nonlocal isoperimetric inequalities for kolmogorov fokker planck operators abstract in this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of kolmogorov fokker planck operators which are of interest in physics these operators are very degenerate and do not possess a variational structure the prototypical example was introduced by kolmogorov in his 1938 paper on brownian motion and the theory of gases our work has been influenced by ideas of m ledoux in the local case

surfaces immersed in lie algebras obtained from the sigma models we study some geometrical aspects of two dimensional orientable surfaces arising from the study of sigma models to this aim we employ an identification of with the lie algebra by means of which we construct a generalized weierstrass formula for immersion of such surfaces the structural elements of the surface like its moving frame the gauss weingarten and the gauss codazzi ricci equations are expressed in terms of the solution of the model defining it further the first and second fundamental forms the gaussian curvature the mean curvature vector the willmore functional and the topological charge of surfaces are expressed in terms of this solution we present detailed implementation of these results for surfaces immersed in and lie algebras

on functions of bounded p variation abstract we obtain estimates of the total p variation 1 p and other related functionals for a periodic function f l p 0 1 in terms of its l p modulus of continuity f p these estimates are sharp for any rate of the decay of f p moreover the constant coefficients in them depend on parameters in an optimal way

on the convergence rate of some nonlocal energies abstract we study the rate of convergence of some nonlocal functionals recently considered by bourgain brezis and mironescu and after a suitable rescaling we establish the convergence of the corresponding rate functionals to a limit functional of second order

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

the first non zero neumann p fractional eigenvalue abstract in this work we study the asymptotic behavior of the first non zero neumann p fractional eigenvalue 1 s p as s 1 and as p we show that there exists a constant k such that k 1 s 1 s p goes to the first non zero neumann eigenvalue of the p laplacian while in the limit case p we prove that 1 1 s 1 p goes to an eigenvalue of the holder laplacian

transport and diffusion of material quantities on propagating interfaces via level set methods we develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework level set methods are computational techniques for tracking moving interfaces they work by embedding the propagating interface as the zero level set of a higher dimensional function and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes the traditional level set method works in the trace space of the evolving interface and hence disregards any parameterization in the interface description consequently material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework we develop model equations and algorithmic techniques to extend the level set method to include these problems we demonstrate the accuracy of our approach through a series of test examples and convergence studies

full gradient stabilized cut finite element methods for surface partial differential equations we propose and analyze a new stabilized cut finite element method for the laplace beltrami operator on a closed surface the new stabilization term provides control of the full r 3 gradient on the

the dirichlet problem for nonlocal operators in this note we set up the elliptic and the parabolic dirichlet problem for linear nonlocal operators as opposed to the classical case of second order differential operators here the boundary data are prescribed on the complement of a given bounded set we formulate the problem in the classical framework of hilbert spaces and prove unique solvability using standard techniques like the fredholm alternative

convergence of solutions for the fractional cahn hilliard system abstract this paper deals with the cauchy dirichlet problem for the fractional cahn hilliard equation the main results consist of global in time existence of weak solutions characterization of parabolic smoothing effects implying under proper condition eventual boundedness of trajectories and convergence of each solution to a single equilibrium in particular to prove the convergence result a variant of the so called ojasiewicz simon inequality is provided for the fractional dirichlet laplacian and possibly non analytic but c 1 nonlinearities

maximizers for the variational problems associated with sobolev type inequalities under constraints we propose a new approach to study the existence and non existence of maximizers for the variational problems associated with sobolev type inequalities both in the subcritical case and critical case under the equivalent constraints the method is based on an useful link between the attainability of the supremum in our variational problems and the attainablity of the supremum of some special functions defined on 0 infty our approach can be applied to the same problems related to the fractional laplacian operators our main results are new in the critical case and in the setting of the fractional laplacian operator which was left open in the work of ishiwata and wadade math ann 364 1043 1068 2016 and ishiwata on variational problems associated with sobolev type inequalities and related topics http www rism it doc ishiwata pdf 2015 in the subcritical case our approach provides a new and elementary proof of the results of ishiwata and wadade 2016 and ishiwata 2015

