coercivity condition
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2020 ◽  
Vol 32 (3) ◽  
pp. 723-738
Author(s):  
Wei Liu ◽  
Rongchan Zhu

AbstractIn this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDEs, where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.



2019 ◽  
Vol 75 (1) ◽  
pp. 163-176 ◽  
Author(s):  
M. Castellani ◽  
M. Giuli


2018 ◽  
Vol 30 (5) ◽  
pp. 1237-1243
Author(s):  
Hongya Gao ◽  
Miaomiao Jia

AbstractThis paper deals with the problem\displaystyle u\in{\cal K}_{u_{*},\psi}(\Omega),\displaystyle\forall v\in{\cal K}_{u_{*},\psi}(\Omega):\int_{\Omega}\sum_{i=1}% ^{n}[a_{i}(x,Du)-f^{i}]D_{i}(u-v)\,dx\leqslant\int_{\Omega}f(u-v)\,dx,where\left\{\begin{aligned} &\displaystyle{\cal K}_{u_{*},\psi}(\Omega)=\biggl{\{}v% \in u_{*}+W_{0}^{1,(p_{i})}(\Omega):\sum_{i=1}^{n}a_{i}(x,Du)D_{i}v\in L^{1}(% \Omega)\text{ and }v\geqslant\psi,\text{ a.e. }\Omega\biggr{\}},\\ &\displaystyle u_{*}\in W^{1,(p_{i})}(\Omega),\quad\theta=\max\{u_{*},\psi\}% \in u_{*}+W_{0}^{1,(p_{i})}(\Omega),\\ &\displaystyle f\in L^{(\bar{p}^{*})^{\prime}}(\Omega),\quad f^{i}\in L^{p_{i}% ^{\prime}}(\Omega),\,i=1,\dots,n,\end{aligned}\right.and the Carathéodory functions {a_{i}:\Omega\times{\mathbb{R}}^{n}\to{\mathbb{R}}}, {i=1,\dots,n}, satisfy some coercivity condition. We assume that the function {\theta=\max\{u_{*},\psi\}} makes {a_{i}(x,D\theta)} to be more integrable than {L^{p_{i}^{\prime}}(\Omega)}, {i=1,\dots,n}, and then we prove that the solution u enjoys higher integrability.



2016 ◽  
Vol 34 (1) ◽  
pp. 225-252 ◽  
Author(s):  
Youssef Akdim ◽  
Nezha El Gorch ◽  
Mounir Mekkour

In this article, we study the existence of a renormalized solution for the nonlinear $p(x)$-parabolic problem associated to the equation: $$\frac{\partial b(x,u)}{\partial t} - \mbox{div} (a(x,t,u,\nabla u)) + H(x,t,u,\nabla u) = f - \mbox{div}F \;\mbox{in }\;Q= \Omega\times(0,T)$$with $ f $ $ \in L^{1} (Q),$\; $b(x,u_{0}) \in L^{1} (\Omega)$ and $ F \in (L^{P'(.)}(Q))^{N}. $The main contribution of our work is to prove the existence of a renormalized solution in the Sobolev space with variable exponent. The critical growth condition on $ H(x,t,u,\nabla u)$\; is with respect to$ \nabla u$, no growth with respect to $u$ and no sign condition or the coercivity condition.



2014 ◽  
Vol 25 (01) ◽  
pp. 1-38 ◽  
Author(s):  
V. Agostiniani ◽  
T. Blass ◽  
K. Koumatos

Linearized elasticity models are derived, via Γ-convergence, from suitably rescaled nonlinear energies when the corresponding energy densities have a multiwell structure and satisfy a weak coercivity condition, in the sense that the typical quadratic bound from below is replaced by a weaker p bound, 1 < p < 2, away from the wells. This study is motivated by, and our results are applied to, energies arising in the modeling of nematic elastomers.



2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yiran He

This paper aims to establish the Tikhonov regularization theory for set-valued variational inequalities. For this purpose, we firstly prove a very general existence result for set-valued variational inequalities, provided that the mapping involved has the so-called variational inequality property and satisfies a rather weak coercivity condition. The result on the Tikhonov regularization improves some known results proved for single-valued mapping.



1986 ◽  
Vol 37 (5) ◽  
pp. 500-505
Author(s):  
S. N. Samborskii ◽  
M. A. Fel'dman


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