On Systems of Parabolic Variational Inequalities with Multivalued Terms

Siegfried Carl; Vy Khoi Le

doi:10.1007/s00605-020-01477-6

On Systems of Parabolic Variational Inequalities with Multivalued Terms

Siegfried Carl
, Vy Khoi Le

Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain [Formula] where K k is a closed and convex subset of [Formula], A k is a time-dependent quasilinear elliptic operator, and f k : Q x &reals; m → 2 &reals; is an upper semicontinuous multivalued function with respect to s ∈ &reals; m . We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints K k , systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.

Original language	American English
Journal	Monatshefte fur Mathematik
Volume	194
DOIs	https://doi.org/10.1007/s00605-020-01477-6
State	Published - Feb 1 2021

Keywords

Evolutionary variational-hemivariational inequalities
Multivalued parabolic variational inequality
Obstacle problem
Pseudomonotone multivalued operator
Sub-supersolution
System of parabolic variational inequalities
Upper semicontinuous multivalued operator

Disciplines

Mathematics
Statistics and Probability

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https://doi.org/10.1007/s00605-020-01477-6

Cite this

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title = "On Systems of Parabolic Variational Inequalities with Multivalued Terms",

abstract = " In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain [Formula] where K k is a closed and convex subset of [Formula], A k is a time-dependent quasilinear elliptic operator, and f k : Q x \&reals; m \→ 2 \&reals; is an upper semicontinuous multivalued function with respect to s \∈ \&reals; m . We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints K k , systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.",

keywords = "Evolutionary variational-hemivariational inequalities, Multivalued parabolic variational inequality, Obstacle problem, Pseudomonotone multivalued operator, Sub-supersolution, System of parabolic variational inequalities, Upper semicontinuous multivalued operator",

author = "Siegfried Carl and Le, \{Vy Khoi\}",

year = "2021",

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doi = "10.1007/s00605-020-01477-6",

language = "American English",

volume = "194",

journal = "Monatshefte fur Mathematik",

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T1 - On Systems of Parabolic Variational Inequalities with Multivalued Terms

AU - Carl, Siegfried

AU - Le, Vy Khoi

PY - 2021/2/1

Y1 - 2021/2/1

N2 - In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain [Formula] where K k is a closed and convex subset of [Formula], A k is a time-dependent quasilinear elliptic operator, and f k : Q x &reals; m → 2 &reals; is an upper semicontinuous multivalued function with respect to s ∈ &reals; m . We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints K k , systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.

AB - In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain [Formula] where K k is a closed and convex subset of [Formula], A k is a time-dependent quasilinear elliptic operator, and f k : Q x &reals; m → 2 &reals; is an upper semicontinuous multivalued function with respect to s ∈ &reals; m . We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints K k , systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.

KW - Evolutionary variational-hemivariational inequalities

KW - Multivalued parabolic variational inequality

KW - Obstacle problem

KW - Pseudomonotone multivalued operator

KW - Sub-supersolution

KW - System of parabolic variational inequalities

KW - Upper semicontinuous multivalued operator

UR - https://scholarsmine.mst.edu/math_stat_facwork/1041

UR - https://doi.org/10.1007/s00605-020-01477-6

U2 - 10.1007/s00605-020-01477-6

DO - 10.1007/s00605-020-01477-6

M3 - Article

VL - 194

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

ER -

On Systems of Parabolic Variational Inequalities with Multivalued Terms

Abstract

Keywords

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