UGO GIANAZZA's LIST OF PUBLICATIONS

[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), 2029–2064.
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 - (2012), 1-21, Nonlinear Analysis Series A: Theory, Methods & Applications.
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 V. Vespri - Harnack's Inequality for Degenerate and Singular Parabolic Equations - Springer Monographs in Mathematics, 2012.

[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 - Regularity Estimates for Parabolic De Giorgi Classes of order p - 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).

[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 - NLA - TMA, 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.

[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.