stability of holomorphic foliations with split tangent sheaf we show that the set of singular holomorphic foliations on projective spaces with split tangent sheaf and good singular set is open in the space of holomorphic foliations we also give a cohomological criterion for the rigidity of holomorphic foliations induced by group actions and prove the existence of rigid codimension one foliations of degree n 1 on p n for every n 3

stefan problems with nonlinear diffusion and convection abstract we consider a class of stefan type problems having a convection term and a pseudomonotone nonlinear diffusion operator assuming data in l 1 we prove existence uniqueness and stability in the framework of renormalized solutions existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to l 1 convergence of the data uniqueness is proved through a classical l 1 contraction principle obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators

comparison and stability of solutions to a class of quasilinear parabolic problems on presente des resultats de comparaison et d unicite pour les solutions de problemes aux valeurs limites quasilineaires paraboliques avec et sans obstacles un resultat de stabilite dans l 1 donne la stabilisation asymptotique dans cet espace quand t vers le probleme elliptique correspondant

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

renormalized entropy solutions of scalar conservation laws with boundary condition we study an initial boundary value problem for a scalar conservation law u t div phi u f on a bounded domain existence and uniqueness of a renormalized entropy solution is established for general l 1 data phi is an element ofc r r n

renormalised solutions of nonlinear parabolic problems with l1 data existence and uniqueness in this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem where the data f and u 0 belong to l 1 0 t and l 1 and where the function a 0 t n n is monotone but not necessarily strictly monotone and defines a bounded coercive continuous operator from the space into its dual space the renormalised solution is an element of c 0 0 t l 1 such that its truncates t k u belong to with this solution satisfies the equation formally obtained by using in the equation the test function s u where belongs to and where s belongs to c with

parabolic equations with nonlinear singularities abstract we show the existence of positive solutions u l 2 0 t h 0 1 for nonlinear parabolic problems with singular lower order terms of the asymptote type more precisely we shall consider both semilinear problems whose model is u t u u 1 u f x t in 0 t u x 0 u 0 x in u x t 0 on 0 t and quasilinear problems having natural growth with respect to the gradient whose model is u t u u 2 u f x t in 0 t u x 0 u 0 x in u x t 0 on 0 t with 0 moreover we prove a comparison principle and as an application we study the asymptotic behavior of the solution as t goes to infinity

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

on the minimal solution for some variational inequalities abstract applying an asymptotic method the existence of the minimal solution to some variational elliptic inequalities defined on bounded or unbounded domains is established the minimal solution is obtained as limit of solutions to some classical variational inequalities defined on domains becoming unbounded when some parameter tends to infinity the considered quasilinear operators are only monotone not strictly and noncoercive some related comparison principles are also investigated

completely positive measures and feller semigroups on donne d abord une approche unifiee aux mesures completement positives qui jouent un role important dans la theorie des equations de volterra a travers le theoreme de bernstein puis on etudie la solution d une equation de transport liee par une mesure non negative qu elle definit a l equation qui caracterise les noyaux completement positifs on obtient une desorption complete des generateurs de semigroupes de feller induits par de telles mesures on donne une application aux equations de flux thermique lineaire dans des materiaux avec memoire

optimal decay rate of solutions for cahn hilliard equation with inertial term in multi dimensions abstract the cauchy problem of the cahn hilliard equation with inertial term is considered based on green s function method together with energy estimates we get the global in time existence and optimal decay rate of solutions furthermore the viscous case is investigated in the last section

on some nonlinear elliptic equations involving derivatives of the nonlinearity the existence of weak solutions of the dirichlet problem for nonlinear elliptic equations of the type x j a ij x u x i b j u a x u f is proved for nonlinearities b j which are only continuous the proof is based on the observation that an l a priori estimate is valid which is independent of b j the second part contains several uniqueness results and counterexamples showing that uniqueness may fail in general

quasi linear degenerate elliptic problems with l1 data abstract we prove existence and uniqueness of a solution for a class of quasi linear problems with l 1 data the diffusion matrix a x u is allowed to degenerate with respect to the unknown u we obtain uniqueness of the solution under a weak assumption on a x u that permits to consider highly oscillating or and increasing coefficients with respect to u

existence uniqueness and homogenization of the second order slip reynolds equation the decreasing of the distance between the head and the magnetic disk surface leads to a model of the behaviour of the air by using a modified reynolds equation the existence and the uniqueness of this stationary equation is under some conditions on the data proved using both fixed point and monotonicity techniques double scale analysis allows us to obtain the associated homogenized equation

