new solutions for effective elastic moduli of microcracked solids abstract a new energy balance equation is proposed to evaluate the effective moduli of solids with randomly dispersed cracks in this study it is assumed that the potential energy released by embedding a circular or spherical rve with microcracks into an infinite matrix is equal to that induced by introducing its effective medium into the identical infinite matrix new non interacting and self consistent solutions for the effective moduli of linear elastic isotropic solids with randomly oriented microcracks were evaluated on the basis of the present energy balance equation the linearization of the two present solutions leads to the dilute solution in the limit of small crack density comparison studies with various existing solutions are also presented in detail

nodal sets and morse indices of solutions of super linear elliptic pdes in this paper we develop a sturm liouville type theory for the nodal sets and morse indices of solutions of super linear elliptic pdes with dirichlet boundary condition it shows that there are some relationships between analytic properties e g lp norm vanishing order of the nodal point andn 1 dimensional hausdorff measure of the nodal set of the solutions as the functions on the domain and morse indices of the solutions as the critical points of the functionalj u 12 u 2 f x u dxonh10

morse index of some min max critical points i application to multiplicity results on donne un resultat general sur des bornes inferieures pour des indices de morse de points critiques obtenus par des principes de min max en combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d equations elliptiques semi lineaires d ordre 2

the numerical calculation of two dimensional effective moduli for microcracked solids abstract there exist several micromechanics models for the determination of the effective moduli of microcracked solids and crack density is the only parameter in these models that characterizes the effect of microcracking a numerical hybrid bem method in conjunction with a unit cell model is proposed in the present paper to evaluate these micromechanics models a unit cell which can be considered as a representative block in the solid contains randomly distributed microcracks the unit cell is then assumed to be periodic in the solid so as to account for interactions between cracks inside and outside the cell there are stochastic variations of the estimated moduli for different microcrack distributions two groups of microcracks with the same crack density one with a low number of large cracks and the other with a large number of small cracks show the same range of stochastic variations and the same mean of effective moduli for random distributions of microcracks the effective moduli based on this numerical method for randomly distributed cracks and parallel cracks are compared with those from various micromechanics models while the differential method provides the closest estimation to the mean of the numerical results at low crack density the generalized self consistent method is much more accurate at relatively high crack density

morse indices of solutions for super linear elliptic pde s abstract in this paper we establish l estimates for solutions of some general superlinear and subcritical elliptic equations via the morse index our results generalize and improve the previous works bahri and lions 1992 yang 1998

solutions of superlinear elliptic equations and their morse indices we consider here solutions of semilinear second order elliptic equations with superlinear nonlinearities and we present some relationships between their morse indices and some qualitative properties in particular we show that for uo subcritical nonlinearities bounds on solutions are equivalent to bounds on their morse indices

a bahri lions theorem revisited abstract in 1988 a bahri and p l lions a bahri p l lions morse index of some min max critical points i application to multiplicity results comm pure appl math 41 1988 1027 1037 studied the following elliptic problem u u p 2 u f x u u h 0 1 where is a bounded smooth domain of r n 2 p 2 n 2 n 2 and f x u is not assumed to be odd in u they proved the existence of infinitely many solutions under an appropriate growth restriction on f in the present paper we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign changing solutions in addition we derive an estimate on the number of their nodal domains we also deal with the corresponding fourth order equation 2 u u p 2 u f x u with both dirichlet and navier boundary conditions as well as with strongly coupled elliptic systems

existence of radial solutions with prescribed number of zeros for elliptic equations and their morse index abstract in this paper we consider radially symmetric solutions of the nonlinear dirichlet problem u f x u 0 in where is a ball in r n n 3 and f satisfies some appropriate assumptions we prove existence of radially symmetric solutions with k prescribed number of zeros moreover when f x u k x u p 1 u using the uniqueness result due to tanaka 2008 21 we verify that these solutions are non degenerate and we prove that their radial morse index is exactly k

on the problem of linear elasticity for an infinite region containing a finite number of non intersecting spherical inhomogeneities abstract the problem of linear elasticity for an infinite number of non intersecting spherical and more generally ellipsoidal inhomogeneities is attacked the approach taken does not misrepresent geometry of inhomogeneities although the continuity conditions at the intefaces are only approximately satisfied the principal idea of the approach is to extend the method of kachanov 1985 int j fracture28 r11 r19 1987 int j solids structures23 23 43 for interacting cracks to the realm of the eshelby equivalent inclusion method the application to a test problem for two spherical cavities suggests that the approach can be useful for predictions of the overall response of composite materials and interfacial stress concentrations

