Parabolic variational inequalities with generalized reflecting directions

Eduard Rotenstein

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 213-222
  • ISSN: 2391-5455

Abstract

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We study, in a Hilbert framework, some abstract parabolic variational inequalities, governed by reflecting subgradients with multiplicative perturbation, of the following type: y´(t)+ Ay(t)+0.t Θ(t,y(t)) ∂φ(y(t))∋f(t,y(t)),y(0) = y0,t ∈[0,T] where A is a linear self-adjoint operator, ∂φ is the subdifferential operator of a proper lower semicontinuous convex function φ defined on a suitable Hilbert space, and Θ is the perturbing term which acts on the set of reflecting directions, destroying the maximal monotony of the multivalued term. We provide the existence of a solution for the above Cauchy problem. Our evolution equation is accompanied by examples which aim to (systems of) PDEs with perturbed reflection.

How to cite

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Eduard Rotenstein. "Parabolic variational inequalities with generalized reflecting directions." Open Mathematics 13.1 (2015): 213-222. <http://eudml.org/doc/276013>.

@article{EduardRotenstein2015,
abstract = {We study, in a Hilbert framework, some abstract parabolic variational inequalities, governed by reflecting subgradients with multiplicative perturbation, of the following type: y´(t)+ Ay(t)+0.t Θ(t,y(t)) ∂φ(y(t))∋f(t,y(t)),y(0) = y0,t ∈[0,T] where A is a linear self-adjoint operator, ∂φ is the subdifferential operator of a proper lower semicontinuous convex function φ defined on a suitable Hilbert space, and Θ is the perturbing term which acts on the set of reflecting directions, destroying the maximal monotony of the multivalued term. We provide the existence of a solution for the above Cauchy problem. Our evolution equation is accompanied by examples which aim to (systems of) PDEs with perturbed reflection.},
author = {Eduard Rotenstein},
journal = {Open Mathematics},
keywords = {Evolution equations; Oblique reflection; PDEs; forward-backward SDEs; quadratic growth; financial derivatives},
language = {eng},
number = {1},
pages = {213-222},
title = {Parabolic variational inequalities with generalized reflecting directions},
url = {http://eudml.org/doc/276013},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Eduard Rotenstein
TI - Parabolic variational inequalities with generalized reflecting directions
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 213
EP - 222
AB - We study, in a Hilbert framework, some abstract parabolic variational inequalities, governed by reflecting subgradients with multiplicative perturbation, of the following type: y´(t)+ Ay(t)+0.t Θ(t,y(t)) ∂φ(y(t))∋f(t,y(t)),y(0) = y0,t ∈[0,T] where A is a linear self-adjoint operator, ∂φ is the subdifferential operator of a proper lower semicontinuous convex function φ defined on a suitable Hilbert space, and Θ is the perturbing term which acts on the set of reflecting directions, destroying the maximal monotony of the multivalued term. We provide the existence of a solution for the above Cauchy problem. Our evolution equation is accompanied by examples which aim to (systems of) PDEs with perturbed reflection.
LA - eng
KW - Evolution equations; Oblique reflection; PDEs; forward-backward SDEs; quadratic growth; financial derivatives
UR - http://eudml.org/doc/276013
ER -

References

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  3. [3] Lions, P.-L.; Sznitman, A. - Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 1984, Volume 37, no. 4, 511-537. Zbl0598.60060
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  10. [10] Altomare, F.; Milella, S.; Musceo, G. - Multiplicative perturbations of the Laplacian and related approximation problems, J. Evol. Equ., 2011, 11, 771-792. [WoS][Crossref] Zbl1256.47024
  11. [11] Barbu, V. - Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. 
  12. [12] Lions, J.L. - Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod and Gauthier-Villars, 1969. 
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  14. [14] Aubin, J. P. - Un théorème de compacité, C. R. Acad. Sci. Paris, 1963, 256, 5042-5044. 

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