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

on the density of algebraic foliations without algebraic invariant sets let x be a smooth complex projective variety of dimension greater than or equal to 2 l an ample line bundle and k 0 an integer we prove that a generic global section of the twisted tangent sheaf x l k gives rise to a foliation of x without any proper algebraic invariant subvarieties of nonzero dimension as a corollary we obtain a dynamical characterization of ampleness for line bundles over smooth projective surfaces

phase field equations with memory the hyperbolic case we present a phenomenological theory for phase transition dynamics with memory which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential thereafter we focus on the implications of our theory in the hyperbolic case and we derive asymptotically an equation of motion in two dimensions for the interface between two different phases this equation can be considered as a hyperbolic generalization of the classical flow by mean curvature equation as well as a generalization of the born infeld equation we use a crystalline algorithm to study the motion of closed curves for the generalized hyperbolic flow by mean curvature equation our hyperbolic generalization of flow by mean curvature and present some numerical results which indicate that a certain type of two dimensional relaxation damped oscillation may occur

a strongly degenerate quasilinear equation the parabolic case we prove the existence and uniqueness of entropy solutions of the neumann problem for the quasilinear parabolic equation ut a u du where a z f z and f is a convex function of with linear growth as satisfying other additional assumptions in particular this class includes the case where f z z 0 and is a convex function with linear growth as

uniqueness and nonuniqueness for the approximation of quasilinear elliptic equations we investigate the issue of uniqueness and nonuniqueness for the approximate solution of quasilinear elliptic equations in particular we show that even if the continuous problem admits a unique solution its approximation by finite elements may lead to several approximate solutions

on the reynolds lubrication equation let be a bounded open set in r 2 with smooth boundary is the region where two solid bodies are in contact these two solids are moving and the average of their velocities is denoted by v v 1 v 2 the pressure p p x which develops in a fluid layer confined between these two bodies satisfies the so called reynolds lubrication equation 1 h 3 p 6 v h in 2 p p a on here is the density of the fluid 0 its dynamic viscosity h h x is the distance between the two bodies and p a 0 is the given ambient pressure we will assume all along that h is a lipschitz continuous function such that 3 0 h 1 h x h 2 a e x x y 4 h x ha e x x y where h 1 h 2 h are positive constants when the fluid is incompressible i e is a positive constant existence and uniqueness for a solution to 1 2 is a trivial matter since the equation is linear in the compressible case when 5 p the problem becomes more challenging the paper is organized as follows in section 1 we investigate the problem of existence of a solution for 1 2 when is given by 5 in section 2 we consider a case where 1 2 reduces to a one dimensional problem and we study the question of uniqueness as well as the shape of the solution finally in a third section we give some extensions of the results of section 2 in higher dimension

uniqueness for elliptic problems with locally lipschitz continuous dependence on the solution abstract we consider a class of elliptic equations of the type div a x u u f div g with dirichlet boundary conditions and with f belonging to l 1 using the framework of renormalized solutions we prove the uniqueness of a solution when the usual lipschitz condition on the function a is relaxed

convergence of finite volume approximations for a nonlinear elliptic parabolic problem a continuous approach we study the approximation by finite volume methods of the model parabolic elliptic problem b v t rm div dv p 2 dv on 0 t times omega subset r times r d with an initial condition and the homogeneous dirichlet boundary condition because of the nonlinearity in the elliptic term a careful choice of the gradient approximation is needed we prove the convergence of discrete solutions to the solution of the continuous problem as the discretization step h tends to 0 under the main hypotheses that the approximation of the operator rm div dv p 2 dv provided by the finite volume scheme is still monotone and coercive and that the gradient approximation is exact on the affine functions of x in om an example of such a scheme is given for a class of two dimensional meshes dual to triangular meshes in particular for structured rectangular and hexagonal meshes the proof uses the rewriting of the discrete problem under a continuous form this permits us to directly apply the alt luckhaus variational techniques which are known for the continuous case

existence and comparison of maximal and minimal solutions for pseudomonotone elliptic problems in l 1 we consider the nonhomogeneous dirichlet problem div a x u u f in u w on where div a x u u is a pseudomonotone operator of leray lions type defined in w1 p0 w w1 p and f is in l1 under suitable assumptions of locally lipschitz or locally holder continuity of a x s with respect to s we prove the existence of maximal and minimal renormalized solutions and comparison results with respect to data f and w the results include examples of nonmonotone operators of p laplace type for any p 1 for which it is known that uniqueness of solutions does not hold

uniqueness results for nonlinear elliptic equations with a lower order term abstract we prove the uniqueness of the weak solution of noncoercive nonlinear elliptic problems whose model is u w 0 1 p div a x 1 u 2 p 2 2 u b x 1 u 2 1 2 f in d where is a bounded open subset of r n n 2 p satisfies p 2 n n 1 a is a function belonging to l such that a x 0 f belongs to the dual space w 1 p b belongs to some lebesgue space l r with r r n p and belongs to the interval 0 n p r with n p r and r n p functions which are specified below

