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

[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.

[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.

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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.

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[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).

[16]

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

[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.

[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).

[19]

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

[20]

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

[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.

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[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).

[16]

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

[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.

[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).

[19]

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

[20]

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

[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.

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[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.

[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.

[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.

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