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. Suche 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/BF00281291Suche 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. 10.1137/17M1121585Suche 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/s00205-014-0762-9Suche 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.2010.01.023Suche 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.2009.08.021Suche 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/BF00250800Suche 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. 10.2307/2374136Suche 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. Suche in Google Scholar

[10] E. DiBenedetto, Regularity results for the porous media equation, Ann. Mat. Pura Appl. (4) 121 (1979), 249–262. 10.1007/BF02412006Suche 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/BF01761493Suche 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.1983.32.32008Suche 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_7Suche 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)90064-1Suche in Google Scholar

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

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

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

[18] E. DiBenedetto, Partial Differential Equations, 2nd ed., Cornerstones, Birkhäuser, Boston, 2010. 10.1007/978-0-8176-4552-6Suche 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. Suche 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.1984.349.83Suche 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.1985.357.1Suche 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/BF00279843Suche 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/s00229-015-0780-4Suche in Google Scholar

[24] E. DiBenedetto and U. Gianazza, Some properties of De Giorgi classes, Rend. Istit. Mat. Univ. Trieste 48 (2016), 245–260. Suche 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/s11511-008-0026-3Suche 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.2010.59.4004Suche 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. 10.2422/2036-2145.2010.2.06Suche 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/s00229-009-0317-9Suche 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. 10.1007/978-1-4614-1584-8Suche 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.1991.40.40033Suche 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)30424-3Suche 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)80005-7Suche 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/BF00382749Suche in Google Scholar

[34] E. DeGiorgi, Congetture sulla continuità delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati, Typewritten document, Lecce, 1995. Suche 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)30429-2Suche 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/BF02849264Suche 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. Suche 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.2018.4354Suche 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/023Suche in Google Scholar

[40] O. A. Ladyzhenskaya and N. N. Ural’ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Suche 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.sp019111Suche 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.06-0849Suche 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.3160130308Suche 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.3160140329Suche in Google Scholar

[45] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. 10.1002/cpa.3160170106Suche 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.3160240507Suche in Google Scholar

[47] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. 10.2307/2372841Suche 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. Suche 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. Suche 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)90092-5Suche in Google Scholar

[51] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. 10.1007/BF02391014Suche 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.3160200406Suche in Google Scholar

[53] N. S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226. 10.1002/cpa.3160210302Suche 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/1357141214Suche 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. 10.1007/978-3-540-75932-4Suche 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). 10.1007/978-3-030-73778-8Suche 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/S0002-9947-1982-0654859-7Suche 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. Suche 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/. Suche in Google Scholar