February  2014, 34(2): 761-787. doi: 10.3934/dcds.2014.34.761

Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data

1. 

Iowa State University, Department of Mathematics, 396 Carver Hall, Ames, IA 50011, United States

2. 

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

Received  January 2013 Revised  May 2013 Published  August 2013

Let $Ω\subset\mathbb{R}^N$ ($N\ge 2$) be a bounded domain with a boundary $∂Ω$ of class $C^2$ and let $\alpha,\beta$ be maximal monotone graphs in $\mathbb{R}^2$ satisfying $\alpha(0)\cap\beta(0)\ni 0$. Given $f\in L^1(Ω)$ and $g\in L^1(∂Ω)$, we characterize the existence and uniqueness of weak solutions to the semi-linear elliptic equation $-\Delta u+\alpha(u)\ni f$ in $Ω$ with the nonlinear general Wentzell boundary conditions $-\Delta_{\Gamma} u+\frac{\partial u}{\partial\nu}+\beta(u)\ni g$ on $∂Ω$. We also show the well-posedness of the associated parabolic problem on the Banach space $L^1(Ω)\times L^1(∂Ω)$.
Citation: Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761
References:
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T. Aiki, Multi-dimensional two-phase Stefan problems with nonlinear dynamic boundary conditions,, in, 7 (1996), 1.   Google Scholar

[2]

F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions,, Adv. Math. Sci. Appl., 7 (1997), 183.   Google Scholar

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F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, $L^ 1$ existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 61.  doi: 10.1016/j.anihpc.2005.09.009.  Google Scholar

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F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Existence and uniqueness for a degenerate parabolic equation with $L^1$-data,, Trans. Amer. Math. Soc., 351 (1999), 285.  doi: 10.1090/S0002-9947-99-01981-9.  Google Scholar

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Ph. Bénilan, H. Brezis and M. G. Crandall, A semilinear equation in $L^1(\mathbbR^N)$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 523.   Google Scholar

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Ph. Bénilan and M. G. Crandall, Completely accretive operators,, in, 135 (1991), 41.   Google Scholar

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Ph. Bénilan, M. G. Crandall and P. Sacks, Some $L^1$ existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions,, Appl. Math. Optim., 17 (1988), 203.  doi: 10.1007/BF01448367.  Google Scholar

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H. Brézis, Problémes unilatéraux,, J. Math. Pures Appl. (9), 51 (1972), 1.   Google Scholar

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H. Brézis and A. Haraux, Image d'une somme d'opérateurs monotones et applications,, Israel J. Math., 23 (1976), 165.  doi: 10.1007/BF02756796.  Google Scholar

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H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.  doi: 10.2969/jmsj/02540565.  Google Scholar

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M. G. Crandall, An introduction to evolution governed by accretive operators,, in, (1976), 131.   Google Scholar

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M. G. Crandall, Nonlinear semigroups and evolution governed by accretive operators,, in, 45 (1986), 305.   Google Scholar

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J. Crank, "Free and Moving Boundary Problems,", The Clarendon Press, (1987).   Google Scholar

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R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Sciences and Technology. Vol. 1. Physical Origins and Classical Methods,", Springer-Verlag, (1990).   Google Scholar

[16]

E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Tracts in Mathematics, 92 (1989).  doi: 10.1017/CBO9780511566158.  Google Scholar

[17]

E. DiBenedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water,, Trans. Amer. Math. Soc., 282 (1984), 183.  doi: 10.2307/1999584.  Google Scholar

[18]

P. Drábek and J. Milota, "Methods of Nonlinear Analysis. Applications to Differential Equations,", Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], (2007).   Google Scholar

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G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,", Grundlehren der Mathematischen Wissenschaften, 219 (1976).   Google Scholar

[20]

L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations,, in, 40 (1978), 163.   Google Scholar

[21]

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

[22]

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

[23]

C. G. Gal, G. Goldstein, J. A. Goldstein, S. Romanelli and M. Warma, Fredholm alternative, semilinear elliptic problems, and Wentzell boundary conditions,, preprint., ().   Google Scholar

[24]

C. G. Gal and M. Warma, Nonlinear elliptic boundary value problems at resonance with nonlinear Wentzell-Robin type boundary conditions,, preprint, ().   Google Scholar

[25]

N. Igbida and M. Kirane, A degenerate diffusion problem with dynamical boundary conditions,, Math. Ann., 323 (2002), 377.  doi: 10.1007/s002080100308.  Google Scholar

[26]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and their Applications,", Pure and Applied Mathematics, 88 (1980).   Google Scholar

[27]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Mathematical Surveys and Monographs, 49 (1997).   Google Scholar

[28]

M. Warma, An ultracontractivity property for semigroups generated by the $p$-Laplacian with nonlinear Wentzell-Robin boundary conditions,, Adv. Differential Equations, 14 (2009), 771.   Google Scholar

