American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain
  • PROC Home
  • This Issue
  • Next Article
    Parabolic Monge-Ampere equations giving rise to a free boundary: The worn stone model
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. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

[1]

Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021004
[2]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258
[3]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[4]

A. V. Rezounenko. Inertial manifolds with delay for retarded semilinear parabolic equations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 829-840. doi: 10.3934/dcds.2000.6.829
[5]

Vladimir V. Chepyzhov, Anna Kostianko, Sergey Zelik. Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1115-1142. doi: 10.3934/dcdsb.2019009
[6]

Norimichi Hirano, Wen Se Kim. Multiplicity and stability result for semilinear parabolic equations. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 271-280. doi: 10.3934/dcds.1996.2.271
[7]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711
[8]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[9]

Júlia Matos. Unfocused blow up solutions of semilinear parabolic equations. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 905-928. doi: 10.3934/dcds.1999.5.905
[10]

Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155
[11]

Francis Ribaud. Semilinear parabolic equations with distributions as initial data. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 305-316. doi: 10.3934/dcds.1997.3.305
[12]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[13]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013
[14]

Motohiro Sobajima. On the threshold for Kato's selfadjointness problem and its $L^p$-generalization. Evolution Equations & Control Theory, 2014, 3 (4) : 699-711. doi: 10.3934/eect.2014.3.699
[15]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[16]

H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315
[17]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557
[18]

Jiabao Su, Zhaoli Liu. A bounded resonance problem for semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 431-445. doi: 10.3934/dcds.2007.19.431
[19]

Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 55-75. doi: 10.3934/era.2015.22.55
[20]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

 Impact Factor: 

Tools

Metrics

  • PDF downloads (114)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy