# American Institute of Mathematical Sciences

July  2016, 15(4): 1401-1417. doi: 10.3934/cpaa.2016.15.1401

## Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary

 1 Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna

Received  September 2015 Revised  January 2016 Published  April 2016

We show a result of maximal regularity in spaces of Hölder continuous function, concerning linear parabolic systems, with dynamic or Wentzell boundary conditions, with an elliptic diffusion term on the boundary.
Citation: Davide Guidetti. Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1401-1417. doi: 10.3934/cpaa.2016.15.1401
##### References:
 [1] T. Clarke, G.R. Goldstein, J.A. Goldstein and S. Romanelli, The Wentzell telegraph equation: asymptotics and continuous dependence on the boundary conditions, Commun. Pure Appl. Anal., 15 (2011), 313-324. [2] G.M. Coclite, A. Favini, G.R. Goldstein, J.A. Goldstein and S. Romanelli, Continuous dependence on the boundary conditions for the Wentzell Laplacian, Semigroup Forum, 77 (2008), 101-108. doi: 10.1007/s00233-008-9068-2. [3] A. Favini, G.R. Goldstein, J.A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521. doi: 10.1002/mana.200910086. [4] G.R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Diff. Eq., 11 (2006), 457-480. [5] G.R. Goldstein, J.A. Goldstein, D. Guidetti and S. Romanelli, General Wentzell boundary conditions in $L^p$ spaces, work in progress. [6] D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275. doi: 10.1002/mana.19911520120. [7] D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460. doi: 10.1007/BF02571401. [8] D. Guidetti, Abstract elliptic problems depending on a parameter and parabolic problems with dynamic boundary conditions, Chapter 9, Springer INdAM Series, 10 (2014), 161-202. doi: 10.1007/978-3-319-11406-4_9. [9] D. Guidetti, Linear parabolic problems with dynamic boundary conditions in spaces of Hölder continuous functions, Ann. Mat. Pura Appl., 195 (2016), 167-198. doi: 10.1007/s10231-014-0457-8. [10] D. Guidetti, Classical solutions to quasilinear parabolic problems with dynamic boundary conditions, preprint, arXiv:1504.05363. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9234-6. [12] P. Sacks and M. Warma, Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1-$data, Discrete Cont. Dyn. Syst., 34 (2014), 761-787. [13] J.L. Vazquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Diff. Eq., 250 (2011), 2143-2161. doi: 10.1016/j.jde.2010.12.012. [14] M. Warma, Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains, Commun. Pure Appl. Anal., 12 (2013), 1881-1905. doi: 10.3934/cpaa.2013.12.1881. [15] M. Warma, Semi linear parabolic equations with nonlinear general Wentzell boundary conditions, Discrete Cont. Dyn. Systems, 33 (2013), 5493-5506. doi: 10.3934/dcds.2013.33.5493.

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##### References:
 [1] T. Clarke, G.R. Goldstein, J.A. Goldstein and S. Romanelli, The Wentzell telegraph equation: asymptotics and continuous dependence on the boundary conditions, Commun. Pure Appl. Anal., 15 (2011), 313-324. [2] G.M. Coclite, A. Favini, G.R. Goldstein, J.A. Goldstein and S. Romanelli, Continuous dependence on the boundary conditions for the Wentzell Laplacian, Semigroup Forum, 77 (2008), 101-108. doi: 10.1007/s00233-008-9068-2. [3] A. Favini, G.R. Goldstein, J.A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521. doi: 10.1002/mana.200910086. [4] G.R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Diff. Eq., 11 (2006), 457-480. [5] G.R. Goldstein, J.A. Goldstein, D. Guidetti and S. Romanelli, General Wentzell boundary conditions in $L^p$ spaces, work in progress. [6] D. Guidetti, On elliptic problems in Besov spaces, Math. Nachr., 152 (1991), 247-275. doi: 10.1002/mana.19911520120. [7] D. Guidetti, On interpolation with boundary conditions, Math. Z., 207 (1991), 439-460. doi: 10.1007/BF02571401. [8] D. Guidetti, Abstract elliptic problems depending on a parameter and parabolic problems with dynamic boundary conditions, Chapter 9, Springer INdAM Series, 10 (2014), 161-202. doi: 10.1007/978-3-319-11406-4_9. [9] D. Guidetti, Linear parabolic problems with dynamic boundary conditions in spaces of Hölder continuous functions, Ann. Mat. Pura Appl., 195 (2016), 167-198. doi: 10.1007/s10231-014-0457-8. [10] D. Guidetti, Classical solutions to quasilinear parabolic problems with dynamic boundary conditions, preprint, arXiv:1504.05363. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9234-6. [12] P. Sacks and M. Warma, Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1-$data, Discrete Cont. Dyn. Syst., 34 (2014), 761-787. [13] J.L. Vazquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Diff. Eq., 250 (2011), 2143-2161. doi: 10.1016/j.jde.2010.12.012. [14] M. Warma, Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains, Commun. Pure Appl. Anal., 12 (2013), 1881-1905. doi: 10.3934/cpaa.2013.12.1881. [15] M. Warma, Semi linear parabolic equations with nonlinear general Wentzell boundary conditions, Discrete Cont. Dyn. Systems, 33 (2013), 5493-5506. doi: 10.3934/dcds.2013.33.5493.
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