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 & 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,, \emph{Commun. Pure Appl. Anal., 15 (2011), 313.   Google Scholar

[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,, \emph{Semigroup Forum, 77 (2008), 101.  doi: 10.1007/s00233-008-9068-2.  Google Scholar

[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,, \emph{Math. Nachr., 283 (2010), 504.  doi: 10.1002/mana.200910086.  Google Scholar

[4]

G.R. Goldstein, Derivation and physical interpretation of general boundary conditions,, \emph{Adv. Diff. Eq., 11 (2006), 457.   Google Scholar

[5]

G.R. Goldstein, J.A. Goldstein, D. Guidetti and S. Romanelli, General Wentzell boundary conditions in $L^p$ spaces,, work in progress., ().   Google Scholar

[6]

D. Guidetti, On elliptic problems in Besov spaces,, \emph{Math. Nachr., 152 (1991), 247.  doi: 10.1002/mana.19911520120.  Google Scholar

[7]

D. Guidetti, On interpolation with boundary conditions,, \emph{Math. Z., 207 (1991), 439.  doi: 10.1007/BF02571401.  Google Scholar

[8]

D. Guidetti, Abstract elliptic problems depending on a parameter and parabolic problems with dynamic boundary conditions,, Chapter 9, 10 (2014), 161.  doi: 10.1007/978-3-319-11406-4_9.  Google Scholar

[9]

D. Guidetti, Linear parabolic problems with dynamic boundary conditions in spaces of Hölder continuous functions,, \emph{Ann. Mat. Pura Appl., (2016), 167.  doi: 10.1007/s10231-014-0457-8.  Google Scholar

[10]

D. Guidetti, Classical solutions to quasilinear parabolic problems with dynamic boundary conditions,, preprint, ().   Google Scholar

[11]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkh\, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[12]

P. Sacks and M. Warma, Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1-$data,, \emph{Discrete Cont. Dyn. Syst.}, (2014), 761.   Google Scholar

[13]

J.L. Vazquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type,, \emph{J. Diff. Eq.}, 250 (2011), 2143.  doi: 10.1016/j.jde.2010.12.012.  Google Scholar

[14]

M. Warma, Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 1881.  doi: 10.3934/cpaa.2013.12.1881.  Google Scholar

[15]

M. Warma, Semi linear parabolic equations with nonlinear general Wentzell boundary conditions,, \emph{Discrete Cont. Dyn. Systems}, 33 (2013), 5493.  doi: 10.3934/dcds.2013.33.5493.  Google Scholar

show all references

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,, \emph{Commun. Pure Appl. Anal., 15 (2011), 313.   Google Scholar

[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,, \emph{Semigroup Forum, 77 (2008), 101.  doi: 10.1007/s00233-008-9068-2.  Google Scholar

[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,, \emph{Math. Nachr., 283 (2010), 504.  doi: 10.1002/mana.200910086.  Google Scholar

[4]

G.R. Goldstein, Derivation and physical interpretation of general boundary conditions,, \emph{Adv. Diff. Eq., 11 (2006), 457.   Google Scholar

[5]

G.R. Goldstein, J.A. Goldstein, D. Guidetti and S. Romanelli, General Wentzell boundary conditions in $L^p$ spaces,, work in progress., ().   Google Scholar

[6]

D. Guidetti, On elliptic problems in Besov spaces,, \emph{Math. Nachr., 152 (1991), 247.  doi: 10.1002/mana.19911520120.  Google Scholar

[7]

D. Guidetti, On interpolation with boundary conditions,, \emph{Math. Z., 207 (1991), 439.  doi: 10.1007/BF02571401.  Google Scholar

[8]

D. Guidetti, Abstract elliptic problems depending on a parameter and parabolic problems with dynamic boundary conditions,, Chapter 9, 10 (2014), 161.  doi: 10.1007/978-3-319-11406-4_9.  Google Scholar

[9]

D. Guidetti, Linear parabolic problems with dynamic boundary conditions in spaces of Hölder continuous functions,, \emph{Ann. Mat. Pura Appl., (2016), 167.  doi: 10.1007/s10231-014-0457-8.  Google Scholar

[10]

D. Guidetti, Classical solutions to quasilinear parabolic problems with dynamic boundary conditions,, preprint, ().   Google Scholar

[11]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkh\, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[12]

P. Sacks and M. Warma, Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1-$data,, \emph{Discrete Cont. Dyn. Syst.}, (2014), 761.   Google Scholar

[13]

J.L. Vazquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type,, \emph{J. Diff. Eq.}, 250 (2011), 2143.  doi: 10.1016/j.jde.2010.12.012.  Google Scholar

[14]

M. Warma, Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 1881.  doi: 10.3934/cpaa.2013.12.1881.  Google Scholar

[15]

M. Warma, Semi linear parabolic equations with nonlinear general Wentzell boundary conditions,, \emph{Discrete Cont. Dyn. Systems}, 33 (2013), 5493.  doi: 10.3934/dcds.2013.33.5493.  Google Scholar

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