Variational Inequalities with General Multivalued Lower Order Terms Given by Integrals

2011 ◽  
Vol 11 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Vy Khoi
Keyword(s):  
Variational Inequalities ◽  
Lower Order ◽  
Order Term ◽  
Multivalued Function ◽  
Solution Set ◽  
Lower Order Term ◽  
Extremal Solutions ◽  
Lower Order Terms

AbstractThis paper is about the existence and some properties of solutions of variational inequalities associated with the 2nd order inclusiondiv[A(x, ∇u)] + L ∈ f (x, u) in Ω,where the lower order term f (x, u) is a general multivalued function. Both coercive and noncoercive cases are considered. In the noncoercive case, we use a sub-supersolution approach to study the existence, comparison, and other properties of the solution set such as its compactness, directedness, and the existence of extremal solutions.

On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals

2018 ◽  
Vol 18 (2) ◽  
pp. 269-287 ◽  
Author(s):  
Vy Khoi Le ◽  
Klaus Schmitt
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Lower Order ◽  
Order Term ◽  
Growth Conditions ◽  
Principal Term ◽  
Lower Order Term ◽  
Convex Functional ◽  
Parabolic Variational Inequality ◽  
Convex Functionals

Abstract In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find {u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))} and {f\in F(\cdot,\cdot,u)} such that {u(\cdot,0)=u_{0}} and \langle u_{t}+Au,v-u\rangle+\Psi(v)-\Psi(u)\geq\int_{Q}f(v-u)\,dx\,dt\quad% \text{for all }v\in L^{p}(0,T;W^{1,p}_{0}(\Omega)), where A is the principal term; F is a multivalued lower-order term; {\Psi(u)=\int_{0}^{T}\psi(t,u)\,dt} is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.

The Regularizing Effect of Lower Order Terms in Elliptic Problems Involving Hardy Potential

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
A. Adimurthi ◽  
Lucio Boccardo ◽  
G. Rita Cirmi ◽  
Luigi Orsina
Keyword(s):  
Lower Order ◽  
Order Term ◽  
Elliptic Problems ◽  
Lower Order Term ◽  
Hardy Potential ◽  
Lower Order Terms

AbstractWe study existence and summability of solutions for elliptic problems with a power-like lower order term and a Hardy potential. We prove that, due to the presence of the lower order term, solutions exist and are more summable under weaker assumptions than those needed for the existence without it.

scholarly journals A regularity result for a class of elliptic equations with lower order terms

2020 ◽  
Vol 6 (2) ◽  
pp. 751-771 ◽  
Author(s):  
Claudia Capone ◽  
Teresa Radice
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Order Term ◽  
Regularity Result ◽  
Lower Order Term ◽  
Bounded Solutions ◽  
Differentiability Of Solutions ◽  
Higher Differentiability ◽  
Lower Order Terms

Abstract In this paper we establish the higher differentiability of solutions to the Dirichlet problem $$\begin{aligned} {\left\{ \begin{array}{ll} \text {div} (A(x, Du)) + b(x)u(x)=f &{} \text {in}\, \Omega \\ u=0 &{} \text {on} \, \partial \Omega \end{array}\right. } \end{aligned}$$ div ( A ( x , D u ) ) + b ( x ) u ( x ) = f in Ω u = 0 on ∂ Ω under a Sobolev assumption on the partial map $$x \rightarrow A(x, \xi )$$ x → A ( x , ξ ) . The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions.

scholarly journals Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support

2017 ◽  
Vol 153 (6) ◽  
pp. 1196-1216 ◽  
Author(s):  
Daniel Fiorilli ◽  
James Parks ◽  
Anders Södergren
Keyword(s):  
Phase Transition ◽  
Fourier Transform ◽  
Lower Order ◽  
Order Term ◽  
Level Density ◽  
Lower Order Term ◽  
Test Function ◽  
The Fourier Transform ◽  
Lower Order Terms ◽  
Function Field Case

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.

scholarly journals Nonlinear evolution problems with singular coefficients in the lower order terms

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Lower Order ◽  
Time Horizon ◽  
Existence Result ◽  
Nonlinear Evolution ◽  
Order Term ◽  
Time Behavior ◽  
Infinite Time ◽  
Singular Coefficient ◽  
Singular Coefficients ◽  
Lower Order Terms

AbstractWe consider a Cauchy–Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite–time horizon.

scholarly journals Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xiaohua He ◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Yonglin Xu
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Order Term ◽  
Combined Effects ◽  
Convection Term ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Lower Order Terms

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem − div M x ∇ u + u p − 1 u = − div u E x + f x , x ∈ Ω , u x = 0 , x ∈ ∂ Ω , where Ω ⊂ ℝ N N > 2 is a bounded smooth domain with 0 ∈ Ω , f belongs to the Lebesgue space L m Ω with m ≥ 1 , p > 0 . The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.

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