A class of functionals possessing multiple global minima

"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 $$(u,v)\to \frac{1}{2}\left ( \int_{\Omega}|\nabla u(x)|^2dx+\int_{\Omega}|\nabla v(x)|^2dx\right )$$ $$-\int_{\Omega}(\alpha(x)\sin(\Phi(u(x),v(x)))+\beta(x)\cos(\Phi(u(x),v(x))))dx\ .$$"

Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

We 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.