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June  2016, 9(3): 697-715. doi: 10.3934/dcdss.2016023

## Generalized Wentzell boundary conditions for second order operators with interior degeneracy

 1 Department of Mathematics, University of Bari Aldo Moro, Via E.Orabona 4, 70125 Bari, Italy 2 The University of Memphis, Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152-3240 3 Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240, United States 4 Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, via E. Orabona 4, 70125 Bari

Received  April 2015 Revised  September 2015 Published  April 2016

We consider operators in divergence form, $A_1u=(au')'$, and in nondivergence form, $A_2u=au''$, provided that the coefficient $a$ vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators $A_1$ and $A_2$, equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of $L^2$ type.
Citation: Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023
##### References:
 [1] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems,, Monographs in Mathematics, 96 (2001). doi: 10.1007/978-3-0348-5075-9. Google Scholar [2] J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constant formula,, Proc. Amer. Math. Soc., 63 (1977), 370. Google Scholar [3] G. I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy,, Electron. J. Differential Equations, 2014 (2014), 1. Google Scholar [4] G. I. Boutaayamou, G. Fragnelli and L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions,, J. Anal. Math., (). Google Scholar [5] G. I. Boutaayamou, G. Fragnelli and L. Maniar, Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions,, J. Inverse Ill-Posed Probl, (2015). doi: 10.1515/jiip-2014-0032. Google Scholar [6] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, 13 (1998). Google Scholar [7] G. M. Coclite, A. Favini, C. G. Gal, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, The role of Wentzell boundary conditions in linear and nonlinear analysis,, in Advances in nonlinear analysis: Theory, 3 (2009), 277. Google Scholar [8] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary conditions,, J. Evol. Equ., 2 (2002), 1. doi: 10.1007/s00028-002-8077-y. Google Scholar [9] G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of $L^2$ type,, Electron. J. Differential Equations, 2012 (2012), 1. Google Scholar [10] G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, A control approach for an identification problem associated to a strongly degenerate parabolic system with interior degeneracy,, in: New Prospects in direct, 10 (2014), 121. doi: 10.1007/978-3-319-11406-4_7. Google Scholar [11] G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy,, J. Evol. Equ., 15 (2015), 27. doi: 10.1007/s00028-014-0247-1. Google Scholar [12] G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy,, Adv. Nonlinear Anal., 2 (2013), 339. doi: 10.1515/anona-2013-0015. Google Scholar [13] G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations,, Mem. Amer. Math. Soc., 242 (2016). doi: 10.1090/memo/1146. Google Scholar [14] G. R. Goldstein, Derivation and physical interpretation of general Wentzell boundary conditions,, Adv. Differential Equations, 11 (2006), 457. Google Scholar [15] J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Univ. Press, (1985). Google Scholar [16] A. Stahel, Degenerate semilinear parabolic equations,, Differential Integral Equations, 5 (1992), 683. Google Scholar

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##### References:
 [1] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems,, Monographs in Mathematics, 96 (2001). doi: 10.1007/978-3-0348-5075-9. Google Scholar [2] J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constant formula,, Proc. Amer. Math. Soc., 63 (1977), 370. Google Scholar [3] G. I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy,, Electron. J. Differential Equations, 2014 (2014), 1. Google Scholar [4] G. I. Boutaayamou, G. Fragnelli and L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions,, J. Anal. Math., (). Google Scholar [5] G. I. Boutaayamou, G. Fragnelli and L. Maniar, Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions,, J. Inverse Ill-Posed Probl, (2015). doi: 10.1515/jiip-2014-0032. Google Scholar [6] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, 13 (1998). Google Scholar [7] G. M. Coclite, A. Favini, C. G. Gal, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, The role of Wentzell boundary conditions in linear and nonlinear analysis,, in Advances in nonlinear analysis: Theory, 3 (2009), 277. Google Scholar [8] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary conditions,, J. Evol. Equ., 2 (2002), 1. doi: 10.1007/s00028-002-8077-y. Google Scholar [9] G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of $L^2$ type,, Electron. J. Differential Equations, 2012 (2012), 1. Google Scholar [10] G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, A control approach for an identification problem associated to a strongly degenerate parabolic system with interior degeneracy,, in: New Prospects in direct, 10 (2014), 121. doi: 10.1007/978-3-319-11406-4_7. Google Scholar [11] G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy,, J. Evol. Equ., 15 (2015), 27. doi: 10.1007/s00028-014-0247-1. Google Scholar [12] G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy,, Adv. Nonlinear Anal., 2 (2013), 339. doi: 10.1515/anona-2013-0015. Google Scholar [13] G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations,, Mem. Amer. Math. Soc., 242 (2016). doi: 10.1090/memo/1146. Google Scholar [14] G. R. Goldstein, Derivation and physical interpretation of general Wentzell boundary conditions,, Adv. Differential Equations, 11 (2006), 457. Google Scholar [15] J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Univ. Press, (1985). Google Scholar [16] A. Stahel, Degenerate semilinear parabolic equations,, Differential Integral Equations, 5 (1992), 683. Google Scholar
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