normal convexity and equations over groups a subgroups of a grouph is said to be normal convex inh if for any subsetr s the natural maps r s h r his injective

the φ harmonic approximation and the regularity of φ harmonic maps abstract we extend the p harmonic approximation lemma proved by duzaar and mingione for p harmonic functions to harmonic functions where is a convex function the proof is direct and is based on the lipschitz truncation technique we apply the approximation lemma to prove holder continuity for the gradient of a solution of a harmonic system with critical growth

asymptotically regular problems i higher integrability abstract we consider weak solutions u of non linear systems of partial differential equations assuming that the system exhibits a certain kind of elliptic behavior near infinity we prove higher integrability results for the gradient du in particular we establish holder continuity of u in low dimensions moreover we obtain analogous results for vectorial minimizers of multi dimensional variational integrals finally we discuss an extension to minimizing sequences and applications to generalized minimizers

restrictions of representations of a surface group to a pair of free subgroups let be the fundamental group of a compact orientable genus m surface and let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero define two free rank m subgroups of by a a 1 a m and b b 1 b m where a 1 a m b 1 b m j 1 m a j b j is the standard presentation of we consider representations of of a and of b into g restriction of representations induces a morphism from c g the variety of closed conjugacy classes of representations of to c a g c b g we prove that if m is greater than the semisimple rank of g then this morphism is dominant and almost all fibers are finite

reductive subgroups of reductive groups in nonzero characteristic abstract let g be a possibly nonconnected reductive linear algebraic group over an algebraically closed field k and let n n the group g acts on gn by simultaneous conjugation let h be a reductive subgroup of g we prove that if k has nonzero characteristic then the natural map of quotient varieties hn h gn g is a finite morphism we use methods introduced by vinberg who proved the same result in characteristic zero as an application we show that if is a finite group then the character variety c g of closed conjugacy classes of representations from to g is finite

completely reducible operatorname sl 2 homomorphisms keywords semisimple groups group scheme homomorphisms frobenius twists optimal homomorphisms geometrically completely reducible images twisted product homomorphisms cocharacters reference ctg article 2007 002doi 10 1090 s0002 9947 07 04289 4 record created on 2008 12 16 modified on 2017 05 12

complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 1 abstract let k be a nonperfect field of characteristic 2 let g be a k split simple algebraic group of type e 6 or g 2 defined over k in this paper we present the first examples of nonabelian non g completely reducible k subgroups of g which are g completely reducible over k our construction is based on that of subgroups of g acting non separably on the unipotent radical of a proper parabolic subgroup of g in our previous work we also present examples with the same property for a non connected reductive group g along the way several general results concerning complete reducibility over nonperfect fields are proved using the recently proved tits center conjecture for spherical buildings in particular we show that under mild conditions a connected k subgroup of g is pseudo reductive if it is g completely reducible over k

regularity on the interior for some class of nonlinear second order elliptic systems abstract the interior c 1 regularity is proved for weak solutions to a class of nonlinear second order elliptic systems it is typical for the system belonging to the class that the continuity moduli of the gradients of its coefficients become slow growing sufficiently far from zero this property guarantees the regularity of the gradients of solutions to such system in a case when the ellipticity constant is big enough some characteristic features of the obtained result are illustrated by examples at the end of the paper

cocharacter closure and the rational hilbert mumford theorem for a field k let g be a reductive k group and v an affine k variety on which g acts using the notion of cocharacter closed g k orbits in v we prove a rational version of the celebrated hilbert mumford theorem from geometric invariant theory we initiate a study of applications stemming from this rationality tool a number of examples are discussed to illustrate the concept of cocharacter closure and to highlight how it differs from the usual zariski closure

the singular set of lipschitzian minima of multiple integrals the singular set of any lipschitzian minimizer of a general quasiconvex functional is uniformly porous and hence its hausdorff dimension is strictly smaller than the space dimension