extension of sampling inequalities to sobolev semi norms of fractional order and derivative data this paper devoted to sampling inequalities provides some extensions of previous results by arcangeli et al numer math 107 2 181 211 2007 j approx theory 161 198 212 2009 given a function u in a suitable sobolev space defined on a domain in mathbb r n sampling inequalities typically yield bounds of integer order sobolev semi norms of u in terms of a higher order sobolev semi norm of u the fill distance d between overline omega and a discrete set a subset overline omega and the values of u on a the extensions established in this paper allow us to bound fractional order semi norms and to incorporate if available values of partial derivatives on the discrete set both the cases of a bounded domain and omega mathbb r n are considered

long time behavior of solutions to the 3d hall magneto hydrodynamics system with one diffusion abstract this paper studies the asymptotic behavior of smooth solutions to the generalized hall magneto hydrodynamics system 1 1 with one single diffusion on the whole space r 3 we establish that in the inviscid resistive case the energy b t 2 2 vanishes and u t 2 2 converges to a constant as time tends to infinity provided the velocity is bounded in w 1 3 r 3 in the viscous non resistive case the energy u t 2 2 vanishes and b t 2 2 converges to a constant provided the magnetic field is bounded in w 1 r 3 in summary one single diffusion being as weak as b or u with small enough is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system

soliton surfaces associated with sigma models differential and algebraic aspects in this paper we consider both differential and algebraic properties of surfaces associated with sigma models it is shown that surfaces defined by the generalized weierstrass formula for immersion for solutions of the sigma model with finite action defined in the riemann sphere are themselves solutions of the euler lagrange equations for sigma models on the other hand we show that the euler lagrange equations for surfaces immersed in the lie algebra with conformal coordinates that are extremals of the area functional subject to a fixed polynomial identity are exactly the euler lagrange equations for sigma models in addition to these differential constraints the algebraic constraints in the form of eigenvalues of the immersion functions are systematically treated the spectrum of the immersion functions for different dimensions of the model as well as its symmetry properties and its transformation under the action of the ladder operators are discussed another approach to the dynamics is given i e description in terms of the unitary matrix which diagonalizes both the immersion functions and the projectors constituting the model

an adaptive finite element method for the laplace beltrami operator on implicitly defined surfaces we present an adaptive finite element method for approximating solutions to the laplace beltrami equation on surfaces in mathbb r 3 which may be implicitly represented as level sets of smooth functions residual type a posteriori error bounds which show that the error may be split into a residual part and a geometric part are established in addition implementation issues are discussed and several computational examples are given

analytic expressions of transfer function responses and choice of numerator coefficients zeros general analytic expressions of transfer function responses are derived in this paper the analytic forms include the numerator coefficients of the transfer function a vandermonde like matrix and a vector containing the transfer function eigenresponses the latter two only depending on the eigenvalues several characteristics of basic transient responses are derived e g their exact number of zeros an upper bound on the number of extrema of the step response is obtained an algorithm is suggested for the selection of the numerator coefficients assuming fixed eigenvalues using the derived characteristics of the basic transient responses effectively leading to a suboptimal choice furthermore the analytic formulas are used to calculate the numerator coefficients optimally by minimizing the step transient the ramp transient the parabolic transient etc simulations are presented indicating the properties of the basic transient responses and the transfer function response properties based on optimal and suboptimal choices of the numerator coefficients

the maz ya shaposhnikova limit in the magnetic setting abstract we prove a magnetic version of the maz ya shaposhnikova singular limit of nonlocal norms with vanishing fractional parameter this complements a general convergence result recently obtained by authors when the parameter approaches one

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

transient properties of type m continuous time scalar systems abstract it is shown that every asymptotically stable strictly proper type m continuous time scalar system tracks every polynomial trajectory of up to order m 2 with an instantaneous tracking error whose integral over the interval 0 identically vanishes this determines that the system will always exhibit overshoot in each of the above responses this phenomenon constitutes an essential manifestation of the existing trade off between steady state and transient accuracies

soliton surfaces associated with generalized symmetries of integrable equations in this paper based on the fokas et al approach fokas and gel fand 1996 commun math phys 177 203 20 fokas et al 2000 sel math 6 347 75 we provide a symmetry characterization of continuous deformations of soliton surfaces immersed in a lie algebra using the formalism of generalized vector fields their prolongation structure and links with the frechet derivatives we express the necessary and sufficient condition for the existence of such surfaces in terms of the invariance criterion for generalized symmetries and identify additional sufficient conditions which admit an explicit integration of the immersion functions of 2d surfaces in lie algebras we discuss in detail the su n valued immersion functions generated by conformal symmetries of the cpn 1 sigma model defined on either the minkowski or euclidean space we further show that the sufficient conditions for explicit integration of such immersion functions impose additional restrictions on the admissible conformal symmetries of the model defined on minkowski space on the other hand the sufficient conditions are identically satisfied for arbitrary conformal symmetries of finite action solutions of the cpn 1 sigma model defined on euclidean space

