-Dirichlet problem for these type of equations, with homogeneous boundary conditions on the parabolic boundary of the domain $E_T$, and we establish the existence of\ a solution in the sense of distributions. Finally, we show that the constructed solution satisfies linear pointwise estimates via linear Riesz potentials.

[53] U. Gianazza - Degenerate and Singular Porous Medium Type Equations with Measure Data - in J. Escher et al., (eds.), Elliptic and Parabolic Equations, Springer Proc. Math. Stat., 119, (2015), 139--158.
Abstract: We consider the inhomogeneous porous medium equation $$\partial_t u -\Delta u^m=\mu,\qquad m>\frac{(N-2)_+}{N},$$ and more general equations of porous medium type with a non-negative Radon measure $\mu$ on the right-hand side. In a first step, we prove a priori estimates for weak solutions in terms of a linear Riesz potential of the right-hand side measure, which takes exactly the same form as the one for the classical heat equation. Then, we give an optimal criterium for the continuity of weak solutions, again in terms of a Riesz potential. Finally, we prove the existence of non-negative very weak solutions, and show that these constructed very weak solutions satisfy the same estimates. We deal with both the degenerate case $m>1$, and the singular case $m\in(\frac{(N-2)_+}{N},1)$.

[52] V. Bögelein, F. Duzaar and U. Gianazza - Continuity estimates for porous medium type equations with measure data - Journal of Functional Analysis, 267(9), (2014), 3351-3396.
Abstract: We consider parabolic equations of porous medium type of the form $$ u_t-\operatorname{div} \mathbf A(x,t,u,Du)=\mu\qquad\mbox{in $E_T$,} $$ in some space time cylinder $E_T$. The most prominent example covered by our assumptions is the classical porous medium equation $$ u_t-\Delta u^m =\mu\qquad\mbox{in $E_T$,} $$ {with $m\ge1$.} We establish a sufficient condition for the continuity of $u$ in terms of a natural Riesz potential of the right-hand side measure $\mu$. As an application we come up with a borderline condition ensuring the continuity of $u$: more precisely, if $\mu\in L\big(\frac{N+2}2,1\big)$, then $u$ is continuous in $E_T$.

[51] E. DiBenedetto, U. Gianazza and N. Liao - Two Remarks on the Local Behavior of Solutions to Logarithmically Singular Diffusion Equations and its Porous-Medium Type Approximations - Riv. Mat. Univ. Parma, vol. 5(1), (2014), 139-182.
Abstract For the logarithmically singular parabolic equation $$u_t-\Delta\ln u=0,$$ we establish a Harnack type estimate in the $L^1_{loc}$ topology, and we show that the solutions are locally analytic in the space variables and differentiable in time. The main assumption is that $\ln u$ possesses a sufficiently high degree of integrability. These two properties are known for solutions of singular porous medium type equations ($m\in(0,1)$), which formally approximate the logarithmically singular equation. However, the corresponding estimates deteriorate as $m\to0$. It is shown that these estimates become stable and carry to the limit as $m\to0$, provided the indicated sufficiently high order of integrability is in force. The latter then appears as the discriminating assumption between solutions of parabolic equations with power-like singularities and logarithmic singularities to insure such solutions to be regular.

[50] A. Björn, J. Björn, U. Gianazza and M. Parviainen - Boundary regularity for degenerate and singular parabolic equations - Calc. Var., 52(3) (2015), 797-827.
Abstract: We characterise regular boundary points of the parabolic $p$-Laplacian in terms of a family of barriers, both when $p>2$ and $p\in(1,2)$. Due to the fact that $p\not=2$, it turns out that one can multiply the $p$-Laplace operator by a positive constant, without affecting the regularity of a boundary point. By constructing suitable families of barriers, we give some simple geometric conditions that ensure the regularity of boundary points.

[49] V. Bögelein, F. Duzaar and U. Gianazza - Porous medium type equations with measure data and potential estimates - SIAM J. Math. Anal., 45(6), (2013), 3283-3330.
Abstract: We consider non-homogeneous, degenerate ($m>1$) porous medium type equations with a non-negative Radon measure $\mu$ having finite total mass $\mu(E_T)$ on the right-hand side, and we derive linear pointwise estimates for solutions via Riesz potentials. Then, we deal with a Cauchy-Dirichlet problem for the same equation, with homogeneous boundary conditions on the parabolic boundary of the domain $E_T$, and we establish the existence of a solution in the sense of distributions.

