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September  2013, 12(5): 1881-1905. doi: 10.3934/cpaa.2013.12.1881

## Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains

 1 University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377, United States

Received  April 2011 Revised  September 2012 Published  January 2013

We show that on a bounded domain $\Omega\subset R^N$ with Lipschitz continuous boundary $\partial \Omega$, weak solutions of the elliptic equation $\lambda u-Au=f$ in $\Omega$ with the boundary conditions $-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ are globally Hölder continuous on $\bar \Omega$. Here $A$ is a uniformly elliptic operator in divergence form with bounded measurable coefficients, $\Delta_\Gamma$ is the Laplace-Beltrami operator on $\partial \Omega$, $\partial_\nu^a u$ denotes the conormal derivative of $u$, $\lambda,\gamma>0$ are real numbers and $\beta$ is a bounded measurable function on $\partial Omega$. We also obtain that a realization of the operator $A$ in $C(\bar \Omega)$ with the general Wentzell boundary conditions $(Au)|_{\partial \Omega}-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ generates a strongly continuous compact semigroup. Some analyticity results of the semigroup are also discussed.
Citation: Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881
##### References:
 [1] H. Amann, Dual semigroup and second order linear elliptic boundary value problems,, Israel J. Math., 45 (1983), 225. doi: 10.1007/BF02774019. [2] H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura Appl., 171 (1996), 41. doi: 10.1007/BF01759381. [3] W. Arendt, G. Metafune, D. Pallara and S. Romanelli, The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions,, Semigroup Forum, 67 (2003), 247. doi: 10.1007/s00233-002-0010-8. [4] R. F. Bass and P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains,, Ann. Probab., 19 (1991), 486. doi: 10.1214/aop/1176990437. [5] M. Biegert and M. Warma, The heat equation with nonlinear generalized Robin boundary conditions,, J. Differential Equations, 247 (2009), 1949. doi: 10.1016/j.jde.2009.07.017. [6] G. M. Coclite, G. R. Goldstein and J. A. Goldstein, Stability of parabolic problems with nonlinear Wentzell boundary conditions,, J. Differential Equations, 246 (2009), 2434. doi: 10.1016/j.jde.2008.10.004. [7] E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1989). doi: 10.1017/CBO9780511566158. [8] E. De Giorgi, Sulla differenziabilità e analiticità delle estremali degli integral multipli regolari,, Men. Accad. Sci. Torino, 3 (1957), 25. [9] K. J. Engel, The Laplacian on $C(\bar \Omega)$ with generalized Wentzell boundary conditions,, Arch. Math. (Basel), 81 (2003), 548. doi: 10.1007/s00013-003-0557-y. [10] 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. doi: 10.1002/mana.200910086. [11] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with Wentzell boundary conditions,, J. Evol. Eq., 2 (2002), 1. doi: 10.1007/s00028-002-8077-y. [12] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with nonlinear general Wentzell boundary condition,, Adv. Differential Equations, 11 (2006), 