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A Duality Approach to Nonlinear Diffusion Equations

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Abstract

We provide new existence results for a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity, under minimal growth and coercivity conditions. The results given in this paper prove that a generalized solution to the nonlinear equation is provided by a solution to an equivalent minimization problem for a convex functional involving the potential of the nonlinearity and its conjugate, in the case when the potential is time and space depending. If the potential is time depending only and it has a symmetry at infinity, the null minimizer in the minimization problem is found to coincide with a weak solution to the nonlinear equation.

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References

  1. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Comm. Pure Appl. Math 12, 623–727 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  2. Auchmuty, G.: Variational principles for operator equations and initial value problems. Nonlinear Anal. 12, 531–564 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Auchmuty, G.: Saddle points and existence-uniqueness for evolution equations. Diff. Integr. Eqs. 6, 1161–1171 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Barbu, V.: Nonlinear Differential Equations of Monotone Type in Banach Spaces. Springer, New York (2010)

    Book  Google Scholar 

  5. Barbu, V.: A variational approach to stochastic nonlinear parabolic problems. J. Math. Anal. Appl. 384, 2–15 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barbu, V.: Optimal control approach to nonlinear diffusion equations driven by Wiener Noise. J. Optim. Theory Appl. 153, 1–26 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Barbu, V., Da Prato, G., Röckner, M.: Existence of strong solutions for stochastic porous media equation under general monotonicity conditions. Ann. Probab. 37, 428–452 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brezis, H., Ekeland, I.: Un Principe Variationnel Associé à Certaines Equations Paraboliques. Le cas Independant du temps. C.R. Acad. Sci. Paris Sér. A 282, 971–974 (1976)

    MATH  MathSciNet  Google Scholar 

  9. Brezis, H., Ekeland, I.: Un Principe Variationnel Associé à Certaines Equations Paraboliques. Le cas Dependant du temps. C.R. Acad. Sci. Paris Sér. A 282, 1197–1198 (1976)

    MATH  MathSciNet  Google Scholar 

  10. Brezis, H., Strauss, W.A.: Semi-linear second elliptic equations in L 1. J. Math. Soc. Japan 25, 565–590 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ciutureanu, C.,Marinoschi, G.: Convergence of the finite difference scheme for a fast diffusion equation in porous media. Numer. Func. Anal. Optim. 29, 1034–1063 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Crandall, M.G., Pazy, A.: Nonlinear evolution equations in banach spaces. Israel J. Math 11, 57–94 (1971)

    Article  MathSciNet  Google Scholar 

  13. Cusulin, C., Iannelli, M., Marinoschi, G.: Convergence in a multi-layer population model with agestructure. Nonlinear Anal. Real World Appl. 8, 887–902 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Favini, A., Marinoschi, G.: Degenerate Nonlinear Diffusion Equations.Lecture Notes in Mathematics 2049. Springer, Berlin (2012)

    Book  Google Scholar 

  15. Fenchel, W.: Convex Cones, Sets and Functions. Princeton Univ (1953)

  16. Ghoussoub, N.: A variational theory for monotone vector fields. J. Fixed Point Theory Appl. 4, 107–135 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ghoussoub, N.: Self-dual partial differential systems and their variational principles. Springer, New York (2009)

    MATH  Google Scholar 

  18. Ghoussoub, N., Tzou, L.: A variational principle for gradient flows. Math. Ann. 330, 519–549 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lions, J.L.: Quelques Méthodes de Résolution des Problémes aux Limites non linéaires. Dunod, Paris (1969)

  20. Marinoschi, G.: Existence to time-dependent nonlinear diffusion equations via convex optimization. J. Optim. Theory Appl. 154, 792–817 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Marinoschi, G.: Variational solutions to nonlinear diffusion equations with singular diffusivity. J. Optim. Theory Appl. 161, 430–445 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nečas, J.: Les Méthodes Directes en Théorie des équations elliptiques. Masson. Paris-Academia, Praha (1967)

    Google Scholar 

  23. Stefanelli, U.: The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim. 47, 1615–1642 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Stefanelli, U.: The discrete Brezis-Ekeland principle. J. Convex Anal. 16, 71–87 (2009)

    MATH  MathSciNet  Google Scholar 

  25. Visintin, A.: Extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv.Math. Sci. Appl. 18, 633–650 (2008)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, 050711, Bucharest, Romania

    Gabriela Marinoschi

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  1. Gabriela Marinoschi
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Correspondence to Gabriela Marinoschi.

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Marinoschi, G. A Duality Approach to Nonlinear Diffusion Equations. Set-Valued Var. Anal 22, 783–807 (2014). https://doi.org/10.1007/s11228-014-0288-1

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  • DOI: https://doi.org/10.1007/s11228-014-0288-1

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