[48] U. Gianazza - List of Mathematical Works Authored or Edited by Enrico Magenes - in F. Brezzi et al., (eds), Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol 4. Springer, Milano, 2013.
Abstract:The following list of mathematical works authored or edited by Enrico Magenes in his long career has been compiled relying on a number of different sources, and it is probably the first one ever prepared.

[47] E. Dallago, D.G. Finarelli, U. Gianazza, A. Lazzarini Barnabei and A. Liberale - Theoretical and experimental analysis of an MPP detection algorithm employing a single voltage sensor only and a noisy signal - IEEE Transactions on Power Electronics, vol.28(11), (2013), 5088-5097.
Abstract: In this paper, a maximum power point (MPP) detection algorithm for photovoltaic (PV) systems is introduced, which uses the experimental information obtained from a single-voltage sensor, measured on a capacitor load, either linked at the output of a solar cell (SC), a PV module, or a PV string. The voltage signal is naturally affected by the noise which has a relevant effect on the process necessary for MPP determination, such as voltage first- and second-order derivatives. The aim of this study is to demonstrate the technical feasibility of a maximum power point tracker (MPPT) based on the present MPP detection algorithm employing a single-voltage sensor acquiring a signal affected by the significant noise. Theoretical evaluation, numerical simulations, and experimental measurements are carried out. Excellent agreement between the theoretical and experimental behavior is observed. Conditions for correct MPP detection are shown and good performances are obtained.

[46] E. DiBenedetto, U. Gianazza and V. Vespri - Continuity of the Saturation in the Flow of Two Immiscible Fluids in a Porous Medium - Indiana Univ. Math. J. 59 No. 6 (2010), 2041-2076.
Abstract: The weakly coupled system \begin{equation*} \left\{ \begin{array}{l} {\displaystyle v_t-{\operatorname{div}}[A(v)\nabla v+{\bf B}(v)]={\bf V}\cdot\nabla C(v)}\\ {\displaystyle {\operatorname{div}}{\bf V}=0} \end{array}\right. \qquad\text{ in }\>E_T. \end{equation*} consists of an elliptic equation and a degenerate parabolic equation, and it arises in the theory of flow of immiscible fluids in a porous medium. The unknown functions $u$ and $v$ and the equations they satisfy, represent the pressure and the saturation respectively, subject to Darcy's law and the Buckley--Leverett coupling. Due to the empirical nature of these laws no determination is possible on the structure of the degeneracy exhibited by the system. It is established that the saturation is a locally continuous function in its space--time domain of definition, irrespective of the nature of the degeneracy of the principal part of the system.

[45] E. DiBenedetto, U. Gianazza and N. Liao - Logarithmically Singular Parabolic Equations as Limits of the Porous Medium Equation - Nonlinear Analysis Series A: Theory, Methods & Applications, 75(12), (2012), 4513-4533.
Abstract: Let $\{u_m\}$ be a local, weak solution to the porous medium equation \begin{equation*} u_{m,t}-\Delta w_m=0 \end{equation*} where $w_m=\frac{u_m^m-1}{m}$. It is shown that if $\{u_m\}$ is locally in $L^r_{loc}$ for $r>\frac12N$ uniformly in $m$ and if $w_m$ is in $L^p_{loc}$ for $p>N+2$ in the space variables, uniformly in time, then $\{u_m\}$ contains a subsequence converging in $C^{\alpha,\frac12\alpha}_{loc}$ to a local, weak solution to the logarithmically singular equation $u_t=\Delta\ln u$. The result is based on local upper and lower bounds on $\{u_m\}$, uniform in $m$. The uniform, local lower bounds are realized by a Harnack type inequality.

[44] E. DiBenedetto, U. Gianazza and N. Liao - On the Local Behavior of Non-Negative Solutions to a Logarithmically Singular Equation - Discrete Continuous Dynamical Systems Ser. B, 17(6), (2012), 1841-1858.
Abstract: The local positivity of solutions to logarithmically singular diffusion equations is investigated in some open space-time domain $E\times(0,T]$. It is shown that if at some time level $t_o\in(0,T]$ and some point $x_o\in E\,$ the solution $u(\cdot,t_o)$ is not identically zero in a neighborhood of $x_o$, in a measure-theoretical sense, then it is strictly positive in a neighborhood of $(x_o,t_o)$. The precise form of this statement is by an intrinsic Harnack-type inequality, which also determines the size of such a neighborhood.

