An explicit example of a Reich sequence for a uniquely extremal quasiconformal mapping in a borderline case between uniqueness and non-uniqueness is given.

In this paper, the author studies the existence of the minimal nonnegative solutions of some elliptic variational inequalities in Orlicz-Sobolev spaces on bounded or unbounded domains. She gets some comparison results between different solutions as tools to pass to the limit in the problems and to show the existence of the minimal solutions of the variational inequalities on bounded domains or unbounded domains. In both cases, coercive and noncoercive operators are handled. The sufficient and necessary conditions for the existence of the minimal nonnegative solution of the noncoercive variational inequality on bounded domains are established.

The authors construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both 2ω + 1, where ω is the smallest infinite ordinal number. Therefore, an example of a metric space with asymptotic property C is obtained.

In this paper, the authors consider a class of generalized curve flow for convex curves in the plane. They show that either the maximal existence time of the flow is finite and the evolving curve collapses to a round point with the enclosed area of the evolving curve tending to zero, i.e., $\mathop {\lim}\limits_{t \to T} A(t) = 0$, or the maximal time is infinite, that is, the flow is a global one. In the case that the maximal existence time of the flow is finite, they also obtain a convergence theorem for rescaled curves at the maximal time.

In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S ⁶, which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S ⁶. This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore (i.e., has no dual surface) in S⁶. This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S ⁶.

In this paper, the authors establish the existence and uniqueness theorem of L ^p (1 < p ≤ 2) solutions for multidimensional backward doubly stochastic differential equations (BDSDEs for short) under the p-order globally (locally) weak monotonicity conditions. Comparison theorem of L ^p solutions for one-dimensional BDSDEs is also proved. These conclusions unify and generalize some known results.

In this paper, the authors study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and they have natural GIT constructions for the corresponding moduli spaces and they show that the K3 surfaces with at worst ADE singularities are GIT stable. They give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion. They compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga’s compactifications from arithmetic perspective. One of the main ingredients is the study of the projective models of K3 surfaces in terms of Noether-Lefschetz divisors.

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces. And the corresponding commutators generated by BMO function are also considered.

In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existence of optimal solution to the optimal control problem, and finally the first order necessary condition is obtained by constructing an appropriate penalty function.