In memoriam Emmanuele DiBenedetto (1947–2021)

References

[1] H. W. Alt and E. DiBenedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 12 (1985), no. 3, 335–392. Search in Google Scholar

[2] D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal. 25 (1967), 81–122. 10.1007/BF00281291Search in Google Scholar

[3] P. Baroni, T. Kuusi, C. Lindfors and J. M. Urbano, Existence and boundary regularity for degenerate phase transitions, SIAM J. Math. Anal. 50 (2018), no. 1, 456–490. 10.1137/17M1121585Search in Google Scholar

[4] P. Baroni, T. Kuusi and J. M. Urbano, A quantitative modulus of continuity for the two-phase Stefan problem, Arch. Ration. Mech. Anal. 214 (2014), no. 2, 545–573. 10.1007/s00205-014-0762-9Search in Google Scholar

[5] M. Bonforte, R. G. Iagar and J. L. Vázquez, Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation, Adv. Math. 224 (2010), no. 5, 2151–2215. 10.1016/j.aim.2010.01.023Search in Google Scholar

[6] M. Bonforte and J. L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations, Adv. Math. 223 (2010), no. 2, 529–578. 10.1016/j.aim.2009.08.021Search in Google Scholar

[7] L. A. Caffarelli and L. C. Evans, Continuity of the temperature in the two-phase Stefan problem, Arch. Ration. Mech. Anal. 81 (1983), no. 3, 199–220. 10.1007/BF00250800Search in Google Scholar

[8] L. A. Caffarelli and A. Friedman, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), no. 3, 361–391. 10.2307/2374136Search in Google Scholar

[9] E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43. Search in Google Scholar

[10] E. DiBenedetto, Regularity results for the porous media equation, Ann. Mat. Pura Appl. (4) 121 (1979), 249–262. 10.1007/BF02412006Search in Google Scholar

[11] E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. (4) 130 (1982), 131–176. 10.1007/BF01761493Search in Google Scholar

[12] E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), no. 1, 83–118. 10.1512/iumj.1983.32.32008Search in Google Scholar

[13] E. DiBenedetto, The flow of two immiscible fluids through a porous medium; regularity of the saturation, Theory and Applications of Liquid Crystals, IMA Vol. Math. Appl. 5, Springer, New York (1985), 123–143. 10.1007/978-1-4613-8743-5_7Search in Google Scholar

[14] E. DiBenedetto, A boundary modulus of continuity for a class of singular parabolic equations, J. Differential Equations 63 (1986), no. 3, 418–447. 10.1016/0022-0396(86)90064-1Search in Google Scholar

[15] E. DiBenedetto, Harnack estimates in certain function classes, Atti Sem. Mat. Fis. Univ. Modena 37 (1989), no. 1, 173–182. Search in Google Scholar

[16] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer, New York, 1993. 10.1007/978-1-4612-0895-2Search in Google Scholar

[17] E. DiBenedetto, Real Analysis, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser/Springer, New York, 1993. Search in Google Scholar

[18] E. DiBenedetto, Partial Differential Equations, 2nd ed., Cornerstones, Birkhäuser, Boston, 2010. 10.1007/978-0-8176-4552-6Search in Google Scholar

[19] E. DiBenedetto, Tribute to Ennio De Giorgi, Pisa, September 20, 2016, https://www.youtube.com/watch?v=CREr0Z6tSro&list=PLZ4vyOIVxsn1RRGqDaik18EQN6g-D15bl&index=7. Search in Google Scholar

[20] E. DiBenedetto and A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 349 (1984), 83–128. 10.1515/crll.1984.349.83Search in Google Scholar

[21] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. 10.1515/crll.1985.357.1Search in Google Scholar

[22] E. DiBenedetto and R. Gariepy, Local behavior of solutions of an elliptic-parabolic equation, Arch. Ration. Mech. Anal. 97 (1987), no. 1, 1–17. 10.1007/BF00279843Search in Google Scholar

[23] E. DiBenedetto and U. Gianazza, A Wiener-type condition for boundary continuity of quasi-minima of variational integrals, Manuscripta Math. 149 (2016), no. 3–4, 339–346. 10.1007/s00229-015-0780-4Search in Google Scholar

[24] E. DiBenedetto and U. Gianazza, Some properties of De Giorgi classes, Rend. Istit. Mat. Univ. Trieste 48 (2016), 245–260. Search in Google Scholar

[25] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math. 200 (2008), no. 2, 181–209. 10.1007/s11511-008-0026-3Search in Google Scholar

[26] 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 (2010), no. 6, 2041–2076. 10.1512/iumj.2010.59.4004Search in Google Scholar

[27] 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. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 2, 385–422. 10.2422/2036-2145.2010.2.06Search in Google Scholar

[28] 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 Math. 131 (2010), no. 1–2, 231–245. 10.1007/s00229-009-0317-9Search in Google Scholar

[29] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer Monogr. Math., Springer, New York, 2012. 10.1007/978-1-4614-1584-8Search in Google Scholar