[29]

M. Warma, Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains,, Nonlinear Analysis, 75 (2012), 5561.  doi: 10.1016/j.na.2012.05.004.  Google Scholar

[30]

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

[31]

M. Warma, Semi linear parabolic equations with nonlinear general Wentzell boundary conditions,, Discrete Contin. Dynam. Systems, 33 (2013), 5493.  doi: 10.3934/dcds.2013.33.5493.  Google Scholar

show all references

References:
[1]

T. Aiki, Multi-dimensional two-phase Stefan problems with nonlinear dynamic boundary conditions,, in, 7 (1996), 1.   Google Scholar

[2]

F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions,, Adv. Math. Sci. Appl., 7 (1997), 183.   Google Scholar

[3]

F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions,, Interfaces Free Bound. 8 (2006), 8 (2006), 447.  doi: 10.4171/IFB/151.  Google Scholar

[4]

F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, $L^ 1$ existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 61.  doi: 10.1016/j.anihpc.2005.09.009.  Google Scholar

[5]

F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Existence and uniqueness for a degenerate parabolic equation with $L^1$-data,, Trans. Amer. Math. Soc., 351 (1999), 285.  doi: 10.1090/S0002-9947-99-01981-9.  Google Scholar

[6]

Ph. Bénilan, H. Brezis and M. G. Crandall, A semilinear equation in $L^1(\mathbbR^N)$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 523.   Google Scholar

[7]

Ph. Bénilan and M. G. Crandall, Completely accretive operators,, in, 135 (1991), 41.   Google Scholar

[8]

Ph. Bénilan, M. G. Crandall and P. Sacks, Some $L^1$ existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions,, Appl. Math. Optim., 17 (1988), 203.  doi: 10.1007/BF01448367.  Google Scholar

[9]

H. Brézis, Problémes unilatéraux,, J. Math. Pures Appl. (9), 51 (1972), 1.   Google Scholar

[10]

H. Brézis and A. Haraux, Image d'une somme d'opérateurs monotones et applications,, Israel J. Math., 23 (1976), 165.  doi: 10.1007/BF02756796.  Google Scholar

[11]

H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.  doi: 10.2969/jmsj/02540565.  Google Scholar

[12]

M. G. Crandall, An introduction to evolution governed by accretive operators,, in, (1976), 131.   Google Scholar

[13]

M. G. Crandall, Nonlinear semigroups and evolution governed by accretive operators,, in, 45 (1986), 305.   Google Scholar

[14]

J. Crank, "Free and Moving Boundary Problems,", The Clarendon Press, (1987).   Google Scholar

[15]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Sciences and Technology. Vol. 1. Physical Origins and Classical Methods,", Springer-Verlag, (1990).   Google Scholar

[16]

E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Tracts in Mathematics, 92 (1989).  doi: 10.1017/CBO9780511566158.  Google Scholar

[17]

E. DiBenedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water,, Trans. Amer. Math. Soc., 282 (1984), 183.  doi: 10.2307/1999584.  Google Scholar

[18]

P. Drábek and J. Milota, "Methods of Nonlinear Analysis. Applications to Differential Equations,", Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], (2007).   Google Scholar

[19]

G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,", Grundlehren der Mathematischen Wissenschaften, 219 (1976).   Google Scholar

[20]

L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations,, in, 40 (1978), 163.   Google Scholar

[21]

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

[22]

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

[23]

C. G. Gal, G. Goldstein, J. A. Goldstein, S. Romanelli and M. Warma, Fredholm alternative, semilinear elliptic problems, and Wentzell boundary conditions,, preprint., ().   Google Scholar

[24]

C. G. Gal and M. Warma, Nonlinear elliptic boundary value problems at resonance with nonlinear Wentzell-Robin type boundary conditions,, preprint, ().   Google Scholar

[25]

N. Igbida and M. Kirane, A degenerate diffusion problem with dynamical boundary conditions,, Math. Ann., 323 (2002), 377.  doi: 10.1007/s002080100308.  Google Scholar

[26]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and their Applications,", Pure and Applied Mathematics, 88 (1980).   Google Scholar

[27]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Mathematical Surveys and Monographs, 49 (1997).   Google Scholar

[28]

M. Warma, An ultracontractivity property for semigroups generated by the $p$-Laplacian with nonlinear Wentzell-Robin boundary conditions,, Adv. Differential Equations, 14 (2009), 771.   Google Scholar

[29]

M. Warma, Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains,, Nonlinear Analysis, 75 (2012), 5561.  doi: 10.1016/j.na.2012.05.004.  Google Scholar

[30]

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

[31]

M. Warma, Semi linear parabolic equations with nonlinear general Wentzell boundary conditions,, Discrete Contin. Dynam. Systems, 33 (2013), 5493.  doi: 10.3934/dcds.2013.33.5493.  Google Scholar

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