481. [13] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Classification of general Wentzell boundary conditions for fourth order operators in one space dimension,, J. Math. Anal. Appl., 333 (2007), 219. doi: 10.1016/j.jmaa.2006.11.058. [14] M. Fukushima and M. Tomisaki, Reflecting diffusions on Lipschitz domains with cusps: Analytic construction and Skorohod representation,, Potential Anal., 4 (1995), 377. doi: 10.1007/BF01053454. [15] M. Fukushima and M. Tomisaki, Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps,, Probab. Theory Relat. Fields, 106 (1996), 521. doi: 10.1007/s004400050074. [16] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (2001). doi: 10.1007/978-3-642-61798-0. [17] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions,, Adv. Differential Equations, 11 (2006), 457. [18] D. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains,, MAA Stud. Math., 23 (1982), 1. [19] J. Jost, "Riemannian Geometry and Geometric Analysis,", Fifth edition. Universitext. Springer-Verlag, (2008). doi: 10.1007/978-3-642-21298-7. [20] O. A. Ladyzhenskaya and N. N. Ural鈥檛seva, "Linear and Quasilinear Elliptic Equations,", Mathematics in Science and Engineering. 46. New York-London: Academic Press, (1968). [21] J. Maly and W. P. Ziemer, "Fine Regurality of Solutions of Elliptic Partial Differential Equations,", Providence, (1997). [22] C. B. Morrey Jr, Second order elliptic equations in several variables and Hölder continuity,, Math. Z., 72 (1959), 146. doi: 10.1007/BF01162944. [23] J. Moser, A new proof of the de Giorgi's theorem concerning the regularity problem for elliptic differential equations,, Commu. Pure Appl. Math., 13 (1960), 457. doi: 10.1002/cpa.3160130308. [24] M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators,, Ann. Mat. Pura Appl., 80 (1968), 1. doi: 10.1007/BF02413623. [25] R. Nittka, Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains,, J. Differential Equations, 251 (2011), 860. doi: 10.1016/j.jde.2011.05.019. [26] El M. Ouhabaz, "Analysis of Heat Equations on Domains,", Lond. Math. Soc. Monographs Series, (2005). [27] R. S. Phillips, The adjoint semi-group,, Pacific J. Math., 5 (1955), 269. doi: 10.2140/pjm.1955.5.269. [28] G. Stampacchia, Problemi al contorno ellittici con dati discontinui dotati di soluzioni Hölderiane,, Ann. Mat. Pura Appl., 51 (1960), 1. doi: 10.1007/BF02410941. [29] M. E. Taylor, "Partial Differential Equations. I. Basic Theory,", Texts Appl. Math., 23 (1996). doi: 10.1007/978-1-4684-9320-7. [30] J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type,, J. Differential Equations, 250 (2011), 2143. doi: 10.1016/j.jde.2010.12.012. [31] M. Warma, "The Laplacian with General Robin Boundary Conditions,", Ph.D Dissertation, (2002). [32] M. Warma, Wentzell-Robin boundary conditions on $C[0,1]$,, Semigroup Forum, 66 (2003), 162. doi: 10.1007/s002330010124. [33] M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains,, Semigroup Forum, 73 (2006), 10. doi: 10.1007/s00233-006-0617-2. [34] M. Warma, Analyticity on $L^1$ of the heat semigroup with Wentzell boundary conditions,, Arch. Math. (Basel), 94 (2010), 85. doi: 10.1007/s00013-009-0068-6.

