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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.  Google Scholar

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura Appl., 171 (1996), 41.  doi: 10.1007/BF01759381.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1989).  doi: 10.1017/CBO9780511566158.  Google Scholar

[8]

E. De Giorgi, Sulla differenziabilità e analiticità delle estremali degli integral multipli regolari,, Men. Accad. Sci. Torino, 3 (1957), 25.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions,, Adv. Differential Equations, 11 (2006), 457.   Google Scholar

[18]

D. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains,, MAA Stud. Math., 23 (1982), 1.   Google Scholar

[19]

J. Jost, "Riemannian Geometry and Geometric Analysis,", Fifth edition. Universitext. Springer-Verlag, (2008).  doi: 10.1007/978-3-642-21298-7.  Google Scholar

[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).   Google Scholar

[21]

J. Maly and W. P. Ziemer, "Fine Regurality of Solutions of Elliptic Partial Differential Equations,", Providence, (1997).   Google Scholar

[22]

C. B. Morrey Jr, Second order elliptic equations in several variables and Hölder continuity,, Math. Z., 72 (1959), 146.  doi: 10.1007/BF01162944.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

El M. Ouhabaz, "Analysis of Heat Equations on Domains,", Lond. Math. Soc. Monographs Series, (2005).   Google Scholar

[27]

R. S. Phillips, The adjoint semi-group,, Pacific J. Math., 5 (1955), 269.  doi: 10.2140/pjm.1955.5.269.  Google Scholar

[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.  Google Scholar

[29]

M. E. Taylor, "Partial Differential Equations. I. Basic Theory,", Texts Appl. Math., 23 (1996).  doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[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.  Google Scholar

[31]

M. Warma, "The Laplacian with General Robin Boundary Conditions,", Ph.D Dissertation, (2002).   Google Scholar

[32]

M. Warma, Wentzell-Robin boundary conditions on $C[0,1]$,, Semigroup Forum, 66 (2003), 162.  doi: 10.1007/s002330010124.  Google Scholar

[33]

M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains,, Semigroup Forum, 73 (2006), 10.  doi: 10.1007/s00233-006-0617-2.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura Appl., 171 (1996), 41.  doi: 10.1007/BF01759381.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1989).  doi: 10.1017/CBO9780511566158.  Google Scholar

[8]

E. De Giorgi, Sulla differenziabilità e analiticità delle estremali degli integral multipli regolari,, Men. Accad. Sci. Torino, 3 (1957), 25.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions,, Adv. Differential Equations, 11 (2006), 457.   Google Scholar

[18]

D. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains,, MAA Stud. Math., 23 (1982), 1.   Google Scholar

[19]

J. Jost, "Riemannian Geometry and Geometric Analysis,", Fifth edition. Universitext. Springer-Verlag, (2008).  doi: 10.1007/978-3-642-21298-7.  Google Scholar

[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).   Google Scholar

[21]

J. Maly and W. P. Ziemer, "Fine Regurality of Solutions of Elliptic Partial Differential Equations,", Providence, (1997).   Google Scholar

[22]

C. B. Morrey Jr, Second order elliptic equations in several variables and Hölder continuity,, Math. Z., 72 (1959), 146.  doi: 10.1007/BF01162944.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

El M. Ouhabaz, "Analysis of Heat Equations on Domains,", Lond. Math. Soc. Monographs Series, (2005).   Google Scholar

[27]

R. S. Phillips, The adjoint semi-group,, Pacific J. Math., 5 (1955), 269.  doi: 10.2140/pjm.1955.5.269.  Google Scholar

[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.  Google Scholar

[29]

M. E. Taylor, "Partial Differential Equations. I. Basic Theory,", Texts Appl. Math., 23 (1996).  doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[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.  Google Scholar

[31]

M. Warma, "The Laplacian with General Robin Boundary Conditions,", Ph.D Dissertation, (2002).   Google Scholar

[32]

M. Warma, Wentzell-Robin boundary conditions on $C[0,1]$,, Semigroup Forum, 66 (2003), 162.  doi: 10.1007/s002330010124.  Google Scholar

[33]

M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains,, Semigroup Forum, 73 (2006), 10.  doi: 10.1007/s00233-006-0617-2.  Google Scholar

[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.  Google Scholar

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