existence of global solutions for a chemotaxis fluid system with nonlinear diffusion we consider a coupled system consisting of the navier stokes equations and a porous medium type of keller segel system that model the motion of swimming bacteria living in fluid and consuming oxygen we establish the global in time existence of weak solutions for the cauchy problem of the system in dimension three in addition if the stokes system instead navier stokes system is considered for the fluid equation we prove that bounded weak solutions exist globally in time

holder continuity of keller segel equations of porous medium type coupled to fluid equations abstract we consider a coupled system consisting of a degenerate porous medium type of keller segel system and stokes system modeling the motion of swimming bacteria living in fluid and consuming oxygen we establish the global existence of weak solutions and holder continuous solutions in dimension three under the assumption that the power of degeneracy is above a certain number depending on given parameter values to show holder continuity of weak solutions we consider a single degenerate porous medium equation with lower order terms and via a unified method of proof expanded to generalized porous medium equations we obtain holder regularity which is of independent interest

analysis of two dimensional finite solids withmicrocracks a general method is presented for solving the plane elasticity problem of finite plates with multiple microcracks the method directly accounts for the interactions between different microcracks and the effect of outer boundary of a finite plate analysis is based on a superposition scheme and series expansions of the complex potentials by using the traction free conditions on each crack surface and resultant forces relations along outer boundary a set of governing equations is formulated the governing equations are solved numerically on the basis of a boundary collocation procedure the effective young s moduli for randomly oriented cracks and parallel cracks are evaluated for rectangular plates with microcracks the numerical results are compared with those from various micromechanics models and experimental data these results show that the present method provides a direct and efficient approach to deal with finite solids containing multiple microcracks

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

partial regularity of finite morse index solutions to the lane emden equation we prove regularity and partial regularity results for finite morse index solutions u h1 lp to the lane emden equation u u p 1u in

characterization of solutions having finite morse index for some nonlinear pde with supercritical growth abstract in this paper we study solutions of the equation u u p in r n where n 2 n 2 p p c n see 3 for the definition of p c n we first classify all solutions not necessarily bounded having finite morse index for r n or r n with dirichlet boundary condition when is a regular bounded domain of r n for a family of solutions with dirichlet boundary condition u n we prove that u n l is bounded if and only if the morse index sequence i u n of u n is bounded finally we prove the same result for other nonlinearities f x u which have similar growth respectively at

global existence and decay properties for a degenerate keller segel model with a power factor in drift term abstract the following degenerate parabolic system modelling chemotaxis is considered ks u t u m u q 1 v x r n t 0 v t v v u x r n t 0 u x 0 u 0 x v x 0 v 0 x x r n where m 1 q 2 0 or 1 and n 1 the aim of this paper is to prove the existence of a time global weak solution u v of ks with the l 0 l r n bound such a global bound is obtained in the case of i m q 2 n for large initial data and ii 1 m q 2 n for small initial data in the case of ii the decay properties of the solution u v are also discussed

global existence of weak solutions to quasilinear degenerate keller segel systems of parabolic parabolic type abstract this paper deals with the quasilinear degenerate keller segel system ks of parabolic parabolic type the global existence of weak solutions to ks is established when q m 2 n m denotes the intensity of diffusion and q denotes the nonlinearity without restriction on the size of initial data note that q m 2 n corresponds to generalized fujita s exponent the result improves both sugiyama 2007 14 theorem 1 and sugiyama and kunii 2006 15 theorem 1 in which it is assumed that q m

uniqueness of solutions for keller segel system of porous medium type coupled to fluid equations abstract we prove the uniqueness of holder continuous weak solutions via duality argument and vanishing viscosity method for the keller segel system of porous medium type equations coupled to the stokes system in dimensions three an important step is the estimate of the green function of parabolic equations with lower order terms of variable coefficients which seems to be of independent interest

an effective inclusion model for effective moduli of heterogeneous materials with ellipsoidal inhomogeneities abstract in the present study an effective inclusion model for effective elastic moduli of heterogeneous materials is proposed to analyze the problem of an infinite matrix containing an ellipsoidal rve it is assumed that the strain energy changes of the infinite matrix due to embedding the rve and its effective inclusion into the matrix are identical a system of equations for effective moduli is then formulated using the new energy balance equation that interrelate effective moduli to a problem of an infinite matrix containing n inhomogeneities in an ellipsoidal sub region a generalized non interacting solution derived based on the present formulation coincides with the estimates of the hashin shtrikman type obtained by ponte casta n eda and willis j mech phys solids 43 1995 1919 the effect of the shapes of rves on the approximate solution is also discussed in detail as further application of the present formulation the numerical models for the effective moduli of solids with cracks and heterogeneous materials with spherical inhomogeneities are proposed which account for the interactions among many cracks or spherical inhomogeneities the numerical results are then compared with the existing micromechanics models and experimental data