quasi minimizing surfaces in hyperbolic space let h 3 g denote 3 dimensional hyperbolic space with constant curvature 1 we are interested in surfaces f in h 3 with the following quasi minimality property if it f h 3 0 1 which does not depend on the homotopy if this condition is satisfied with q 1 then f is called homotopically minimizing a surface f c h 3 with this property is not necessarily area minimizing in contrast to the situation in higher dimensions cf 13 our main result is

multiplicity of solutions to a nonlinear boundary value problem of concave convex type abstract problem p p u u p 2 u u r 1 u x u p 2 u u s 1 u x where r n is a bounded smooth domain is the unit outward normal at p is the p laplacian operator and 0 is a parameter was studied in sabina de lis 2011 and sabina de lis and segura de leon in press among other features it was shown there that when exponents lie in the regime 1 s p r a minimal positive solution exists if 0 for a certain finite while no positive solutions exist in the complementary range furthermore in the radially symmetric case a second positive solution exists for varying in the same full range 0 provided r p our main achievement in this work just asserts that such global multiplicity feature holds true when is a general domain to show such result the well known brezis nirenberg variational result in brezis and nirenberg 1993 must be extended to the framework of p this is the second main contribution in the present work

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

a finite volume method with linearisation in time for the solution of advection reaction diffusion systems the numerical solution in one space dimension of advection reaction diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models this work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest the basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method the resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations

a finite volume formulation of compact central schemes on arbitrary structured grids in the present paper a formulation allowing the use of compact schemes in the finite volume context on arbitrary meshes is presented the proposed formulation is based on the use of an implicit formula to evaluate the fluxes on the cell faces a series of numerical experiments for a 2d model convection equation a flat plate a subsonic vortical problem as well as the les simulation of channel flow has been carried out the results indicate an important improvement in obtained accuracy compared to a standard central finite volume formulation

complete linear weingarten surfaces of bryant type a plateau problem at infinity in this paper we study a large class of weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3 space we show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the euclidean 3 space and we use it to study their completeness we also establish some existence and uniqueness theorems by studing the plateau problem at infinity when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family and how many embedded solutions are there

extremal domains on hadamard manifolds abstract we investigate the geometry and topology of extremal domains in a hadamard manifold i e domains that support a positive solution to an overdetermined elliptic problem oep first we study narrow properties of such domains and characterize the boundary at infinity we give an upper bound for the hausdorff dimension of its boundary at infinity and how the domain behaves at infinity this shows interesting relations with the singular yamabe problem later we focus on extremal domains in the hyperbolic space symmetry and boundedness properties will be shown in a certain sense we extend levitt rosenberg s theorem to oeps which suggests a strong relation with constant mean curvature hypersurfaces in particular we are able to prove the berestycki caffarelli nirenberg conjecture under certain assumptions either on the boundary at infinity of the extremal domain or on the oep itself also a height estimate for solutions on extremal domains in a hyperbolic space will be given

hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity in this paper we study the problem of finding smooth complete hypersurfaces of constant mean curvature in hyperbolic space with a prescribed asymptotic boundary at infinity our main results are proved by deriving a priori global gradient estimates based on the classical maximum principle for elliptic equations

erratum on the dirichlet problem for minimal graphs in hyperbolic space r n 0 that can be identified with the asymptotic infinity of hn 1 let nn 1 hn 1 be a closed embedded n 1 dimensional submanifold then there exists a complete absolutely area minimizing integral n current asymptotic to nn 1 at infinity see for example 1 2 it was also noted in 1 2 that in the case n 6 currents obtained are smoothly embedded complete hypersurfaces in the case n 7 as in the euclidean case there can be a singular set of hausdorff dimension at most n 7

minimal and weakly trapped submanifolds in standard static spacetimes abstract in this article we obtain some rigidity results for spacelike submanifolds of arbitrary codimension in standard static spacetimes in particular under appropriate assumptions we prove that they must be contained in a spacelike slice which enables us to characterize minimal closed submanifolds of codimension two we also prove new rigidity results for complete non compact weakly trapped submanifolds of arbitrary codimension in these ambient spacetimes