asphericity and finiteness for certain group presentations abstract we study diagrammatic reducibility for the relative group presentations r n k l e h x t 3 x k t 2 x e k l where h t t n n 7 k 1 l 0 and e 1 we apply our results to classify finiteness for the group g n k l e defined by r n k l e apart from the two exceptional cases n k l e 7 2 1 1 and 9 1 1 1

partial regularity for non autonomous degenerate quasi convex functionals with general growth abstract we study partial c 1 regularity of minimizers of quasi convex variational integrals with non standard growth we assume in particular that the relevant integrands satisfy an orlicz s type growth condition i e a so called general growth condition moreover the functionals are supposed to be non autonomous and possibly degenerate

complete reducibility and separability let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 a subgroup of g is said to be separable in g if its global and infinitesimal centralizers have the same dimension we study the interaction between the notion of separability and serre s concept of g complete reducibility for subgroups of g the separability hypothesis appears in many general theorems concerning g complete reducibility we demonstrate that many of these results fail without this hypothesis on the other hand we prove that if g is a connected reductive group and p is very good for g then any subgroup of g is separable we deduce that under these hypotheses on g a subgroup h of g is g completely reducible provided the lie algebra of g is semisimple as an h module recently guralnick has proved that if h is a reductive subgroup of g and c is a conjugacy class of g then the intersection of c and g is a finite union of h conjugacy classes for generic p when certain extra hypotheses hold including separability this follows from a well known tangent space argument due to richardson but in general it rests on lusztig s deep result that a connected reductive group has only finitely many unipotent conjugacy classes we show that the analogue of guralnick s result is false if one considers conjugacy classes of n tuples of elements from h for n 1

on one homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions we extend a result from phillips by showing that one homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine the result is novel in two ways firstly the system is allowed to depend in a sufficiently smooth way on the spatial variable x secondly phillips s original result is shown to apply to w 1 w 2 one homogeneous solutions from which his treatment of lipschitz solutions follows as a special case a singular one homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method

a normal subgroup of a strongly reductive subgroup is strongly reductive abstract let h be a strongly reductive subgroup of a reductive linear algebraic group g over an algebraically closed field k we prove that any closed normal subgroup n of h is also a strongly reductive subgroup of g if g gl n k then this is a consequence of clifford s theorem from representation theory

explicit examples of lipschitz one homogeneous solutions of log singular planar elliptic systems abstract we give examples of systems of partial differential equations that admit non trivial lipschitz and one homogeneous solutions in the form u r r g where r are plane polar coordinates and g takes values in r m m 2 the systems are singular in the sense that they arise as the euler lagrange equations of the functionals i u b w x u x d x where d f w x f behaves like 1 x as x 0 and w satisfies an ellipticity condition such solutions cannot exist when x d f w x f 0 as x 0 so the condition is optimal the associated analysis exploits the well known fefferman stein duality fefferman and stein 1972 we also discuss conditions for the uniqueness of these one homogeneous solutions and demonstrate that they are minimizers of certain variational functionals

harmonic type approximation lemmas we give a survey of known and not known harmonic type approximation lemmas which are descendants of the classical de giorgi s one and we outline some of their recent or possible applications

regularity results for vectorial minimizers of a class of degenerate convex integrals abstract we establish the higher differentiability and the higher integrability for the gradient of vectorial minimizers of integral functionals with p q growth conditions we assume that the non homogeneous densities are uniformly convex and have a radial structure with respect to the gradient variable only at infinity the results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand

partial regularity for minimizers of discontinuous quasi convex integrals with degeneracy abstract in this paper we are concerned with the regularity of minimizers u w 1 p r n of quasi convex integral functionals of the type f u f x u d u d x the crucial point here is that the integrand f admits very weak regularity properties with respect to the gradient variable it satisfies degenerate singular p growth conditions without necessarily possessing a quasi diagonal uhlenbeck type structure and with respect to the x variable the integrand might be even discontinuous it is only assumed that a certain vmo condition holds under those assumptions we prove partial holder continuity of minimizers i e holder continuity of u for any holder exponent 0 1 outside a set of measure zero under such weak assumptions regularity results for the gradient of minimizers is not expected to hold since even in the scalar case counterexamples to c 1 regularity are known

