Opuscula Math. 39, no. 2 (2019), 145-157
https://doi.org/10.7494/OpMath.2019.39.2.145

 
Opuscula Mathematica

Some remarks on the coincidence set for the Signorini problem

Miguel de Benito Delgado
Jesus Ildefonso Díaz

Abstract. We study some properties of the coincidence set for the boundary Signorini problem, improving some results from previous works by the second author and collaborators. Among other new results, we show here that the convexity assumption on the domain made previously in the literature on the location of the coincidence set can be avoided under suitable alternative conditions on the data.

Keywords: Signorini problem, coincidence set, location estimates, free boundary problem, contact problems.

Mathematics Subject Classification: 35J86, 35R35, 35R70, 35B60.

Full text (pdf)

  1. M. de Benito Delgado, Sobre Un Problema de Contacto En Elasticidad, Bachelor's Thesis, Universidad Complutense de Madrid, Madrid, 2012.
  2. H. Brézis, Problèmes unilatéraux, Journal de Mathematiques Pures et Appliquées 51 (1972), 1-168.
  3. L. Caffarelli, L. Silvestre, An Extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) 8, 1245-1260.
  4. L.A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations 4 (1979) 9, 1067-1075.
  5. J. Dávila, M. Montenegro, Nonlinear problems with solutions exhibiting a free boundary on the boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 3, 303-330.
  6. M. Delfour, J. Zolésio, Shapes and Geometries, Advances in Design and Control, Society for Industrial and Applied Mathematics, 2011.
  7. J. Díaz, Special Finite Time Extinction, Nonlinear Evolution Systems: Dynamic Boundary Conditions and Coulomb Friction Type Problems, [in:] H. Brezis, M. Chipot, J. Escher (eds), Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann, Birkhäuser Basel, Progress in Nonlinear Differential Equations and Their Applications, 71-97, 2005.
  8. J.I. Díaz, Localización de fronteras libres en inecuaciones variacionales estacionarias dadas por obstáculos, 3. Congreso de ecuaciones diferenciales y Aplicaciones, Universidad de Santander y Universidad Complutense de Madrid, Santiago, 1980, 1-12.
  9. J.I. Díaz, R.F. Jiménez, Aplicación de la teoría no lineal de semigrupos a un operador pseudodiferencial, in Actas del VII CEDYA, SEMA, Granada, 1985, 137-142.
  10. J.I. Díaz, R.F. Jiménez, Behaviour on the boundary of solutions of parabolic equations with nonlinear boundary condition: The evolutionary Signorini problem, [in:] Análisis No Lineal, PNUD/UNESCO, Univ. de Concepción, 69-80, 1986.
  11. J.I. Díaz, R.F. Jiménez, Boundary behaviour of solutions of the Signorini problem, Bolletino U.M.I 7 (1988) 2-B, 127-139.
  12. J.I. Díaz, T. Mingazzini, Free boundaries touching the boundary of the domain for some reaction-diffusion problems, Nonlinear Anal. 119 (2015), 275-294.
  13. J.I. Díaz, D. Gómez-Castro, A.V. Podolskiy, T.A. Shaposhnikova, Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis "nano-composite" membranes, Advances in Nonlinear Analysis (2018), https://doi.org/10.1515/anona-2018-0158
  14. J.I. Díaz, D. Gómez-Castro, J.L. Vázquez, The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach, Nonlinear Anal. 177 (2018), 325-360.
  15. G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, vol. 219, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1976.
  16. G. Fichera, Unilateral constraints in elasticity, Actes, Congrès international de Mathematiques 3 (1970), 79-84.
  17. A. Friedman, Boundary behavior of solutions of variational inequalities for elliptic operators, Arch. Rational Mech. Anal. 27 (1967), 95-107.
  18. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 2nd ed., Springer-Verlag, Berlin-Heidelberg, 2001.
  19. D. Kinderlehrer, Remarks about Signorini's problem in linear elasticity, Ann. Sc. Norm. Super. Pisa 8 (1981) 4, 605-645.
  20. D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, volume 31 of SIAM's classics in applied mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 2000.
  21. J.L. Lions, G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967) 3, 493-519.
  22. A. Petrosyan, H. Shahgholian, N. Uraltseva, Regularity of Free Boundaries in Obstacle-Type Problems, vol. 136, Graduate Studies in Mathematics, American Mathematical Society, 2012.
  23. M. Schatzmann, Problemes aux limites non linéaires non coercifs, Ann. Sc. Norm. Super. Pisa Cl. Sci. 27 (1973), 1-686.
  24. A. Signorini, Questioni di elasticità non linearizzata e semilinearizzata, Rend. Mat. Appl. 18 (1959), 95-139.
  25. R.P. Sperb, Maximum Principles and Their Applications, Mathematics in Science and Engineering, Academic Press, 1981.
  • Miguel de Benito Delgado
  • ORCID iD https://orcid.org/0000-0002-3045-3786
  • University of Augsburg, Department of Mathematics, 86159 Augsburg, Germany
  • appliedAI @ UnternehmerTUM GmbH, Lichtenbergstraße 6, D-85748 Garching, Germany
  • Communicated by Giovanni Molica Bisci.
  • Received: 2018-11-05.
  • Accepted: 2018-11-11.
  • Published online: 2018-12-07.
Opuscula Mathematica - cover

Cite this article as:
Miguel de Benito Delgado, Jesus Ildefonso Díaz, Some remarks on the coincidence set for the Signorini problem, Opuscula Math. 39, no. 2 (2019), 145-157, https://doi.org/10.7494/OpMath.2019.39.2.145

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise that this website uses cookies to help us understand how the site is used. All data is anonymized. Recent versions of popular browsers provide users with control over cookies, allowing them to set their preferences to accept or reject all cookies or specific ones.