show all references

##### References:
 [1] H. Amann, Dual semigroup and second order linear elliptic boundary value problems,, Israel J. Math., 45 (1983), 225. doi: 10.1007/BF02774019. [2] H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura Appl., 171 (1996), 41. doi: 10.1007/BF01759381. [3] W. Arendt, G. Metafune, D. Pallara and S. Romanelli, The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions,, Semigroup Forum, 67 (2003), 247. doi: 10.1007/s00233-002-0010-8. [4] R. F. Bass and P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains,, Ann. Probab., 19 (1991), 486. doi: 10.1214/aop/1176990437. [5] M. Biegert and M. Warma, The heat equation with nonlinear generalized Robin boundary conditions,, J. Differential Equations, 247 (2009), 1949. doi: 10.1016/j.jde.2009.07.017. [6] G. M. Coclite, G. R. Goldstein and J. A. Goldstein, Stability of parabolic problems with nonlinear Wentzell boundary conditions,, J. Differential Equations, 246 (2009), 2434. doi: 10.1016/j.jde.2008.10.004. [7] E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1989). doi: 10.1017/CBO9780511566158. [8] E. De Giorgi, Sulla differenziabilità e analiticità delle estremali degli integral multipli regolari,, Men. Accad. Sci. Torino, 3 (1957), 25. [9] K. J. Engel, The Laplacian on $C(\bar \Omega)$ with generalized Wentzell boundary conditions,, Arch. Math. (Basel), 81 (2003), 548. doi: 10.1007/s00013-003-0557-y. [10] 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. doi: 10.1002/mana.200910086. [11] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with Wentzell boundary conditions,, J. Evol. Eq., 2 (2002), 1. doi: 10.1007/s00028-002-8077-y. [12] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with nonlinear general Wentzell boundary condition,, Adv. Differential Equations, 11 (2006), 481. [13] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Classification of general Wentzell boundary conditions for fourth order operators in one space dimension,, J. Math. Anal. Appl., 333 (2007), 219. doi: 10.1016/j.jmaa.2006.11.058. [14] M. Fukushima and M. Tomisaki, Reflecting diffusions on Lipschitz domains with cusps: Analytic construction and Skorohod representation,, Potential Anal., 4 (1995), 377. doi: 10.1007/BF01053454. [15] M. Fukushima and M. Tomisaki, Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps,, Probab. Theory Relat. Fields, 106 (1996), 521. doi: 10.1007/s004400050074. [16] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (2001). doi: 10.1007/978-3-642-61798-0. [17] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions,, Adv. Differential Equations, 11 (2006), 457. [18] D. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains,, MAA Stud. Math., 23 (1982), 1. [19] J. Jost, "Riemannian Geometry and Geometric Analysis,", Fifth edition. Universitext. Springer-Verlag, (2008). doi: 10.1007/978-3-642-21298-7. [20] O. A. Ladyzhenskaya and N. N. Ural鈥檛seva, "Linear and Quasilinear Elliptic Equations,", Mathematics in Science and Engineering. 46. New York-London: Academic Press, (1968). [21] J. Maly and W. P. Ziemer, "Fine Regurality of Solutions of Elliptic Partial Differential Equations,", Providence, (1997). [22] C. B. Morrey Jr, Second order elliptic equations in several variables and Hölder continuity,, Math. Z., 72 (1959), 146. doi: 10.1007/BF01162944. [23] J. Moser, A new proof of the de Giorgi's theorem concerning the regularity problem for elliptic differential equations,, Commu. Pure Appl. Math., 13 (1960), 457. doi: 10.1002/cpa.3160130308. [24] M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators,, Ann. Mat. Pura Appl., 80 (1968), 1. doi: 10.1007/BF02413623. [25] R. Nittka, Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains,, J. Differential Equations, 251 (2011), 860. doi: 10.1016/j.jde.2011.05.019. [26] El M. Ouhabaz, "Analysis of Heat Equations on Domains,", Lond. Math. Soc. Monographs Series, (2005). [27] R. S. Phillips, The adjoint semi-group,, Pacific J. Math., 5 (1955), 269. doi: 10.2140/pjm.1955.5.269. [28] G. Stampacchia, Problemi al contorno ellittici con dati discontinui dotati di soluzioni Hölderiane,, Ann. Mat. Pura Appl., 51 (1960), 1. doi: 10.1007/BF02410941. [29] M. E. Taylor, "Partial Differential Equations. I. Basic Theory,", Texts Appl. Math., 23 (1996). doi: 10.1007/978-1-4684-9320-7. [30] J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type,, J. Differential Equations, 250 (2011), 2143. doi: 10.1016/j.jde.2010.12.012. [31] M. Warma, "The Laplacian with General Robin Boundary Conditions,", Ph.D Dissertation, (2002). [32] M. Warma, Wentzell-Robin boundary conditions on $C[0,1]$,, Semigroup Forum, 66 (2003), 162. doi: 10.1007/s002330010124. [33] M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains,, Semigroup Forum, 73 (2006), 10. doi: 10.1007/s00233-006-0617-2. [34] M. Warma, Analyticity on $L^1$ of the heat semigroup with Wentzell boundary conditions,, Arch. Math. (Basel), 94 (2010), 85. doi: 10.1007/s00013-009-0068-6.
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