gradient estimates for solutions to the conductivity problem in this paper we derive very precise gradient estimates for solutions to the conductivity problem in the case where two circular conductivity inclusions are very close but not touching the novelty of these estimates is that they give very specific information about the blow up of the gradient as the conductivities of the inclusions degenerate

existence and nonexistence of blow up boundary solutions for sublinear elliptic equations abstract in this paper we consider the semilinear elliptic problem u a x f u u 0 in with the boundary blow up condition u where is a bounded domain in r n n 2 a x c may blow up on and f is assumed to satisfy f1 and f2 below which include the sublinear case f u u m m 0 1 for the radial case that b the unit ball and a x is radial we show that a solution exists if and only if 0 1 1 r a r d r for a general domain we obtain an optimal nonexistence result the existence for nonradial solutions is also studied by using sub supersolution method

on abelian subgroups ofp groups abstract this is a continuation of our note j algebra 186 1996 120 131 we generalize and present simplified proofs almost all elementary lemmas from section 8 of the odd order paper in many places we use hall s enumeration principle and other simple combinatorial arguments a number of related results are proved as well some open questions are posed

on the existence of positive solutions for a class of semilinear elliptic equations we prove some existence and nonexistence results for the semilinear elliptic equation u q x f u on rn n 2 where u is required to blow up on the boundary of and f is a nonnegative function which is assumed to be lipschitz continuous and bounded away from zero on each interval e and have at worst linear growth in particular we extend some results already obtained in the case where f u u 0 1

boundary blow up solutions for a class of elliptic equations on a bounded domain this paper deals with existence of solutions of the quasilinear elliptic equation du f x u u in w satisfying the boundary blow up condition u x x email protected where email protected r n is a bounded domain with smooth boundary email protected our main result applies to the existence of blow up solutions of a class of equations which appear in stochastic control theory namely du a x g u email protected u email protected x in w where a x g u j x are suitable functions and l s are nonnegative constants our approach employs an approximation procedure combined with both a non monotone iteration variant of the method of lower and upper solutions and fixed point arguments

interior regularity estimates in high conductivity homogenization and application in this paper uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by an e periodic lattice of highly conducting thin rods the estimates are derived only at a distance e1 for some 0 away from the fibres this distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in lp as soon as p 2 one key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a fourier series expansion with respect to the fibres direction and weighted by the high contrast conductivity the dependence on powers of e of these two dimensional estimates is shown to be sharp the initial motivation for this work comes from imaging and enhanced resolution phenomena observed experimentally in the presence of micro structures lerosey et al science 315 1120 1124 2007 we use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium assuming that the heterogeneities stay at a distance e1 away from the fibres

boundary blow up for differential equations with indefinite weight we obtain results of existence and multiplicity of solutions for the second order equation x q t g x 0 with x t defined for all tis an element of 0 1 and such that x t infinity as t 0 and t 1 we assume g having superlinear growth at infinity and q t possibly changing sign on 0 1 c 2002 elsevier science usa all rights reserved

characterization of electric fields between two spherical perfect conductors with general radii in 3d abstract in composite materials the inclusions are frequently spaced very closely the electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large in this paper we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions an explicit blowup factor relying on radii is obtained which also involves the digamma function and euler mascheroni constant and so the role of inclusions radii played in such blowup analysis is identified

an asymptotic formula for the voltage potential in a perturbed e periodic composite medium containing misplaced inclusions of size e we consider composite media made of homogeneous inclusions with c1 boundaries our goal is to compare the potential ue in a perfectly periodic composite with the potential ue d of a perturbed periodic medium where the periodicity defects consist of misplaced inclusions we give an asymptotic expansion of the difference ue d ue away from the defects and show that to first order a misplaced inclusion manifests itself via a polarization tensor which is characterized

gradient estimates for solutions of the lame system with partially infinite coefficients we establish upper bounds on the blow up rate of the gradients of solutions of the lame system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero

gradient estimates for the perfect conductivity problem we establish both upper and lower bounds of the gradient estimates for the perfect conductivity problem in the case where perfect stiff conductors are closely spaced inside an open bounded domain and away from the boundary these results give the optimal blow up rates of the stress for conductors with arbitrary shape and in all dimensions

a necessary and sufficient condition for existence of large solutions to semilinear elliptic equations we consider the semilinear equation u p x f u on a domain rn n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in such that p is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists if and only if the function s s0 f t dt satisfies 1 s 1 2 ds for unbounded including rn we show that a similar result holds where u x as x within and u x as x if p x decays to zero rapidly as x

a classification of finite 2 groups with exactly three involutions we finish here a classification of finite 2 groups with exactly three involutions we clear up this remaining very difficult situation and determine completely the structure of g in terms of two generators and relations

gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients in this paper we derive global w1 and piecewise c1 estimates for solutions to divergence form elliptic equations with piecewise holder continuous coefficients the novelty of these estimates is that even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients they are independent of the distance between these surfaces

gradient estimates for solutions of the lame system with partially infinite coefficients in dimensions greater than two abstract we establish upper bounds on the blow up rate of the gradients of solutions of the lame system with partially infinite coefficients in dimensions greater than two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero

blow up boundary solutions of semilinear elliptic problems we consider the following semilinear elliptic equation u p x f u in u 0 u 0 in 1 where rn n 3 is a smooth domain bounded or possibly unbounded with compact possibly empty boundary we assume throughout this paper that p is a non negative function such that p c0 if is bounded and p c loc otherwise the nonlinearity f is assumed to ful ll f 1 f c1 0 f 0 f 0 0 and f 0 on 0 and the keller osserman condition see 7 13

finite 2 groups with exactly four cyclic subgroups of order 2 n in this paper finite 2 groups g with exactly four cyclic subgroups of order 2 n n 2 n fixed are classified it turns out that these four cyclic subgroups of order 2 n generate always a subgroup of order 2 n 2 in the second part of the paper a kind of converse of this result is proved if g is a finite 2 group all of whose elements of order 2 n generate a subgroup of order 2 n 2 then the structure of g is completly determined at the end of the paper we completely classify so called u 2 groups and give a new non computational proof of the famous blackburn s theorem in which minimal non metacyclic 2 groups are completely classified

linear elliptic equations in composite media with anisotropic fibres abstract linear elliptic equations in composite media with anisotropic fibres are concerned the media consist of a periodic set of anisotropic fibres with low conductivity included in a connected matrix with high conductivity inside the anisotropic fibres the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions the coefficients of the elliptic equations depend on the conductivity this work is to derive the holder and the gradient l p estimates uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix for the solutions of the elliptic equations furthermore it is shown that inside the fibres the solutions have higher regularity along the fibres than in the transverse directions

existence and nonexistence of entire positive solutions of semilinear elliptic systems abstract we show that entire positive solutions exist for the semilinear elliptic system u p x v v q x u on r n n 3 for positive and provided that the nonnegative functions p and q are continuous and satisfy appropriate decay conditions at infinity we also show that entire solutions fail to exist if the functions p and q are of slow decay

boundary blow up elliptic problems of bieberbach and rademacher type with nonlinear gradient terms abstract by a perturbation method and constructing comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x e u x u where is a bounded domain with smooth boundary in r n q 0 b is non negative and non trivial in which may be vanishing on the boundary

the asymptotic behaviour of solutions with blow up at the boundary for semilinear elliptic problems abstract by constructing the comparison functions and the perturbed method it is showed that any solution u c 2 to the semilinear elliptic problems u k x g u x u satisfies lim d x 0 u x z d x 2 2 2 c 0 2 1 where is a bounded domain with smooth boundary in r n lim d x 0 k x d x c 0 2 c 0 0 2 2 g c 1 0 g 0 and g s s is increasing on 0 there exists 0 such that lim s g s g s 0 z s d t 2 g t s g t 0 t g s d s

free transmission problems we study transmission problems with free interfaces from one random medium to another solutions are required to solve distinct partial differential equations rm l and rm l within their positive and negative sets respectively a corresponding flux balance from one phase to another is also imposed we establish existence and l infty bounds of solutions we also prove that variational solutions are non degenerate and develop the regularity theory for solutions of such free boundary problems

existence non existence and asymptotic behavior of blow up entire solutions of semilinear elliptic equations abstract we are mainly concerned with existence non existence and the behavior at infinity of non negative blow up entire solutions of the equation u x f u in r n no monotonicity condition is assumed upon f and in fact we obtain solutions with a prescribed behavior both at infinity and at the origin the method used to get existence is based upon lower and upper solutions techniques while for non existence we explore radial symmetry estimates on an associated integral equation and the keller osserman condition

on finite nonabelian 2 groups all of whose minimal nonabelian subgroups are of exponent 4 abstract in theorem 2 3 we determine finite 2 groups all of whose minimal nonabelian subgroups are of order 8 i e they are isomorphic to d 8 or q 8 in corollary 2 4 we determine finite 2 groups all of whose minimal nonabelian subgroups are isomorphic and have order 8 in corollary 2 5 we show that a minimal non dedekindian finite 2 group is either minimal nonabelian or is isomorphic to q 16 in further three theorems we classify finite 2 groups all of whose minimal nonabelian subgroups are pairwise isomorphic and have order 8 and exponent 4 this solves some problems stated by berkovich y berkovich groups of prime power order parts i and ii with z janko in preparation