notes on limits of sobolev spaces and the continuity of interpolation scales we extend lemmas by bourgain brezis mironescu 2001 and maz ya shaposhnikova 2002 on limits of sobolev spaces to the setting of interpolation scales this is achieved by means of establishing the continuity of real and complex interpolation scales at the end points a connection to extrapolation theory is developed and a new application to limits of sobolev scales is obtained we also give a new approach to the problem of how to recognize constant functions via sobolev conditions

small data scattering of the inhomogeneous cubic quintic nls in 2 dimensions abstract the aim of this paper is to show the small data scattering for 2d icqnls i u t u k 1 x u 2 u k 2 x u 4 u under the assumption that j k l x b l j for j 0 1 2 l 1 2 and 0 b l l 2 3 we prove the small data scattering in an angularly regular sobolev space h 1 1 we use the decaying property of angularly regular functions which are defined as functions in sobolev space h 1 1 h 1 with angular regularity such that f h 1 and also use the recently developed angularly averaged strichartz estimates tao 2000 cho and lee 2013 guo et al 2018 in addition we suggest a sufficient condition for non existence of scattering

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

on a stack of surfaces obtained from the sigma models under the assumption that the sigma model is defined on the riemann sphere and its action functional is finite we derive surfaces induced by surfaces and we prove that the stacked surfaces coincide with each other which demonstrates the idempotency of the recurrent procedure in the process of finding the solutions of the model equations we prove that the euler lagrange equations for the projectors admit larger classes of solutions than the ones corresponding to rank 1 projectors

regularity results and harnack inequalities for minimizers and solutions of nonlocal problems a unified approach via fractional de giorgi classes abstract we study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a gagliardo type fractional seminorm we prove that minimizers of such non differentiable functionals are locally bounded holder continuous and that they satisfy a suitable harnack inequality hence we provide an extension of celebrated results of m giaquinta and e giusti to the nonlocal setting to do this we introduce a particular class of fractional sobolev functions reminiscent of that considered by e de giorgi in his seminal paper of 1957 the flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations

an adaptive octree finite element method for pdes posed on surfaces abstract the paper develops a finite element method for partial differential equations posed on hypersurfaces in r n n 2 3 the method uses traces of bulk finite element functions on a surface embedded in a volumetric domain the bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures the cartesian structure of the bulk mesh leads to easy and efficient adaptation process while the trace finite element method makes fitting the mesh to the surface unnecessary the number of degrees of freedom involved in computations is consistent with the two dimension nature of surface pdes no parametrization of the surface is required it can be given implicitly by a level set function in practice a variant of the marching cubes method is used to recover the surface with the second order accuracy we prove the optimal order of accuracy for the trace finite element method in h 1 and l 2 surface norms for a problem with smooth solution and quasi uniform mesh refinement experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom if grid adaptation is based on an appropriate error indicator the paper shows results of numerical experiments for a variety of geometries and problems including advection diffusion equations on surfaces analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted trace finite elements provide simple efficient and reliable tool for numerical treatment of pdes posed on surfaces

an eulerian space time finite element method for diffusion problems on evolving surfaces in this paper we study numerical methods for the solution of partial differential equations on evolving surfaces the evolving hypersurface in bbb r d defines a d dimensional space time manifold in the space time continuum bbb r d 1 we derive and analyze a variational formulation for a class of diffusion problems on the space time manifold for this variational formulation new well posedness and stability results are derived the analysis is based on an inf sup condition and involves some natural but nonstandard anisotropic function spaces based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous galerkin dg space time finite element discretization this dg space time method is explained and results of numerical experiments are presented that illustrate its properties

higher order meshing of implicit geometries part ii approximations on manifolds abstract a new concept for the higher order accurate approximation of partial differential equations on manifolds is proposed where a surface mesh composed by higher order elements is automatically generated based on level set data thereby it enables a completely automatic workflow from the geometric description to the numerical analysis without any user intervention a master level set function defines the shape of the manifold through its zero isosurface which is then restricted to a finite domain by additional level set functions it is ensured that the surface elements are sufficiently continuous and shape regular which is achieved by manipulating the background mesh the numerical results show that optimal convergence rates are obtained with a moderate increase in the condition number compared to handcrafted surface meshes