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

dynamics of layered interfaces arising from phase boundaries the dynamics of a material in two phases is studied in the context of phase field models based on a landau ginzburg free energy functional they consist of a system of two nonlinear diffusion equat

comparison principle for some classes of nonlinear elliptic equations abstract we prove a comparison principle for second order quasilinear elliptic operators in divergence form when a first order term appears we deduce uniqueness results for weak solutions to dirichlet problems when data belong to the natural dual space

pointwise estimates of solutions for the cahn hilliard equation with inertial term in multi dimensions abstract in this paper we study the pointwise estimates of solutions for the cahn hilliard equation with inertial term in multi dimensions by the green function method furthermore we obtain the l p convergence rate of the solution

a conserved phase field system with memory and relaxed chemical potential abstract in this paper we propose a conserved phase field model with memory and relaxed chemical potential our model could be considered also in the frame of cahn hilliard s theory of spinodal decomposition of a binary mixture where there is a coupling to a slowly relaxing variable e g in phase separation of polymer mixtures near a glass transition the global well posedness in an l p setting is obtained under suitable relaxation functions and under homogeneous conditions

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

on the uniqueness of the solution of the evolution dam problem the purpose here is to investigate some aspects of the evolution dam problem partial chi alpha u partial t delta u rm div chi e in q chi in h u in q u geq 0 in q u phi in sigma 2 partial u partial v chi ev leq 0 on sigma 2 cap phi 0 e in bf r n e 0 cdots 0 1 partial u partial v chi ev 0 on sigma 1 here h is the heaviside maximal graph h s 1 if s 0 h s 0 1 if s 0 alpha is the storativity constant alpha 0 if the fluid is incompressible and alpha 0 if the fluid is compressible also q omega times 0 t where omega is a bounded domain in bf r n omega represents a porous medium separating a finite number of reservoirs with lipschitz boundary g where g is divided into two parts g 1 which is the impervious part and g 2 which is the pervious part with the following assumptions g 2 is an open subset of g g 2 not emptyset g 2 cap g 1 emptyset g 2 cup g 1 g and e nu leq 0 a e on g 1 sigma i denotes g i times 0 t i 1 2 the pressure at sigma 2 is known and is represented by a nonnegative function phi in c 0 1 overline q the above problem can be completed by prescribing an initial value to chi alpha u the existence of a weak solution of the evolution dam problem is well known however many properties of the solutions of the initial value dam problem in particular comparison and uniqueness theorems continue to be open for a general domain omega the results of this paper give precise information about the comparison theorem 4 7 and the uniqueness corollary 4 9 statements the key point to prove comparison and uniqueness results is that the function h mapsto chi x x n h t h x in bf r n 1 is a e nonincreasing

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

irreducible components of the space of holomorphic foliations of degree two in cp n n 3 in this paper we will prove that the space of holomorphic fo liations of codimension 1 and degree 2 in cp n n 3 has six irreducible components

anisotropic tempered diffusion equations abstract we introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type under an amenable set of hypothesis involving a very natural potential function these models can be shown to belong to the entropy solution framework devised by andreu et al 2005 therefore ensuring well posedness we connect the properties of this potential with those of the associated cost function thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions moreover we characterize the anisotropic spreading properties of these models and we determine the rankine hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows

uniqueness of the bounded solution to a strongly degenerate parabolic problem abstract this paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem the equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies by using holmgren s approach we prove that the bounded solution of the associated initial boundary value problem is unique under some essentially necessary condition on the growth of the convection

local in time regularity results for some flux limited diffusion equations of porous media type abstract we prove some regularity results for solutions of some flux limited diffusion equations of porous media type for lipschitz initial data or assuming a uniform gradient bound on some power of the data including the fast diffusion case in which the results are global in time we also develop the existence and uniqueness theory for solutions of the fast diffusion case which was not covered in the current literature

the equivalence of weak solutions and entropy solutions of nonlinear degenerate second order equations abstract we prove the equivalence of weak solutions and entropy solutions of an elliptic parabolic hyperbolic degenerate equation g t t b u div u f with homogeneous dirichlet conditions and initial conditions as a result of the equivalence we obtain the l1 contraction principle and uniqueness of weak solutions of elliptic parabolic degenerate equations

well posedness results for triply nonlinear degenerate parabolic equations we study well posedness of triply nonlinear degenerate elliptic parabolic hyperbolic problems of the kind b u t div a u u u f u t 0 u0 in a bounded domain with homogeneous dirichlet boundary conditions the nonlinearities b and are supposed to be continuous non decreasing and the nonlinearity a falls within the leray lions framework some restrictions are imposed on the dependence of a u u on u and also on the set where degenerates a model case is a u u f b u u u k u a0 u with a nonlinearity which is strictly increasing except on a locally finite number of segments and the nonlinearity a0 which is of the leray lions kind we are interested in existence uniqueness and stability of l entropy solutions for the parabolic hyperbolic equation b id we obtain a general continuous dependence result on data u0 f and nonlinearities b a similar result is shown for the degenerate elliptic problem which corresponds to the case of b 0 and general non decreasing surjective existence uniqueness and continuous dependence on data u0 f are shown in more generality for instance the assumptions b r r and the continuity of b 1 permit to achieve the well posedness result for bounded entropy solutions of this triply nonlinear evolution problem