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2015, 2015(special): 379-386. doi: 10.3934/proc.2015.0379

Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem

J.I. Díaz 1, and  D. Gómez-Castro 1,

1. 

Instituto de Matemática Interdisciplinar and Dpto. de Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias, 3, 28040 Madrid, Spain, Spain

Received  September 2014 Revised  January 2015 Published  November 2015

We extend some previous results in the literature on the Steiner rearrangement of linear second order elliptic equations to the semilinear concave parabolic problems and the obstacle problem.
Keywords: Trotter-Kato formula, obstacle problem., Semilinear concave parabolic equations, Steiner rearrangement.
Mathematics Subject Classification: 35J61, 35J65, 35J4.
Citation: J.I. Díaz, D. Gómez-Castro. Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. Conference Publications, 2015, 2015 (special) : 379-386. doi: 10.3934/proc.2015.0379
References:
[1]

A. Alvino, J.I. Díaz, P.L. Lions and G.Trombetti, Elliptic Equations and Steiner Symmetrization, Communications on Pure and Applied Mathematics, Vol. XLIX (1996), 217-236.

[2]

H. Attouch, H. and A. Damlamian, Problemes d'evolution dans les Hilbert et applications, J. Math. Pures Appl., 54 (1975), 53-74.

[3]

C. Bandle, Isoperimetric Inequalities and Applications, Pitman, London, 1980.

[4]

P. Benilan, M. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces., Book in preparation., (). 

[5]

H. Brezis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. Notes de Matematica, 5, North-Holland, Amsterdam. 1973.

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.

[7]

F. Chiacchio and V.M. Monetti, Comparison results for solutions of elliptic problems via Steiner symmetrization. Differential and Integral Equations 14(11) (2001), 1351-1366.

[8]

F. Chiacchio, Steiner symmetrization for an elliptic problem with lower-order terms. Ricerche di Matematica, 53, 1, (2004) 87-106.

[9]

F. Chiacchio, Estimates for the first eigenfunction of linear eigenvalue problems via Steiner symmetrization, Publ. Mat., 1 (2009), 47-71.

[10]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Pitman, London, 1985.

[11]

J.I. Díaz, Simetrización de problemas parablicos no lineales: Aplicación a ecuaciones de reacción difusión. Memorias de la Real Acad. de Ciencias Exactas, Físicas y Naturales, Tomo XXVII. 1991.

[12]

J.I. Díaz and D. Gómez-Castro, On the effectiveness of wastewater cylindrical reactors: an analysis through Steiner symmetrization. Pure and Applied Geophysics. DOI: 10.1007/s00024-015-1124-8 (2015).

[13]

V. Ferone and A. Mercaldo, A second order derivation formula for functions defined by integrals, C.R. Acad. Sci. Paris, 236, Série I, (1998) 549-554.

[14]

J. Mossino, and J.M. Rakotoson, Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1986), 51-73.

[15]

T. Nagai, Global existence and decay estimates of solutions to a parabolic-elliptic system of drift-diffusion type in $\mathbb{ R}^{2}$, Differential Integral Equations 24(1/2) (2011), 29-68.

[16]

P.-A. Vuillermot, W.F. Wreszinski and V.A.Zagrebnov, A Trotter-Kato Product Formula for a Class of Non-Autonomous Evolution Equations, Trends in Nonlinear Analysis: in Honour of Professor V. Lakshmikantham, Nonlinear Analysis, Theory, Methods and Applications 69 (2008), 1067-1072.

show all references

References:
[1]

A. Alvino, J.I. Díaz, P.L. Lions and G.Trombetti, Elliptic Equations and Steiner Symmetrization, Communications on Pure and Applied Mathematics, Vol. XLIX (1996), 217-236.

[2]

H. Attouch, H. and A. Damlamian, Problemes d'evolution dans les Hilbert et applications, J. Math. Pures Appl., 54 (1975), 53-74.

[3]

C. Bandle, Isoperimetric Inequalities and Applications, Pitman, London, 1980.

[4]

P. Benilan, M. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces., Book in preparation., (). 

[5]

H. Brezis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. Notes de Matematica, 5, North-Holland, Amsterdam. 1973.

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.

[7]

F. Chiacchio and V.M. Monetti, Comparison results for solutions of elliptic problems via Steiner symmetrization. Differential and Integral Equations 14(11) (2001), 1351-1366.

[8]

F. Chiacchio, Steiner symmetrization for an elliptic problem with lower-order terms. Ricerche di Matematica, 53, 1, (2004) 87-106.

[9]

F. Chiacchio, Estimates for the first eigenfunction of linear eigenvalue problems via Steiner symmetrization, Publ. Mat., 1 (2009), 47-71.

[10]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Pitman, London, 1985.

[11]

J.I. Díaz, Simetrización de problemas parablicos no lineales: Aplicación a ecuaciones de reacción difusión. Memorias de la Real Acad. de Ciencias Exactas, Físicas y Naturales, Tomo XXVII. 1991.

[12]

J.I. Díaz and D. Gómez-Castro, On the effectiveness of wastewater cylindrical reactors: an analysis through Steiner symmetrization. Pure and Applied Geophysics. DOI: 10.1007/s00024-015-1124-8 (2015).

[13]

V. Ferone and A. Mercaldo, A second order derivation formula for functions defined by integrals, C.R. Acad. Sci. Paris, 236, Série I, (1998) 549-554.

[14]

J. Mossino, and J.M. Rakotoson, Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1986), 51-73.

[15]

T. Nagai, Global existence and decay estimates of solutions to a parabolic-elliptic system of drift-diffusion type in $\mathbb{ R}^{2}$, Differential Integral Equations 24(1/2) (2011), 29-68.

[16]

P.-A. Vuillermot, W.F. Wreszinski and V.A.Zagrebnov, A Trotter-Kato Product Formula for a Class of Non-Autonomous Evolution Equations, Trends in Nonlinear Analysis: in Honour of Professor V. Lakshmikantham, Nonlinear Analysis, Theory, Methods and Applications 69 (2008), 1067-1072.

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