This article is concerned with the problem of prescribing Gaussian curvature K and geodesic curvature h in a compact surface with boundary with conical singularities {p1, ... , pn} and corners {q1, ... , qn}. This is equivalent to solving the Liouville-type equation: { - Delta u+2k(0) = 2ke(u) - 4 pi Sigma (n)( i=1)alpha( i) (delta( pi) - 1 /|Sigma|) -2 pi Sigma (m )(j=1) beta(j) (delta(qi) -1 /|Sigma|) in Sigma partial derivative(v)u+2h(0) = 2he (u /2) on partial derivative Sigma where K0, h0 are the pre-existing Gaussian curvature and the geodesic curvature, respectively, and alpha(i), beta(j )> -1 are given. Solutions are obtained using a new variational formulation, first introduced in Thierry (1998) in [1] for the regular counterpart of the problem and extended here to the singular case. As far as we know, this is the first result for the problem of prescribed curvatures in surfaces with the two types of singularities. Key ingredients are a blow-up analysis around a sequence of points different from local maxima and Morse index estimates.
This article is concerned with the interaction between a planar traveling front and a compact obstacle for monotone bistable reaction-diffusion systems in exterior domains. By constructing appropriate sub- and supersolutions, we first establish the existence, uniqueness and monotonicity of the entire solution emanating from a planar traveling front. In particular, we verify that regardless of the shape of the obstacle, the entire solution locally converges to a stationary solution as time tends to infinity. Under the complete propagation assumption, we further show that the entire solution recovers to the same planar traveling front as time tends to infinity after passing the obstacle, and it constitutes a transition front. In addition, we provide some geometric conditions on the obstacle to ensure that the complete propagation assumption is nonempty. Without the complete propagation assumption, we prove that the entire solution still propagates in the form of the planar traveling front far behind the obstacle for large time. Finally, we apply our theoretical results to the Lotka-Volterra competition-diffusion system and the buffered bistable system.