partial regularity for bv minimizers we establish an varepsilon regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth and as a consequence the partial regularity of such bv minimizers this result extends the regularity theory for minimizers of quasiconvex integrals on sobolev spaces to the context of maps of bounded variation previous partial regularity results for bv minimizers in the linear growth set up were confined to the convex situation

lorentz estimates for degenerate and singular evolutionary systems we prove estimates of calderon zygmund type for evolutionary p laplacian systems in the setting of lorentz spaces we suppose the coefficients of the system to satisfy only a vmo condition with respect to the space variable our results hold true mutatis mutandis also for stationary p laplacian systems

on a certain class of cyclically presented groups abstract we continue the study of the cyclically presented groups g g n w where w x 0 x i x j x i x j we prove that under certain conditions on the parameters n i j the group g is infinite these results support our conjecture which states for n 4 exactly when g is trivial and that if g is non trivial then g is infinite

a trace preserving operator and applications abstract we construct a trace preserving operator which improves the integrability of functions in sobolev classes refining the ones available in literature as applications we prove a c 1 partial regularity result for local minimizers of quasiconvex integral functionals satisfying non standard p q growth conditions in the borderline case p n 1 and q n and a global integrability result for weak solutions to a nonlinear elliptic system

partial regularity of strong local minimizers of quasiconvex integrals with p q growth defined on vector valued maps u r n 1 here denotes a bounded open set in r n 2 and f r r is a c function satisfying suitable assumptions described below the existence and the partial regularity of minimizers of f are classical issues in the modern calculus of variations and they have been extensively studied especially over the last twenty years specifically in the present paper we will focus on gradient regularity under the basic assumption that f is quasiconvex that is

a geometric approach to complete reducibility let g be a connected reductive linear algebraic group we use geometric methods to investigate g completely reducible subgroups of g giving new criteria for g complete reducibility we show that a subgroup of g is g completely reducible if and only if it is strongly reductive in g this allows us to use ideas of r w richardson and hilbert mumford kempf from geometric invariant theory we deduce that a normal subgroup of a g completely reducible subgroup of g is again g completely reducible thereby providing an affirmative answer to a question posed by j p serre and conversely we prove that the normalizer of a g completely reducible subgroup of g is again g completely reducible some rationality questions and applications to the spherical building of g are considered many of our results extend to the case of non connected g

scalar minimizers with fractal singular sets lack of regularity of local minimizers for convex functionals with non standard growth conditions is considered it is shown that for every 0 there exists a function ac such that the functional admits a local minimizer uw1 p whose set of non lebesgue points is a closed set with dim n p and where 1 p n n q

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

complete reducibility in good characteristic let g be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p 0 a closed subgroup h of g is called g completely reducible g cr if whenever h is contained in a parabolic subgroup p of g it is contained in a levi subgroup of p in this paper we determine the g conjugacy classes of non g cr simple connected subgroups of g when p is good for g for each such subgroup x we determine the action of x on the adjoint module l g and the connected centraliser of x in g as a consequence we classify all non g cr reductive subgroups of g and we additionally show the existence of and classify the following restricted non g cr semisimple subgroups non g cr reductive subgroups with trivial connected centraliser and subgroups which are maximal among reductive subgroups of g

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

the solution of length four equations over groups let g be a group f the free group generated by t and let r t e g f the equation r t 1 is said to have a solution over g if it has a solution in some group that contains g this is equivalent to saying that the natural map g g flr t is injective there is a conjecture attributed to m kervaire and f laudenbach that injectivity fails only if the exponent sum of t in r t is zero in this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of t in r t is equal to four

