We study the boundary value problem of the quasi-linear elliptic equationdiv(|∇u|m−2∇u)+f x,u,∇u)= 0in Ω,u= 0on ∂Ω, where Ω⊂Rn (n⩾2) is a connected smooth domain, and the exponent m∈(1,n) is a positive number.
For n≥3 and 0<ϵ≤1, let Ω⊂Rn be a bounded, simply connected, smooth domain, and uϵ:Ω⊂Rn→R2 solve the Ginzburg Landau equation under the weak anchoring boundary condition:{−Δuϵ="1ϵ2(1−|uϵinΩuϵinΩ,∂uϵ∂ν+λϵ uϵ−gϵ)= 0on∂Ω, where the anchoring strength parameter λϵ="Kϵ−α for some K>0 and α∈[0,1), and gϵ∈C2(∂Ω,S1).
We now specialize to the case when (X) is a connected smooth manifold and (G) a topological group acting on (X) properly and effectively by diffeomorphisms.
It is interesting to observe that the previous result was extended by Donaldson [138] also to the case where (X_1, X_2) are simply connected smooth projective surfaces.
We start with the following important fact: Let (H) be a locally compact group acting effectively by diffeomorphisms on a connected smooth manifold (Y).
where Ω is a bounded regular domain of R n ( n ≥ 1 ), i.e., Ω ¯ is an n-dimensional compact connected smooth submanifold of R n with boundary ∂ Ω.
We can now summarize our discussion as follows: describing all proper effective actions of topological groups by diffeomorphisms on a connected smooth paracompact manifold (X) means describing all closed subgroups of the groups of isometries (mathrm{Isom}[X,{mathtt g}]), where ({mathtt g}) runs over all smooth Riemannian metrics on (X) (cf. [19], p. 106]).
There is a smooth proper family with connected smooth base manifold T, ( p : {mathcal {X}}rightarrow T) having two fibres respectively isomorphic to X, and Y, where Y is one of the 4 surfaces ( S = (C_1 times C_2) / G), ( S_ : = (overline{C_1} times C_2) / G), ( bar{S }= (overline{C_1} times overline{C_2} ) / G), ( S_ : = (C_1 times overline{C_2} ) / G = overline{ S_).
Our broad aim is to classify, whenever possible, pairs ((X,G)), where (X) is a connected smooth paracompact manifold and (Gsubset mathrm{Diff}(X)) a group acting properly, with two such pairs ((X_1,G_1)) and ((X_2,G_2)) called equivalent if there exists a diffeomorphism (f:X_1rightarrow X_2) satisfying (fcirc G_1circ f^{-1}=G_2).
The data were interpolated by using cubic spline lines to connect and smooth the curves.
