Kaplan's penalty operator in approximation of a diffusion-absorption problem with a one-sided constraint

T. V. Sazhenkova ; S. A. Sazhenkov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 236-248.

Résumé

We consider the homogeneous Dirichlet problem for the nonlinear diffusion-absorption equation with a one-sided constraint imposed on diffusion flux values. The family of approximate solutions constructed by means of Alexander Kaplan's integral penalty operator is studied. It is shown that this family converges weakly in the first-order Sobolev space to the solution of the original problem, as the small regularization parameter tends to zero. Thereafter, a property of uniform approximation of solutions is established in Hölder's spaces via systematic study of structure of the penalty operator.

Détail

Keywords: penalty method, p-Laplace operator, one-sided constraint.
Mots-clés : diffusion-absorption equation

Bibliographie

[1] S.N. Antontsev, J.I. Diaz, S. Shmarev, Energy Methods for Free Boundary Problems. Applications to Nonlinear PDEs and Fluid Mechanics, Birkhäuser, Boston, 2002 | MR | Zbl

[2] S.N. Antontsev, S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up, Atlantis Press, Paris, 2015 | MR | Zbl

[3] C. Baiocci, V.V. Pukhnachev, “Problems with one-sided constraints for Navier–Stokes equations and the dynamic contact angle”, J. Appl. Mech. Tech. Phys., 31:2 (1990), 185–197 | DOI | MR

[4] A.Yu. Chebotarev, “Variational inequalities for Navier–Stokes type operators and one-sided problems for equations of viscous heat-conducting fluids”, Math. Notes, 70:1–2 (2001), 264–274 | DOI | MR | Zbl

[5] J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, X, Academic Press, New York–London, 1960 | MR | Zbl

[6] I. Ekeland, R. Témam, Convex analysis and variational problems, Classics in Applied Mathematics, 28, SIAM, 1999 | MR

[7] R. Glowinski, J.-L. Lions, R. Tremolier, Numerical Analysis of Variational Inequalities, Studies in Mathematics and its Applications, 8, North-Holland, Amsterdam, 1981 | MR | Zbl

[8] I. Gómez, J. Rossi, “Tug-of-war games and the infinity Laplacian with spatial dependence”, Commun. Pure Appl. Anal., 12:5 (2013), 1959–1983 | DOI | MR

[9] A.V. Goncharova, T.V. Sazhenkova, “Application of penalty functions in solutions of extremal problems with constraints”, MAK: ‘Mathematicians contribute to Altai Region’, Proceedings of All-Russian mathematical conference, Altai State University Publishing House, Barnaul, 2016, 33–35 (in Russian)

[10] J.D. Griffin, T.G. Kolda, “Nonlinearly constrained optimization using heuristic penalty methods and asynchronous parallel generating set search”, Appl. Math. Res. eXpress. AMRX, 2010:1 (2010), 36–62 | MR | Zbl

[11] Nauka, Novosibirsk, 1981 | MR | Zbl

[12] A.A. Kaplan, “Convex programming algorithms using smoothing of exact penalty functions”, Siberian Math. J., 23:4 (1982), 491–500 | DOI | MR

[13] A. Kaplan, R. Tichatschke, “Some results about proximal-like methods”, Recent Advances in Optimization, Lecture Notes in Economics and Mathematical Systems, 563, ed. A. Seeger, Springer-Verlag, Berlin–Heidelberg–New York, 2006, 61–86 | DOI | MR | Zbl

[14] D. Kinderlerer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980 | MR

[15] P. Lindqvist, Notes on the $p$-Laplace Equation, Second edition, University Printing House, Jyväskylä, 2017 | MR

[16] Mir, M., 1972 | MR

[17] J. Manfredi, J. Rossi, S. Somersille, “An obstacle problem for tug-of-war games”, Commun. Pure Appl. Anal., 14:1 (2015), 217–228 | DOI | MR | Zbl

[18] I.P. Natanson, Theory of Functions of a Real Variable, Frederick Ungar Publ. Co, New York, 1961 | MR

[19] Yu.G. Reshetnyak, Stability Theorems in Geometry and Analysis, Mathematics and Its Application, 304, Springer, 1994 | MR

[20] J. Rossi, “Tug-of-war games and PDEs”, Proc. Royal Soc. Edinburgh, A141 (2011), 319–369 | DOI | MR | Zbl

[21] A.N. Sazhenkov, T.V. Sazhenkova, S.P. Pron, “Study of a class of penalty functions”, Collection of articles, Proceedings of the seminar on geometry and mathematical modelling, 2, ed. E.D. Rodionov, Altai State University Publishing House, Barnaul, 2016, 86–88 (in Russian)

[22] T.V. Sazhenkova, A.N. Sazhenkov, E.A. Plotnikova, “Application of one class of penalty functions for solving variational problems”, Izvestiya of Altai State University, 1:99 (2018), 123–126 (in Russian, English summary) | DOI

[23] T. Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics, 126, AMS, Providence, RI, 2011 | DOI | MR | Zbl

[24] R. Temam, A. Miranville, Mathematical Modeling in Continuum Mechanics, Second Edition, Cambridge University Press, Cambridge, 2005 | MR

[25] J.L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press Inc, New York, 2006 | MR