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

the p laplace system with right hand side in divergence form inner and up to the boundary pointwise estimates abstract in this note we collect some very recent pointwise bounds for the gradient of solutions and for the solutions themselves to the p laplace system with right hand side in divergence form both estimates inside the domain for local solutions and global estimates for solutions to boundary value problems are discussed their formulation involves sharp maximal operators whose properties enable us to translate some aspects of the elliptic regularity theory into a merely harmonic analytic framework as a consequence a flexible comprehensive approach to estimates for solutions to the p laplace system for a broad class of norms is derived in particular global estimates under minimal boundary regularity are presented

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

nonlinear potential theory of elliptic systems abstract we give a summary of recent results from nonlinear potential theory focusing on linear and nonlinear potential estimates of solutions to non homogeneous equations and systems we start with the cases of quasilinear possibly degenerate equations of p laplacian type and passing through fully nonlinear elliptic equations finally move to systems in this last case we describe recent results implying potential estimates for the p laplacian system finally we describe results bridging nonlinear potential theory and classical partial regularity theory for general elliptic systems the main outcomes are new e regularity criteria involving both excess functionals and nonlinear potentials

complete reducibility and commuting subgroups let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 we study j p serre s notion of g complete reducibility for subgroups of g specifically for a subgroup h and a normal subgroup n of h we look at the relationship between g complete reducibility of n and of h and show that these properties are equivalent if h n is linearly reductive generalizing a result of serre we also study the case when h mn with m a g completely reducible subgroup of g which normalizes n in our principal result we show that if g is connected n and m are connected commuting g completely reducible subgroups of g and p is good for g then h mn is also g completely reducible

higher integrability for minimizers of asymptotically convex integrals with discontinuous coefficients abstract we study the local regularity of vectorial minimizers of integral functionals with standard p growth we assume that the non homogeneous densities are uniformly convex and have a radial structure with respect to the gradient variable only at infinity under a w 1 n sobolev dependence on the spatial variable of the integrand n being the space dimension we show that the minimizers have the gradient locally in l q for every q p as a consequence they are locally holder continuous for every 1

boundary regularity and sufficient conditions for strong local minimizers abstract in this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning c 1 extremals at which the second variation is strictly positive the results are presented in the quasiconvex setting in accordance with the original statement by grabovsky and mengesha 31 the strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts as well as on a sufficiency result according to which smooth extremals are spatially local minimizers furthermore we prove partial regularity up to the boundary for strong local minimizers in the non homogeneous case and a full regularity result for lipschitz extremals with gradient of vanishing mean oscillation as a consequence we also establish a sufficiency result for this class of extremals the regularity results are established via a blow up argument

complete reducibility and conjugacy classes of tuples in algebraic groups and lie algebras let h be a reductive subgroup of a reductive group g over an algebraically closed field k we consider the action of h on g n the n fold cartesian product of g with itself by simultaneous conjugation we give a purely algebraic characterization of the closed h orbits in g n generalizing work of richardson which treats the case h g this characterization turns out to be a natural generalization of serre s notion of g complete reducibility this concept appears to be new even in characteristic zero we discuss how to extend some key results on g complete reducibility in this framework we also consider some rationality questions

orbit closures and invariants let g be a reductive linear algebraic group h a reductive subgroup of g and x an affine g variety let x h denote the set of fixed points of h in x and n g h the normalizer of h in g in this paper we study the natural map of quotient varieties psi x h x h n g h rightarrow x g induced by the inclusion x h subseteq x we show that given g and h psi x h is a finite morphism for all affine g varieties x if and only if h is a g completely reducible subgroup of g in the sense defined by serre this was proved in characteristic 0 by luna in the 1970s we discuss some applications and give a criterion for psi x h to be an isomorphism we show how to extend some other results in luna s paper to positive characteristic and also prove the following theorem let h and k be reductive subgroups of g then the double coset hgk is closed for generic g in g if and only if h cap gkg 1 is reductive for generic g in g