[30] E. DiBenedetto, Y. Kwong and V. Vespri, Local space-analyticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J. 40 (1991), no. 2, 741–765. 10.1512/iumj.1991.40.40033Search in Google Scholar

[31] E. DiBenedetto and N. S. Trudinger, Harnack inequalities for quasiminima of variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 295–308. 10.1016/s0294-1449(16)30424-3Search in Google Scholar

[32] E. DiBenedetto, J. M. Urbano and V. Vespri, Current issues on singular and degenerate evolution equations, Evolutionary Equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam (2004), 169–286. 10.1016/S1874-5717(04)80005-7Search in Google Scholar

[33] E. DiBenedetto and V. Vespri, On the singular equation β ⁢ ( u ) t = Δ ⁢ u , Arch. Ration. Mech. Anal. 132 (1995), no. 3, 247–309. 10.1007/BF00382749Search in Google Scholar

[34] E. DeGiorgi, Congetture sulla continuità delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati, Typewritten document, Lecce, 1995. Search in Google Scholar

[35] M. Giaquinta and E. Giusti, Quasiminima, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 79–107. 10.1016/s0294-1449(16)30429-2Search in Google Scholar

[36] J. Hadamard, Extension à l’équation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. Palermo (2) 3 (1954), 337–346. 10.1007/BF02849264Search in Google Scholar

[37] M. A. Herrero, J. J. Manfredi and J. L. Vazquez, En memoria de Emmanuele DiBenedetto (1947–2021), Bol. Electrónico SEMA 27 (2021), 31–38. Search in Google Scholar

[38] C. Klaus and N. Liao, A short proof of Hölder continuity for functions in Degiorgi classes, Ann. Acad. Sci. Fenn. Math. 43 (2018), no. 2, 931–934. 10.5186/aasfm.2018.4354Search in Google Scholar

[39] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968. 10.1090/mmono/023Search in Google Scholar

[40] O. A. Ladyzhenskaya and N. N. Ural’ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Search in Google Scholar

[41] T. D. Lamb and E. N. Pugh, A quantitative account of the activation steps involved in phototransduction in amphibian photoreceptors, J. Physiology 449 (1992), 719–758. 10.1113/jphysiol.1992.sp019111Search in Google Scholar PubMed PubMed Central

[42] T. D. Lamb and E. N. Pugh, Phototransduction, dark adaptation, and rhodopsin regeneration. The Proctor Lecture, Invest. Ophth. Vis. Sci. 47 (2006), 5138–5152. 10.1167/iovs.06-0849Search in Google Scholar

[43] J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. 10.1002/cpa.3160130308Search in Google Scholar

[44] J. Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. 10.1002/cpa.3160140329Search in Google Scholar

[45] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. 10.1002/cpa.3160170106Search in Google Scholar

[46] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727–740. 10.1002/cpa.3160240507Search in Google Scholar

[47] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. 10.2307/2372841Search in Google Scholar

[48] O. A. Oleĭnik, A method of solution of the general Stefan problem, Soviet Math. Dokl. 135 (1960), no. 5, 1350–1354. Search in Google Scholar

[49] B. Pini, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Mat. Univ. Padova 23 (1954), 422–434. Search in Google Scholar

[50] P. E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409. 10.1016/0362-546X(83)90092-5Search in Google Scholar

[51] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. 10.1007/BF02391014Search in Google Scholar

[52] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. 10.1002/cpa.3160200406Search in Google Scholar

[53] N. S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226. 10.1002/cpa.3160210302Search in Google Scholar

[54] J. M. Urbano, Hölder continuity of local weak solutions for parabolic equations exhibiting two degeneracies, Adv. Differential Equations 6 (2001), no. 3, 327–358. 10.57262/ade/1357141214Search in Google Scholar

[55] J. M. Urbano, The Method of Intrinsic Scaling. A Systematic Approach to Regularity for Degenerate and Singular PDEs, Lecture Notes in Math. 1930, Springer, Berlin, 2008. 10.1007/978-3-540-75932-4Search in Google Scholar

[56] V. Vespri, U. Gianazza, D. D. Monticelli, F. Punzo and D. Andreucci, Harnack Inequalities and Nonlinear Operators. Proceedings of the INdAM Conference to Celebrate the 70th Birthday of Emmanuele DiBenedetto, Springer INdAM Ser. 46, Springer, Cham, (2021). 10.1007/978-3-030-73778-8Search in Google Scholar

[57] W. P. Ziemer, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 271 (1982), no. 2, 733–748. 10.1090/S0002-9947-1982-0654859-7Search in Google Scholar

[58] Society for Industrial and Applied Mathematics, New Editors for Two SIAM Journals, March 9, 2001, https://archive.siam.org/news/news.php?id=517. Search in Google Scholar

[59] Vanderbilt University, DiBenedetto, mathematician who advanced knowledge on differential equations, has died, June 4, 2021, https://news.vanderbilt.edu/2021/06/04/dibenedetto-mathematician-who-advanced-knowledge-on-differential-equations-has-died/. Search in Google Scholar