dynamics of coupled oscillators with convex interaction potential in this paper we study via the monotonicity approach the dynamical behavior of a class of coupled oscillators with periodic on site potential and convex interaction potential it is shown that under periodic or neumann boundary conditions the poincare map admits an invariant curve on which the poincare map is actually an orientation preserving circle homeomorphism some consequences are then obtained including a massera type theorem and the existence of running periodic solutions for the dirichlet boundary conditions each solution is asymptotic to a periodic solution if it is not periodic

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

nonlinear schrodinger equation with unbounded or decaying radial potentials involving exponential critical growth in r2 abstract we study the existence and multiplicity of solutions for the following class of nonlinear schrodinger equations u v x u q x f u in r 2 where v and q are unbounded or decaying radial potentials and the nonlinearity f s has exponential critical growth the approaches used here are based on a version of the trudinger moser inequality and a minimax theorem

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

the spectrum of the locked state for the kuramoto model of coupled oscillators we analyze the linear stability of the phase locked state in the kuramoto model of coupled oscillators the main result is the first rigorous characterization of the spectrum and its associated eigenvectors for any finite number of oscillators all but two of the eigenvalues are negative and merge into a continuous spectrum as the number of oscillators tends to infinity one eigenvalue is always zero by rotational invariance the final eigenvalue corresponding to a collective mode determines the stability of the locked state

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

volume preserving flow by powers of symmetric polynomials in the principal curvatures we study a volume preserving curvature flow of convex hypersurfaces driven by a power of the k th elementary symmetric polynomial in the principal curvatures unlike most of the previous works on related problems we do not require assumptions on the curvature pinching of the initial datum we prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm in the case of the volume preserving scalar curvature flow we can prove that the hypersurfaces converge smoothly to a round sphere

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

rotation number of the overdamped frenkel kontorova model with ac driving abstract the dynamical behavior of the overdamped and overdamped limit cases of the frenkel kontorova model with ac driving is studied the relationship is established between the average velocity of the particles and the rotation number of the orientation preserving homeomorphism on the circle induced from the poincare map the dynamical behavior for both cases is determined completely by the rotation number numerical results are also provided for the second order system to support the theoretical conclusions

pinching estimates and motion of hypersurfaces by curvature functions second derivative pinching estimates are proved for a class of elliptic and parabolic equations including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature the estimates imply convergence of convex hypersurfaces to spheres under these flows improving earlier results of b chow and the author the result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives

anisotropic flow of convex hypersurfaces by the square root of the scalar curvature abstract we show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor when the strictly convex initial hypersurface in euclidean space is suitably pinched we also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere for a general case in dimension two it is shown that with a volume preserving rescaling the limit profile satisfies a soliton equation

massera s theorem for monotone dynamical systems in three dimensions in this paper we give a variant of the classical second massera s theorem for three dimensional periodic systems that satisfy different types of monotonicity assumptions

sharp constant and extremal function for weighted trudinger moser type inequalities in r2 abstract in this note we prove the sharpness and the existence of extremal function for a trudinger moser type inequality in weighted sobolev spaces established by albuquerque alves medeiros in 2014 see 6 theorem 1 1

convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature we consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity we prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero we apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one for smooth strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time recently several papers have considered the flow of convex hypersurfaces by speeds which are homogeneous functions of the principal curvatures of degree 1 under suitable pinching conditions on the curvature of the initial hypersurface the aim is to prove that solutions become spherical as they contract to points this behaviour has been established for a wide range of flows where the speed is homogeneous of degree 1 in the principal curvatures including the mean curvature flow h1 the flow by nth root of gauss curvature ch1 square root of scalar curvature ch2 and a large family of other speeds a1 a3 a4 the first such result with degree of homogeneity higher than 1 was due to ben chow ch1 and concerned flow by powers of the gauss curvature he proved that flow by k with 1 n produces a spherical limiting shape provided the initial hypersurface is sufficiently pinched i n the sense that hij c hgij later such results were proved by schulze s3 for powers of the mean curvature by alessandroni and sinestrari for powers of the scalar curvature a as and by cabezas rivas and sinestrari crs for normalized flows by powers of elementary symmetric functions a feature of the results mentioned above is that the flows all have some divergence structure a point which is used crucially in deriving sufficient regularity of solutions to deduce the existence of a smooth limiting hypersurface in ch1 the divergence structure was used to deduce pinching of the principal curvatures using integral estimates in a manner similar to h1 in s3 a and as the curvature pinching was proved using maximum principle arguments but the divergence structure was still needed because higher regularity of solutions was established using results for divergence form porous medium equations to date we are not aware of any work which provides holder continuity for solutions of porous medium equations in non divergence form without assuming regularity of the coefficients in this paper we provide a geometric estimate which circumvents this difficulty we prove in section 3 that if the ratios of principal curvatures at each point are bounded in terms of the maximum principal curvature by a function which approaches one at infinity then the ratio of circumradius to inradius is as close to one as desired if the circumradius is sufficiently small using