boundary regularity in variational problems we prove that if u omega subset mathbb r n to mathbb r n is a solution to the dirichlet variational problem mathop rm min limits w int omega f x w dw rm d x quad rm subject to quad w equiv u 0 rm on partial omega involving a regular boundary datum u 0 and a regular integrand f x w dw strongly convex in dw and satisfying suitable growth conditions then mathcal h n 1 almost every boundary point is regular for u in the sense that du is holder continuous in a relative neighborhood of the point the existence of even one such regular boundary point was previously not known except for some very special cases treated by jost meier math ann 262 549 561 1983 our results are consequences of new up to the boundary higher differentiability results that we establish for minima of the functionals in question the methods also allow us to improve the known boundary regularity results for solutions to non linear elliptic systems and in some cases to improve the known interior singular sets estimates for minimizers moreover our approach allows for a treatment of systems and functionals with rough coefficients belonging to suitable sobolev spaces of fractional order

global gradient estimates for parabolic equations with measurable nonlinearities abstract we establish a global calderon zygmund theory of nonlinear parabolic equations with measurable nonlinearities in divergence form by proving that the spatial gradient of a weak solution is as integrable as the inhomogeneous term nonlinearity a x t is assumed to be only measurable in one spatial variable and has locally small bmo semi norm in the other spatial and time variables uniformly with respect to variable the boundary of the bounded domain can be beyond the lipschitz category but it is well trapped in two narrow strips at each point and at each scale

non lipschitz minimizers of smooth uniformly convex functionals we construct non lipschitz minimizers of smooth uniformly convex functionals of type i u f du x dx our method is based on the use of null lagrangians

regularity in parabolic systems with continuous coefficients under the only assumption of continuous coefficients we prove a partial holder continuity result for solutions to parabolic systems with polynomial growth a key component throughout the argument is the use of dibenedetto s intrinsic geometry degenerate parabolic equations universitext springer new york 1993 to accommodate the inhomogeneity in the system a main technical point is that although we are proving the holder continuity of solutions we employ the intrinsic geometry using cylinder stretched according to the size of the spatial gradient in an iteration scheme that does not necessarily imply the boundedness of the gradient itself

gradient lq theory for a class of non diagonal nonlinear elliptic systems abstract we consider a class of nonlinear non diagonal elliptic systems with p growth and establish the l q integrability for all q p p 2 of any weak solution provided the corresponding right hand side belongs to the corresponding lebesgue space and the involved elliptic operator asymptotically satisfies the p uniform ellipticity the so called splitting condition and it is continuous with respect to the spatial variable for operators satisfying the uniform p ellipticity condition the higher integrability is known for q p d p d 2 and for operators having the so called uhlenbeck structure the theory is valid for all q p the key novelty of the paper is twofold first the statement uses only the information coming from the asymptotic operator and second and more importantly by using the splitting condition we are able to extend the range of possible q s significantly whenever p d 2

regularity in oscillatory nonlinear elliptic systems we give uniform bounds in morrey sobolev spaces h1 for weak solutions ue to oscillatory nonlinear elliptic systems the main novelty is in the precise qualitative description of the e uniform morrey exponent in terms of the ellipticity ratio of the elliptic systems this is achieved by showing that weak solutions of general nonlinear elliptic systems with lipschitz coefficients are of class w2 p locally or globally depending on assumptions for an exponent p 2 that is independent of dimensions and for which p 2 is inversely proportional to the ellipticity ratio

conjugacy classes of n tuples in lie algebras and algebraic groups let g be a reductive algebraic group over an algebraically closed field f of characteristic zero and let l g be the lie algebra of g then g acts on g by inner automorphisms and it acts on l g by the adjoint representation by taking the diagonal actions we get actions of g on the spaces of n tuples g n and l g in this paper we will prove a number of geometric properties of the orbits of g on these spaces of n tuples