a convex concave problem with a nonlinear boundary condition in this paper we study the existence of nontrivial solutions of the problem u u u p 2uin u u q 2uon with 1 q 2 n 1 n 2 and 1 p 2n n 2 in the concave convex case i e 1 q 2 p if is small there exist two positive solutions while for large there is no positive solution when p is critical and q subcritical we obtain existence results using the concentration compactness method finally we apply the implicit function theorem to obtain solutions for small near u0 1

regularity at infinity for area minimizing hypersurfaces in hyperbolic space equipped with the hyperbolic metric y z dxz dy2 a standard compactification of ih involves adding the boundary r 0 w so that i i is simply the one point compactification of the euclidean closed half space r 0 suppose 0 7 any interior singularities of m must remain in a bounded region of hyperbolic space near points of f in the euclidean topology m w f may thus be described as the graph of a function this function is the solution of an interesting partial differential equation that becomes degenerate along the part of the boundary corresponding to f the second author has recently established l a boundary higher regularity result for this equation which implies in particular that m w f is r if f is cgk for k 2 3 finally in case f bounds a star

a posteriori error analysis of a cell centered finite volume method for semilinear elliptic problems in this paper we conduct a goal oriented a posteriori analysis for the error in a quantity of interest computed from a cell centered finite volume scheme for a semilinear elliptic problem the a posteriori error analysis is based on variational analysis residual errors and the adjoint problem to carry out the analysis we use an equivalence between the cell centered finite volume scheme and a mixed finite element method with special choice of quadrature

a compact finite volume method and its extrapolation for elliptic equations with third boundary conditions a fourth order compact finite volume method is constructed for one and two dimensional elliptic equations with third boundary conditions in this paper taking two point boundary value problem of third kind as an example we derive some useful high accuracy post processing formulas to recover the numerical derivatives at the nodes or midpoints of the elements we also improve the accuracy of the compact finite volume scheme from order 4 to 6 based on richardson extrapolation by rigorously proving the scheme has error asymptotic expansion numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme as well as its post processing formulas and extrapolation

a finite volume method for one dimensional reaction diffusion problems abstract a truly conservative finite volume method for one dimensional reaction diffusion equations is presented the method is based on piecewise second degree polynomials approximations that result in linear fluxes a symmetric mass matrix and a semi discrete system of nonlinear ordinary differential equations in time it is shown that the semi discrete method is linearly stable provided that the problem is mathematically well posed it is also shown that implicit discretizations of the time variable are stable whereas the explicit one is conditionally stable and has a smaller domain of linear stability than the standard explicit finite difference method it is also illustrated that upon lumping the mass matrix the method coincides with a finite difference discretization of the original problem a detailed formulation of the dirichlet neumann and robin boundary conditions is presented such a formulation makes use of control volumes which are here referred to as incomplete ones although the finite volume technique presented here uses a node centered formulation it can be easily extended to cell centered ones the technique is also developed for multidimensional problems

complete minimal varieties in hyperbolic space it is a basic problem in the subject of minimal varieties to determine the class of complete minimal and area minimizing varieties m k in a given riemannian manifold n besides being of their own intrinsic interest the cases r of euclidean space and mn sn of the sphere are fundamental in understanding the local structure of minimal varieties due to the existence of tangent cones to general minimal varieties there has been much study of the complete areaminimizing varieties in these cases in particular relating to the bernstein conjecture and minimizing cones e g l2 f1 nevertheless many deep problems remain unsolved this paper is centered around the question of existence of complete areaminimizing varieties in hyperbolic space h here by way of contrast to the situation in ir n or s we find a comparatively complete solution to the problem of existence in fact the main result of this paper exhibits a wealth of complete area minimizing varieties in h in all dimensions and codimensions more precisely we have