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August  2017, 10(4): 853-866. doi: 10.3934/dcdss.2017043

## A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class

 Dipartimento di Matematica "Tullio Levi Civita, " Università di Padova, via, Trieste 63,35121, Padova, Italy

Received  April 2016 Revised  September 2016 Published  April 2017

We define a homogeneous parabolic De Giorgi class of order 2 which suits a mixed type class of evolution equations whose simplest example is $\mu (x) \frac{\partial u}{\partial t} - \Delta u = 0$ where $\mu$ can be positive, null and negative. The functions belonging to this class are local bounded and satisfy a Harnack type inequality. Interesting by-products are Hölder-continuity, at least in the "evolutionary" part of $\Omega$ and in particular in the interface $I$ where $\mu$ change sign, and an interesting maximum principle.

Citation: Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
##### References:
 [1] E. B. Fabes, C. E. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.  doi: 10.1080/03605308208820218.  Google Scholar [2] J. Garcia Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. Google Scholar [3] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co. Inc. , River Edge, NJ, 2003. doi: 10.1142/5002.  Google Scholar [4] C. E. Gutiérrez and R. L. Wheeden, Mean value and Harnak inequalities for degenerate parabolic equations, Colloq. Math., 60/61 (1990), 157-194.   Google Scholar [5] F. Paronetto, Homogenization of degenerate elliptic-parabolic equations, Asymptotic Anal., 37 (2004), 21-56.   Google Scholar [6] F. Paronetto, A Harnack's inequality and Hölder continuity for solutions of mixed type evolution equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 385-395.  doi: 10.4171/RLM/711.  Google Scholar [7] F. Paronetto, A Harnack's inequality for mixed type evolution equations, J. Differential Equations, 260 (2016), 5259-5355.  doi: 10.1016/j.jde.2015.12.003.  Google Scholar

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##### References:
 [1] E. B. Fabes, C. E. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.  doi: 10.1080/03605308208820218.  Google Scholar [2] J. Garcia Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. Google Scholar [3] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co. Inc. , River Edge, NJ, 2003. doi: 10.1142/5002.  Google Scholar [4] C. E. Gutiérrez and R. L. Wheeden, Mean value and Harnak inequalities for degenerate parabolic equations, Colloq. Math., 60/61 (1990), 157-194.   Google Scholar [5] F. Paronetto, Homogenization of degenerate elliptic-parabolic equations, Asymptotic Anal., 37 (2004), 21-56.   Google Scholar [6] F. Paronetto, A Harnack's inequality and Hölder continuity for solutions of mixed type evolution equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 385-395.  doi: 10.4171/RLM/711.  Google Scholar [7] F. Paronetto, A Harnack's inequality for mixed type evolution equations, J. Differential Equations, 260 (2016), 5259-5355.  doi: 10.1016/j.jde.2015.12.003.  Google Scholar
The sets involved in the estimates of points $i)$ and $ii)$ of Theorem 4.1
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