[43] E. DiBenedetto, U. Gianazza and V. Vespri - Liouville-Type Theorems for Certain Degenerate and Singular Parabolic Equations - C. R. Acad. Sci. Paris, Ser. I 348 (2010) 873-877.

[42] E. DiBenedetto, U. Gianazza and V. Vespri - Forward, Backward and Elliptic Harnack Inequalities for Non-Negative Solutions to Certain Singular Parabolic Partial Differential Equations - Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Vol. IX (2010), 385-422.

[41] E. DiBenedetto, U. Gianazza and V. Vespri - A New Approach to the Expansion of Positivity Set of Non-negative Solutions to Certain Singular Parabolic Partial Differential Equations - Proc. Amer. Math. Soc. 138 (2010), 3521-3529.

[40] U. Gianazza, M. Surnachev and V. Vespri - On a new proof of Hölder continuity of solutions of p-Laplace type parabolic equations - Adv. Calc. Var. 3 (2010), 263-278.

[39] E. DiBenedetto, U. Gianazza and V. Vespri - Harnack Type Estimates and Hölder Continuity for Non-Negative Solutions to Certain Sub-Critically Singular Parabolic Partial Differential Equations - manuscripta mathematica, 131, (1-2), (2010), 231-245.

[38] S. Fornaro, U. Gianazza - Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations - Discrete and Continuous Dynamical Systems A, 26, (2), (2010), 481-492.

[37] E. DiBenedetto, U. Gianazza and V. Vespri - Alternative Forms of the Harnack Inequality for Non-Negative Solutions to Certain Degenerate and Singular Parabolic Equations - Rendiconti Lincei Matematica ed Applicazioni, 20(4), (2009), 369-377.

[36] U. Gianazza, G. Savaré, G. Toscani - The Wasserstein gradient flow of the Fisher information and the Quantum Drift-Diffusion equation - Arch. Rational Mech. Analysis, 194, (1), (2009) 133-220.

[35] E. DiBenedetto, U. Gianazza, V. Vespri - Harnack Estimates for Quasi-Linear Degenerate Parabolic Differential Equation - Acta Mathematica, 200 (2008), 181-209.

[34] L. Corazzini, U. Gianazza - Unequal contributions from identical agents in a local interaction model - Journal of Public Economic Theory, 10 (3), 2008, 351-370.

[33] E. DiBenedetto, U. Gianazza, V. Vespri - Sub-Potential Lower Bounds for Non-Negative Solutions to Certain Quasi-Linear Degenerate Parabolic Differential Equations - Duke Mathematical Journal, Vol. 143, 1, (2008), 1-15.

[32] U. Gianazza, S. Polidoro - Lower Bounds for Solutions of Degenerate Parabolic Equations - Lecture Notes of Seminario Interdisciplinare di Matematica, Vol. 6(2007), 157-162.

[31] F. Dinuzzo, M. Neve, U. Gianazza, G. De Nicolao - On the representer theorem and equivalent degrees of freedom of SVR - Journal of Machine Learning Research 8 (2007), 2467-2495.

[30] E. DiBenedetto, U. Gianazza and V. Vespri - Intrinsic Harnack Inequalities for Quasi-linear Singular Parabolic Partial Differential Equations - Rend. Lincei Mat. Appl. 18 (2007), 359-364.

[29] E. DiBenedetto, U. Gianazza, V. Vespri - Intrinsic Harnack estimates for non-negative local solutions of degenerate parabolic equations - Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 95-99.

[28] E. DiBenedetto, U. Gianazza, V. Vespri - Local Clustering of the Non-Zero Set of Functions in W^1,1(E) - Rend. Lincei Mat. Appl. 17, (2006), 223-225.

[27] U. Gianazza, V. Vespri - A Harnack Inequality for a Degenerate Parabolic Equation - Journal of Evolution Equations, 6, 2, (2006), 247-267.

[26] U. Gianazza, V. Vespri - Parabolic De Giorgi classes of order p and the Harnack inequality- Calculus of Variations and Partial Differential Equations, 26, 3, (2006), 379-399.

[25] U. Gianazza, V. Vespri - A Harnack Inequality for Solutions of Doubly Nonlinear Parabolic Equations - Journal Applied Functional Analysis, 1, 3, (2006), 271-284.

[24] U. Gianazza, B. Stroffolini, V. Vespri - Interior and boundary continuity of the solution of the singular equation (β(u))_t=Lu - Nonlinear Anal. 56, 2, (2004) 157-183.

