ﻻ يوجد ملخص باللغة العربية
For non-homotopic maps $u,vin C^{infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $gamma_p(u,v)$ for which there exist paths $u_tin W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $|du_t|_{L^p}^pleq gamma_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows that $gamma_p(u,v)<infty$ for $pin [1,k)$, and using their construction, one can obtain an estimate of the form $gamma_p(u,v)leq frac{C(u,v)}{k-p}$. When $M$ and $N$ are oriented, and $u$ and $v$ induce different maps on real cohomology in degree $k-1$, we show that the growth $gamma_p(u,v)sim frac{1}{k-p}$ as $pto k$ is sharp, and obtain a lower bound for the coefficient $liminf_{pto k}(k-p)gamma_p(u,v)$ in terms of the min-max masses of certain non-contractible loops in the space of codimension-$k$ integral cycles in $M$. In the process, we establish lower bounds for a related smaller quantity $gamma_p^*(u,v)leqgamma_p(u,v)$, for which there exist critical points $u_pin W^{1,p}(M,N)$ of the $p$-energy functional satisfying $gamma_p^*(u,v)leq |du_p|_{L^p}^pleq gamma_p(u,v).$
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Holder continuous. In [3
We discuss the application of the Mountain Pass algorithm to the so-called quasi-linear Schrodinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical algorithm is not directly applicable.
In this paper we consider approximations introduced by Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in [6] about limits of $alpha$-harmonic maps with uniformly bounded energy. Using a recent energy i
Critical points of approximations of the Dirichlet energy `{a} la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such perturbed energi
Let $M$ be a topological space that admits a free involution $tau$, and let $N$ be a topological space. A homotopy class $beta in [ M,N ]$ is said to have {it the Borsuk-Ulam property with respect to $tau$} if for every representative map $f: M to N$