# American Institute of Mathematical Sciences

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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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

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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