fractional sobolev norms and bv functions on manifolds abstract the bounded variation seminorm and the sobolev seminorm on compact manifolds are represented as a limit of fractional sobolev seminorms this establishes a characterization of functions of bounded variation and of sobolev functions on compact manifolds as an application the special case of sets of finite perimeter is considered

elliptic problems involving the fractional laplacian in rn abstract we study the existence and multiplicity of solutions for elliptic equations in r n driven by a non local integro differential operator which main prototype is the fractional laplacian the model under consideration denoted by p depends on a real parameter and involves two superlinear nonlinearities one of which could be critical or even supercritical the main theorem of the paper establishes the existence of three critical values of which divide the real line in different intervals where p admits no solutions at least one nontrivial non negative entire solution and two nontrivial non negative entire solutions

on analytic descriptions of two dimensional surfaces associated with the bb c p n 1 sigma model we study analytic descriptions of conformal immersions of the riemann sphere s2 into the sigma model in particular an explicit expression for two dimensional 2d surfaces obtained from the generalized weierstrass formula is given it is also demonstrated that these surfaces coincide with those obtained from the sym tafel formula these two approaches correspond to parametrizations of one and the same surface in

a quantitative isoperimetric inequality for fractional perimeters abstract recently frank and seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from sobolev seminorms of fractional order this isoperimetric inequality is improved here in a quantitative form

on formulae decoupling the total variation of bv functions abstract in this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non local functionals one special case shows that rescalings of the fractional laplacian of a function u s b v converge strictly to the singular portion of d u

limits of higher order besov spaces and sharp reiteration theorems abstract we compute the limits of higher order besov norms and derive the sharp constants for certain forms of the sobolev embedding theorem our results extend to higher order spaces the recent work by brezis bourgain mironescu and maz ya shaposhnikova the interpolation methods we develop are of interest on their own and could have applications to related inequalities

fractional sobolev spaces from a complex analytic viewpoint abstract for any non zero function u in the schwartz space s r n we prove that s u s 2 2 can be extended to c as a transcendental meromorphic function which establishes a connection between the bourgain brezis mironescu s formula maz ya shaponshikova s formula and the residues of the transcendental meromorphic function of s u s 2 2 at s 0 and s 1 separately moreover we study the function properties of u s 2 2 and obtain the convergence rate version of the bourgain brezis mironescu s formula and maz ya shaponshikova s formula and we also obtain the following sharp interpolation inequality for any u w 1 2 r n and s 0 1 we have u s 2 2 n 2 1 2 2 s 1 n 2 s 1 s sin s u 2 2 1 s u 2 2 s

homogeneous endpoint besov space embeddings by hausdorff capacity and heat equation two embeddings of a homogeneous endpoint besov space are established via the hausdorff capacity and the heat equation meanwhile a co capacity formula and a trace inequality are derived from the besov space

anisotropic fractional perimeters the anisotropic fractional s perimeter with respect to a convex body k in rn is shown to converge as s 1 to the anisotropic perimeter with respect to the moment body of k for anisotropic fractional s seminorms on bv rn the corresponding result is established generalizing results of bourgain brezis mironescu and davila minimizers of the anisotropic fractional isoperimetric inequality with respect to k are shown to converge to the moment body of k as s 1 anisotropic fractional sobolev inequalities are established for a borel set e r and 0 s 1 the fractional s perimeter of e is given by ps e

fractional cahn hilliard allen cahn and porous medium equations abstract we introduce a fractional variant of the cahn hilliard equation settled in a bounded domain r n and complemented with homogeneous dirichlet boundary conditions of solid type i e imposed in the whole of r n after setting a proper functional framework we prove existence and uniqueness of weak solutions to the related initial boundary value problem then we investigate some significant singular limits obtained as the order of either of the fractional laplacians appearing in the equation is let tend to 0 in particular we can rigorously prove that the fractional allen cahn fractional porous medium and fractional fast diffusion equations can be obtained in the limit finally in the last part of the paper we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits

non linear ground state representations and sharp hardy inequalities we determine the sharp constant in the hardy inequality for fractional sobolev spaces to do so we develop a non linear and non local version of the ground state representation which even yields a remainder term from the sharp hardy inequality we deduce the sharp constant in a sobolev embedding which is optimal in the lorentz scale in the appendix we characterize the cases of equality in the rearrangement inequality in fractional sobolev spaces

robust preconditioning for xfem applied to time dependent stokes problems we consider a quasi stationary stokes interface problem with a reaction term proportional to tau 1 delta t geq 0 as obtained by a time discretization of a time dependent stokes problem the mesh used for space discretization is not aligned with the interface we use the p 1 extended finite element space for the pressure approximation and the standard conforming p 2 finite element space for the velocity approximation a pressure stabilization term known from the literature is added since the fe pair is not lbb stable for the stabilized discrete bilinear form we derive a new inf sup stability result a new schur complement preconditioner is proposed and analyzed we present an analysis which proves robustness of the preconditioner with respect to h tau with tau in 0 c 0 cup c 1 h 2 infty and the position of the interface numerical results are included which indicate that the preconditioner is robust for the whole parameter range tau geq 0 and also with respect to the viscos

brief fundamental limitations due to jω axis zeros in siso systems this paper shows that fundamental limitations arise from the presence of stable zeros on or near the j w axis the stability of the closed loop system is used in conjunction with the laplace transform to show that time domain integral constraints must be satisfied by the system response the presence of stable zeros near the j w axis together with upper limits on the permissible overshoot of the output signal place an effective lower bound on the settling time of the closed loop system the constraints are illustrated by an application to the control of a single stand reversing mill

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality

on the bourgain brezis and mironescu theorem concerning limiting embeddings of fractional sobolev spaces the article is concerned with the bourgain brezis and mironescu theorem on the asymptotic behaviour of the norm of the sobolev type embedding operator ws p lpn n sp as s 1 and s n p thei

boundary behavior of nonlocal minimal surfaces abstract we consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary by a series of detailed examples we show that nonlocal minimal surfaces may stick at the boundary of the domain even when the domain is smooth and convex this is a purely nonlocal phenomenon and it is in sharp contrast with the boundary properties of the classical minimal surfaces in particular we show stickiness phenomena to half balls when the datum outside the ball is a small half ring and to the side of a two dimensional box when the oscillation between the datum on the right and on the left is large enough when the fractional parameter is small the sticking effects may become more and more evident moreover we show that lines in the plane are unstable at the boundary namely small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary by a quantity that is proportional to a power of the perturbation in all the examples we present concrete estimates on the stickiness phenomena also we construct a family of compactly supported barriers which can have independent interest

asymptotic stability for nonlinear damped kirchhoff systems involving the fractional p laplacian operator abstract this paper is devoted to the question of global and local asymptotic stability for nonlinear damped kirchhoff systems with homogeneous dirichlet boundary conditions under fairly natural assumptions on the external force f f t x u the distributed damping q q t x u u t the perturbation term u p 2 u and the dissipative term t m u s p u t p 2 u t with 0 and in l loc 1 r 0 when the initial data are in a special region here u u 1 u n u t x represents the vectorial displacement with n 1 particular attention is devoted to the asymptotic behavior of the solutions in the linear case specified in section 5 finally the results are extended to problems where the fractional p laplacian is replaced by a more general elliptic nonlocal integro differential operator the paper extends in several directions recent theorems and covers also the so called degenerate case that is the case in which m is zero at zero

an adaptive surface finite element method based on volume meshes in this paper we define an adaptive version of a recently introduced finite element method for numerical treatment of elliptic pdes defined on surfaces the method makes use of a standard outer volume mesh to discretize an equation on a two dimensional surface embedded in mathbb r 3 extension of the equation from the surface is avoided but the number of degrees of freedom d o f is optimal in the sense that it is comparable to methods in which the surface is meshed directly in previous work it was proved that the method exhibits optimal order of convergence for an elliptic surface pde if the volume mesh is uniformly refined in this paper we extend the method and develop an a posteriori error analysis which admits adaptively refined meshes the reliability of a residual type a posteriori error estimator is proved and both reliability and efficiency of the estimator are studied numerically in a series of experiments a simple adaptive refinement strategy based on the error estimator is numerical

nonlocal problems with critical hardy nonlinearity abstract by means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p laplacian and a combined sobolev and hardy nonlinearity at subcritical and critical growth

