gradient systems
Recently Published Documents


TOTAL DOCUMENTS

173
(FIVE YEARS 5)

H-INDEX

20
(FIVE YEARS 0)

2021 ◽  
Vol 66 (1) ◽  
pp. 75-84
Author(s):  
Biagio Ricceri

"We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $\Omega\subset {\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $\Phi:{\bf R}^2\to {\bf R}$ be a $C^1$ function, with $\Phi(0,0)=0$, such that $$\sup_{(u,v)\in {\bf R}^2}\frac{|\Phi_u(u,v)|+|\Phi_v(u,v)|}{1+|u|^p+|v|^p}<+\infty$$ where $p>0$, with $p=\frac{2}{n-2}$ when $n>2$.\\ Then, for every convex set $S\subseteq L^{\infty}(\Omega)\times L^{\infty}(\Omega)$ dense in $L^2(\Omega)\times L^2(\Omega)$, there exists $(\alpha,\beta)\in S$ such that the problem $$-\Delta u=(\alpha(x)\cos(\Phi(u,v))-\beta(x)\sin(\Phi(u,v)))\Phi_u(u,v)\hskip 5pt \hbox {\rm in}\hskip 5pt \Omega$$ $$-\Delta v= (\alpha(x)\cos(\Phi(u,v))-\beta(x)\sin(\Phi(u,v)))\Phi_v(u,v)\hskip 5pt \hbox {\rm in}\hskip 5pt \Omega$$ $$u=v=0\hskip 5pt \hbox {\rm on}\hskip 5pt \partial\Omega$$ has at least three weak solutions, two of which are global minima in $H^1_0(\Omega)\times H^1_0(\Omega)$ of the functional $$(u,v)\to \frac{1}{2}\left ( \int_{\Omega}|\nabla u(x)|^2dx+\int_{\Omega}|\nabla v(x)|^2dx\right )$$ $$-\int_{\Omega}(\alpha(x)\sin(\Phi(u(x),v(x)))+\beta(x)\cos(\Phi(u(x),v(x))))dx\ .$$"


Author(s):  
Alexander Mielke ◽  
Alberto Montefusco ◽  
Mark A. Peletier

AbstractWe introduce two new concepts of convergence of gradient systems $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) to a limiting gradient system $$({\mathbf{Q}},{{\mathcal {E}}}_0,{{\mathcal {R}}}_0)$$ ( Q , E 0 , R 0 ) . These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.


2020 ◽  
Vol 269 (11) ◽  
pp. 10073-10084
Author(s):  
Jaume Llibre ◽  
Oscar Ramírez

2020 ◽  
Vol 2020 (2) ◽  
pp. 023207
Author(s):  
Jinjie Zhu ◽  
Jiong Wang ◽  
Shang Gao ◽  
Xianbin Liu
Keyword(s):  

2020 ◽  
Vol 58 (4) ◽  
pp. 2079-2092
Author(s):  
J. M. Sanz Serna ◽  
K. C. Zygalakis
Keyword(s):  

Sign in / Sign up

Export Citation Format

Share Document