[23] U. Gianazza, V. Vespri - Continuity of weak solutions of a singular parabolic equation - Advances in Differential Equations, 8, 11, (2003), 1341-1376.

[22] U. Gianazza, V. Vespri - The Heisenberg Laplacian: a survey - Rend. Circolo Matem. Palermo Serie II, Suppl. 52 (1998). pp. 491-512.

[21] U. Gianazza, V. Vespri - Hölder Classes relative to degenerate Elliptic Operators as Interpolation Spaces - Ulmer Seminare 1997, Funktionalanalysis und Differentialgleichungen, Heft2, 367-376, Le Matematiche, Vol LIII, (1998) - Fasc. I, 107 - 121.

[20] U. Gianazza - Existence for a nonlinear problem relative to Dirichlet forms - Rend. Acc. Naz. XL, 115 (1997), Vol. XXI, fasc. 1, 209-234.

[19] U. Gianazza, V. Vespri - Analytic Semigroups generated by Square Hörmander Operators - Rend. Istit. Mat. Univ. Trieste, Suppl. Vol. XXVIII, 199-218 (1997).

[18] U. Gianazza - Regularity for a degenerate obstacle problem - IAN preprint # 997.

[17] U. Gianazza, V. Vespri - Generation of Analytic semigroups by Degenerate Elliptic Operators - NoDEA, 4 (1997) 305-324.

[16] U. Gianazza, S. Marchi - Interior regularity for solutions to some degenerate quasilinear obstacle problems - Nonlinear Anal. 36 (1999), no. 7, Ser. A: Theory Methods, 923-942.

[15] U. Gianazza, G. Savaré - Abstract Evolution Equations on Variable Domains: An Approach by Minimizing Movements - Ann. Scuola Norm. Sup. Pisa, IV, XXIII, 1, (1996), 149-178.

[14] U. Gianazza - Meyer's estimate for Dirichlet forms - Rend. Ist. Lomb. A, 128, (1994) 147-151.

[13] U. Gianazza, G. Savaré - Some results on Minimizing Movements - Rend. Acc. Naz. XL, 112, (1994), XVIII, fasc. 1, 57-80.

[12] U. Gianazza, M. Gobbino, G. Savaré - Evolution Problems and Minimizing Movements - Rend. Mat. Acc. Lincei, s. 9, v. 5:289-296 (1994).
Abstract: We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.

[11] M. P. Bernardi, E. Gagliardo, U. Gianazza - Proprietà di combinazioni lineari intere. Applicazioni - Rend. Ist. Lomb. A, 127, (1993) 33-39.

[10] U. Gianazza - Limit of obstacles for square Hörmander operators - Atti Sem. Mat. Fis. Univ. Modena, XLIII, 467-471 (1995).

[9] U. Gianazza - Sequences of obstacles problems for Dirichlet forms - Diff. Int. Eq., 9, (1996), 89-118.

[8] U. Gianazza - Higher integrability for Quasi-minima of functionals depending on vector fields - Rend. Acc. Naz. XL, 111 (1993), Vol. XVII, fasc. 1, 209-227.

[7] U. Gianazza - Regularity for non linear equations involving square Hörmander operators - Nonlinear Analysis: Theory, Methods & Applications, 23(1), (1994), 49-73.

[6] U. Gianazza - The L^p integrability on homogeneous spaces - Rend. Ist. Lomb. A, 126, (1992) 83-92.

[5] U. Gianazza - Local properties of variational solutions for the two obstacle problem involving square Hörmander operators - Ann. Mat. Pura Appl., (IV), 164, (1994), 301-333.
Abstract: We consider the double obstacle problem involving square Hörmander operators. We prove a Wiener estimate giving a generalization of results obtained both for degenerate elliptic operators and for uniformly elliptic operators with bounded and measurable coefficients.

[4] M. Biroli, U. Gianazza - Wiener criterion for the obstacle problem relative to square Hörmander's operators - Variational and Free Boundary Problems. IMA Volumes in Mathematics and its Applications 53.

[3] U. Gianazza - Potential estimate for the obstacle problem relative to the sum of squares of vector fields - Riv. Mat. Univ. Parma, (4) 17 (1991) 221 - 239.

[2] U. Gianazza - Wiener points and energy decay for a relaxed Dirichlet problem relative to a degenerate elliptic operator - Riv. Mat. Univ. Parma (4) 16 (1990) 297 - 309.

[1] U. Gianazza - Soluzioni forti per un problema ellittico degenere - Rend. Ist. Lomb. A, 124, (1990) 189 - 206.