technical communique linear multivariable servomechanisms revisited system type and accuracy trade offs it is shown that every scalar closed loop output of a linear multivariable type m servomechanism where m is larger than one will always track its corresponding reference trajectory with an instantaneous tracking error whose integral over time identically vanishes this is true for all polynomial reference signals of up to order m 2 the simplest implication of this result is that the step responses of a multivariable servomechanism which is capable of asymptotically following ramp signals must all display overshoot this is analogous to the familiar classical control result new insight into tracking of dependent signals is also provided using the notion of generalized system type

some notes on supersolutions of fractional p laplace equation abstract this paper gives a further understanding of supersolutions to the regional fractional p laplace equation to do so the obstacle problem and the fractional superharmonic functions associated with this equation are further studied in particular we clarify the connections among supersolutions of the fractional p laplace equation the fractional superharmonic functions and solutions to the obstacle problem

fractional order orlicz sobolev spaces abstract in this paper we define the fractional order orlicz sobolev spaces and prove its convergence to the classical orlicz sobolev spaces when the fractional parameter s 1 in the spirit of the celebrated result of bourgain brezis mironescu we then deduce some consequences such as convergence of the modulars and convergence of solutions for some fractional versions of the g operator as the fractional parameter s 1

nonlocal minimal graphs in the plane are generically sticky we prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities more precisely we show that if a nonlocal minimal graph in a slab is continuous up to the boundary then arbitrarily small perturbations of the far away data produce boundary discontinuities hence either a nonlocal minimal graph is discontinuous at the boundary or a small perturbation of the prescribed conditions produces boundary discontinuities the proof relies on a sliding method combined with a fine boundary regularity analysis based on a discontinuity smoothness alternative namely we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is holder continuous up to the boundary in this spirit we prove that the boundary regularity of nonlocal minimal graphs in the plane jumps from discontinuous to c 1 gamma with no intermediate possibilities allowed in particular we deduce that the nonlocal curvature equation is always satisfied up to the boundary as an interesting byproduct of our analysis one obtains a detailed understanding of the switch between the regime of continuous and hence differentiable nonlocal minimal graphs to that of discontinuous and hence with differentiable inverse ones

new results on the canham helfrich membrane model via the generalized weierstrass representation the canham helfrich membrane model is discussed by making use of the generalized weierstrass representation formulae for arbitrary surfaces immersed into the three dimensional euclidean space particular solutions of the shape equation are considered and correlators describing thermal fluctuations in the one loop approximation are computed

surfaces on lie groups on lie algebras and their integrability it is shown that the problem of the immersion of a 2 dimensional surface into a 3 dimensional euclidean space as well as then dimensional generalization of this problem is related to the problem of studying surfaces in lie groups and surfaces in lie algebras a particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 2 matrices using an arbitrary parametrization it is also shown that this generality of parametrization is useful for studying integrable surfaces i e surfaces described by integrable equations in particular starting from a suitable lax pair i e a suitable integrable equation it is possible to construct explicitly large classes of integrable surfaces

existence of entire solutions for schrodinger hardy systems involving two fractional operators abstract this paper deals with the existence of nontrivial nonnegative solutions of schrodinger hardy systems driven by two possibly different fractional laplacian operators via various variational methods the main features of the paper are the presence of the hardy terms and the fact that the nonlinearities do not necessarily satisfy the ambrosetti rabinowitz condition moreover we consider systems including critical nonlinear terms as treated very recently in literature and present radial versions of the main theorems finally we briefly show how to extend the previous results when the fractional laplacian operators are replaced by more general elliptic nonlocal integro differential operators

technical communique on type number concept in fractional order systems in this note the type number concept is defined for fractional order systems based on this definition it is shown that fractional order systems having type numbers more than 1 can not track some classes of reference inputs without any overshoot

invariant recurrence relations for mathbb c p n 1 models in this paper we present invariant recurrence relations for the completely integrable euclidean sigma model in two dimensions defined on the riemann sphere s2 when its action functional is finite we determine the links between successive projection operators wavefunctions of the linear spectral problem and immersion functions of surfaces in the algebra together with the outlines of the proofs our formulation preserves the conformal and scaling invariance of these quantities certain geometrical aspects of these relations are described we also discuss the singularities of the meromorphic solutions of the model and show that they do not affect the invariant quantities we illustrate the construction procedure through the examples of the and models

stabilized cutfem for the convection problem on surfaces we develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms the discrete piec