Quasilinear Parabolic Variational Inequalities with Multi-valued Lower-order Terms

Siegfried Carl; Vy Khoi Le

doi:10.1007/s00033-013-0357-6

Quasilinear Parabolic Variational Inequalities with Multi-valued Lower-order Terms

Siegfried Carl
, Vy Khoi Le

Mathematics and Statistics

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

Abstract

<p> In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain Q=Ω×(0,τ) : Find u∈K and an η∈Lp′(Q) such that η∈f(⋅,⋅,u),&langle;ut+Au,v−u&rangle;+∫Qη(v−u)dxdt≥0,∀v∈K, where K⊂X0=Lp(0,τ;W1,p0(Ω)) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function s&mapsto;f(⋅,⋅,s) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.</p>

Original language	American English
Journal	Zeitschrift Fur Angewandte Mathematik und Physik
DOIs	https://doi.org/10.1007/s00033-013-0357-6
State	Published - Jan 1 2013

Keywords

comparison principle
parabolic variational inequality
peudomonotone multi-valued operator
sub-ubersolution
upper semicontinuous multi-valued operator

Disciplines

Mathematics
Statistics and Probability

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https://doi.org/10.1007/s00033-013-0357-6

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@article{18d041fa8e284573b71eb10818abb839,

title = "Quasilinear Parabolic Variational Inequalities with Multi-valued Lower-order Terms",

abstract = " In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain Q=\Ω\×(0,\τ) : Find u\∈K and an \η\∈Lp\′(Q) such that \η\∈f(\⋅,\⋅,u),\&langle;ut+Au,v\−u\&rangle;+\∫Q\η(v\−u)dxdt\≥0,\∀v\∈K, where K\⊂X0=Lp(0,\τ;W1,p0(\Ω)) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function s\&mapsto;f(\⋅,\⋅,s) is assumed to be upper semicontinuous only, so that Clarke\’s generalized gradient is included as a special case. Thus, parabolic variational\–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational\–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.",

keywords = "comparison principle, parabolic variational inequality, peudomonotone multi-valued operator, sub-ubersolution, upper semicontinuous multi-valued operator",

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

year = "2013",

month = jan,

day = "1",

doi = "10.1007/s00033-013-0357-6",

language = "American English",

journal = "Zeitschrift Fur Angewandte Mathematik und Physik",

}

TY - JOUR

T1 - Quasilinear Parabolic Variational Inequalities with Multi-valued Lower-order Terms

AU - Carl, Siegfried

AU - Le, Vy Khoi

PY - 2013/1/1

Y1 - 2013/1/1

N2 - In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain Q=Ω×(0,τ) : Find u∈K and an η∈Lp′(Q) such that η∈f(⋅,⋅,u),&langle;ut+Au,v−u&rangle;+∫Qη(v−u)dxdt≥0,∀v∈K, where K⊂X0=Lp(0,τ;W1,p0(Ω)) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function s&mapsto;f(⋅,⋅,s) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.

AB - In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain Q=Ω×(0,τ) : Find u∈K and an η∈Lp′(Q) such that η∈f(⋅,⋅,u),&langle;ut+Au,v−u&rangle;+∫Qη(v−u)dxdt≥0,∀v∈K, where K⊂X0=Lp(0,τ;W1,p0(Ω)) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function s&mapsto;f(⋅,⋅,s) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.

KW - comparison principle

KW - parabolic variational inequality

KW - peudomonotone multi-valued operator

KW - sub-ubersolution

KW - upper semicontinuous multi-valued operator

UR - https://scholarsmine.mst.edu/math_stat_facwork/252

UR - https://doi.org/10.1007/s00033-013-0357-6

U2 - 10.1007/s00033-013-0357-6

DO - 10.1007/s00033-013-0357-6

M3 - Article

JO - Zeitschrift Fur Angewandte Mathematik und Physik

JF - Zeitschrift Fur Angewandte Mathematik und Physik

ER -

Quasilinear Parabolic Variational Inequalities with Multi-valued Lower-order Terms

Abstract

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