uniform bounds for strongly competing systems the optimal lipschitz case for a class of systems of semi linear elliptic equations including delta u i f i x u i beta u i sum j neq i a ij u j p quad i 1 dots k for p 2 variational type interaction or p 1 symmetric type interaction we prove that uniform l infty boundedness of the solutions implies uniform boundedness of their lipschitz norm as beta to infty that is in the limit of strong competition this extends known quasi optimal regularity results and covers the optimal case for this class of problems the proofs rest on monotonicity formulae of alt caffarelli friedman and almgren type in the variational setting and on the caffarelli jerison kenig almost monotonicity formula in the symmetric one

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

gradient estimates for variable coefficient parabolic equations and singular perturbation problems in this article we prove via monotonicity formulas interior and boundary gradient esti mates for solutions to second order parabolic equations in divergence form with dini top order coefficients we then prove uniform lipschitz estimates for solutions of singular perturbation prob lems using the previous results and two phase monotonicity formulas introduction the purpose of this paper is to study the regularity of spatial gradients of solutions to second order uniformly parabolic equations in divergence form with bounded lower order terms and dini continuous coefficients and also to establish uniform spatial lipschitz estimates for some singular perturbation problems in the same context in part 1 we obtain interior and boundary estimates for solutions to these linear equations as is well known the weakest assumption on the modulus of continuity of the top order terms to obtain interior continuity of the gradient of solutions is dini continuity see part 1 for the relevant definitions see for instance g w in the elliptic case the usual approach to such results is to use appropriate potentials here we use a monotonicity formula approach theo rem 1 2 9 which we believe is new the approach has the advantage that it allows us to show that dini continuity of the coefficients at a given point yields bound edness of the gradient at the same point using our monotonicity formula the fact that linear functions are almost solutions and a geometric argument from cl we also obtain under uniform dini continuity of the coefficients a uniform modulus of continuity estimate for gradients of solutions theorem 1 3 1 the fact that the monotonicity formula at a point depends only on the dini continuity at that same point enables us to use a simple reflection argument to extend these results up to the boundary theorem 1 4 5 and theorem 1 4 12 we thus obtain a modulus of continuity estimate up to the boundary for gradients of solutions with smooth dirichlet data under uniform dini continuity of the top order coeffi cients once this is established a compactness argument theorem 1 5 10 and a judicious application of harnack s principle allow us to obtain a hopf maximum principle theorem 1 5 10 corollary 1 5 13 and corollary 1 5 16 for our class of equations

uniform holder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames the main result of this paper is a general holder estimate in a class of singularly perturbed elliptic systems this estimate is applied to the study of the well known burke schuman approximation in flame theory after reviewing some classical cases equidiffusional case high activation energy approximation we turn to the non equidiffusional case and to nonlinear diffusion models the limiting problems are nonlinear elliptic equations they have holder or lipschitz maximal global regularity a natural question is then whether this regularity is kept uniformly throughout the approximation process and we show that this is true in general

internal labelling problem an algorithmic procedure combining the decomposition of casimir operators induced by the embedding of a subalgebra into a semisimple lie algebra with the properties of commutators of subgroup scalars an analytical algorithm for the computation of missing label operators with the commutativity requirement is proposed two new criteria for subgroups scalars to commute are given the algorithm is completed with a recursive method to construct orthonormal bases of states as examples to illustrate the procedure four labelling problems are explicitly studied

missing label operators in the reduction o p o p 2 o 2 i consider the missing label problem for basis vectors of an o p representation corresponding to a group reduction chain with links o p o p 2 o 2 a chain with these links is required if the basis vectors are to be of definite weight i obtain two different sets of missing label operators which together with the casimir operators of group and subgroups from a complete set of labeling operators whose eigenvectors provide a canonical basis in the o p representation space the problem is solved for both the even and odd dimensional orthogonal groups

regularity of solutions to regular shock reflection for potential flow the shock reflection problem is one of the most important problems in mathematical fluid dynamics since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete however most of the fundamental issues for shock reflection have not been understood including the regularity and transition of different patterns of shock reflection configurations therefore it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection on the other hand for a regular reflection configuration the potential flow governs the exact behavior of the solution in c 1 1 across the pseudo sonic circle even starting from the full euler flow that is both of the nonlinear systems are actually the same in a physically significant region near the pseudo sonic circle thus it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo sonic circle the transonic boundary from the elliptic to hyperbolic region and at the point where the pseudo sonic circle the degenerate elliptic curve meets the reflected shock a free boundary connecting the elliptic to hyperbolic region in this paper we study the regularity of solutions to regular shock reflection for potential flow in particular we prove that the c 1 1 regularity is optimal for the solution across the pseudo sonic circle and at the point where the pseudo sonic circle meets the reflected shock we also obtain the c 2 regularity of the solution up to the pseudo sonic circle in the pseudo subsonic region the problem involves two types of transonic flow one is a continuous transition through the pseudo sonic circle from the pseudo supersonic region to the pseudo subsonic region the other a jump transition through the transonic shock as a free boundary from another pseudo supersonic region to the pseudo subsonic region the techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties

a parabolic almost monotonicity formula we prove the parabolic counterpart of the almost monotonicity formula of caffarelli jerison and kening for pairs of functions u x s in the strip s 1 r n x 1 0 satisfying u 0

monotonicity theorems for laplace beltrami operator on riemannian manifolds abstract for free boundary problems on euclidean spaces the monotonicity formulas of alt caffarelli friedman and caffarelli jerison kenig are cornerstones for the regularity theory as well as the existence theory in this article we establish the analogs of these results for the laplace beltrami operator on riemannian manifolds as an application we show that our monotonicity theorems can be employed to prove the lipschitz continuity for the solutions of a general class of two phase free boundary problems on riemannian manifolds

an optimal partition problem related to nonlinear eigenvalues we extend to the case of many competing densities the results of the paper ann inst h poincare 6 2002 more precisely we are concerned with an optimal partition problem in n dimensional domains related to the method of nonlinear eigenvalues introduced by nehari acta math 105 1961 we prove existence of the minimal partition and some extremality conditions moreover in the case of two dimensional domains we give an asymptotic formula near the multiple intersection points finally we show some connections between the variational problem and the behavior of competing species systems with large interaction

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

a harnack inequality approach to the regularity of free boundaries part ii flat free boundaries are lipschitz soit u une solution faible d un probleme aux limites libres du type u 0 sur u 0 et sur u 0 et soit le long de la frontiere libre f la relation u v g u x est satisfaite au sens faible alors si f est proche du graphe d une fonction de lipschitz dans la boule unite b 1 de r n f est le graphe d une fonction de lipschitz dans b 1 2

regularity of the free boundary for two phase problems governed by divergence form equations and applications abstract we study a class of two phase inhomogeneous free boundary problems governed by elliptic equations in divergence form in particular we prove that lipschitz or flat free boundaries are c 1 our results apply to the classical prandtl bachelor model in fluiddynamics

a set of commuting missing label operators for so 5 so 3 the two missing label problem for basis vectors of an so 5 irreducible representation reduced according to the principal so 3 subalgebra is considered a pair of commuting hermitian operators which are scalars with respect to the so 3 subalgebra are explicitly constructed within the so 5 enveloping algebra one label generating operator is of fourth order and the other of sixth order in the so 5 basis elements

on the classification of commutator ideals abstract we compute the k theory of the commutator ideals associated with almost periodic toeplitz operators and use the k 0 group to classify these ideals

regularity for shape optimizers the nondegenerate case we consider minimizers of f 1 n 1 where f is a function strictly increasing in each parameter and k is the kth dirichlet eigenvalue of our main result is that the reduced boundary of the minimizer is composed of c1 graphs and exhausts the topological boundary except for a set of hausdorff dimension at most n 3 we also obtain a new regularity result for vector valued bernoulli type free boundary problems 2018 wiley periodicals inc

homology and topological full groups of etale groupoids on totally disconnected spaces for almost finite groupoids we study how their homology groups reflect dynamical properties of their topological full groups it is shown that two clopen subsets of the unit space has the same class in h0 if and only if there exists an element in the topological full group which maps one to the other it is also shown that a natural homomorphism called the index map from the topological full group to h1 is surjective and any element of the kernel can be written as a product of four elements of finite order in particular the index map induces a homomorphism from h1 to k1 of the groupoid c algebra explicit computations of homology groups of af groupoids and etale groupoids arising from subshifts of finite type are also given

perron s solutions for two phase free boundary problems with distributed sources abstract we use perron method to construct a weak solution to a two phase free boundary problem with right hand side we thus extend the results in caffarelli 1988 for the homogeneous case

a new approach to recent constructions of c algebras from modular index theory abstract we present a new approach to c algebras recently constructed in the context of modular index theory by carey phillips putnam and rennie it turns out that their constructions can be identified with full corners of ideals in semigroup c algebras this new point of view leads to a systematic analysis of these algebras and their canonical maximal abelian subalgebras

lipschitz regularity of the eigenfunctions on optimal domains we study the optimal sets omega ast subseteq mathbb r d for spectral functionals of the form f big lambda 1 omega ldots lambda p omega big which are bi lipschitz with respect to each of the eigenvalues lambda 1 omega lambda 2 omega ldots lambda p omega of the dirichlet laplacian on omega a prototype being the problem min big lambda 1 omega cdots lambda p omega omega subseteq mathbb r d omega 1 big we prove the lipschitz regularity of the eigenfunctions u 1 ldots u p of the dirichlet laplacian on the optimal set omega ast and as a corollary we deduce that omega ast is open for functionals depending only on a generic subset of the spectrum as for example lambda k omega our result proves only the existence of a lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved

variational formulas on lipschitz domains a rigorous treatment is given of variational formulas for solutions of certain dirichlet problems for the laplace operator on lipschitz domains under interior variations in particular we extend well known variational formulas for the torsional rigidity and for capacity from the class of c1 domains to the class of lipschitz domains a motivation for this work comes from the use of variational methods in the study of prandtl batchelor flows in fluid mechanics

internal labelling operators and contractions of lie algebras we analyse under which conditions the missing label problem associated with a reduction chain of simple lie algebras can be completely solved by means of an inonu wigner contraction naturally related to the embedding this provides a new interpretation of the missing label operators in terms of the casimir operators of the contracted algebra and shows that the available labelling operators are not completely equivalent further the procedure is used to obtain upper bounds for the number of invariants of affine lie algebras arising as contractions of semi simple algebras

existence and nondegeneracy of ground states in critical free boundary problems abstract existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature the corresponding study of higher critical points was recently initiated in jerison and perera 2018 0000 in particular the existence and nondegeneracy of a mountain pass point in a superlinear and subcritical free boundary problem related to plasma confinement was proved in jerison and perera 2018 in this paper we study ground states of a critical free boundary problem related to the brezis nirenberg problem 1983 we extend the results of jerison and perera 2018 to this problem by combining the method introduced there with the concentration compactness principle to overcome the difficulties arising from lack of compactness

almost monotonicity formulas for elliptic and parabolic operators with variable coefficients in this paper we extend the results of caffarelli jerison and kenig ann of math 2 155 2002 and caffarelli and kenig amer j math 120 1998 by establishing an almost monotonicity estimate for pairs of continuous functions satisfying u 0 lu 1 u u 0 in an infinite strip global version or a finite parabolic cylinder localized version where l is a uniformly parabolic operator lu l a b c u div a x s u b x s u c x s u s u with double dini continuous a and uniformly bounded b and c we also prove the elliptic counterpart of this estimate this closes the gap between the known conditions in the literature both in the elliptic and parabolic case imposed on u in order to obtain an almost monotonicity estimate at the end of the paper we demonstrate how to use this new almost monotonicity formula to prove the optimal c 1 1 regularity in a fairly general class of quasi linear obstacle type free boundary problems

a pair of commuting scalars for g 2 su 2 su 2 g 2 su 2 su 2 is a two missing labels problem and therefore in order to give a complete and orthogonal specification of states of irreducible representations of g 2 in an su 2 su 2 basis one needs to find a pair of commuting hermitian operators which are scalar with respect to the su 2 su 2 subalgebra a theorem due to peccia and sharp states that there are apart from the lie algebra invariants twice as many functionally independent scalars as missing labels here two commuting su 2 su 2 scalars are obtained both of sixth order in the g 2 basis elements they are in fact combinations of five scalars of different tensorial types indicating that the functionally independent ones are in general insufficient to provide the lowest order commuting scalars an expression for the sixth order invariant of g 2 is also obtained

on a class of 6j coefficients with one multiplicity index for groups sp 2n so 2n and so 2n 1 an important class of 6j symbols for the groups sp 2n so 2n and so 2n 1 with one nontrivial multiplicity index is investigated an appropriate choice of a basis in the multiplicity space is made and the so called canonical form for 6j symbols is obtained their expressions depending on the roots of an nth order equation and explicit expressions for some simple class of representations are obtained

two phase anisotropic free boundary problems and applications to the bellman equation in 2d we prove lipschitz continuity of solutions to a class of rather general two phase anisotropic free boundary problems in 2d and we classify global solutions as a consequence we obtain c 2 1 regularity of solutions to the bellman equation in 2d

commutativity of missing label operators in terms of berezin brackets we obtain a criterion on the commutativity of polynomials in the enveloping algebra of a lie algebra in terms of an involution condition with respect to the berezin bracket as an application it is shown that the commutativity requirement of missing label operators for reduction chains in the missing label problem can be solved analytically

c1 1 regularity in semilinear elliptic problems in this paper we give an astonishingly simple proof of c1 1 regularity in elliptic theory our technique yields both new simple proofs of old results as well as new optical results the setting we ll consider is the following let u be a solution to where b is the unit ball in n f x t is a bounded lipschitz function in x and ft is bounded from below then we prove that u c1 1 b1 2 our method is a simple corollary to a recent monotonicity argument due to caffarelli jerison and kenig 2002 wiley periodicals inc

some new monotonicity theorems with applications to free boundary problems the monotonicity theorem of alt caffarelli and friedman acf lemma 5 1 plays a central role in the existence and regularity theory of two phase free boundary problems it has been extended for example to variable coefficient operators c3 lemma 1 and to eigenvalue problems fl in this paper we reformulate the theorem so that it applies to inhomogeneous equations in which the right hand side of the equation need not vanish at the free boundary of particular interest is the case in which au takes two different constant values on the set where u 0 and on the set where u 0 the prandtl batchelor

families of type iii kms states on a class of c algebras containing on and qn abstract we construct a family of purely infinite c algebras q for 0 1 that are classified by their k groups there is an action of the circle t with a unique kms state on each q for 1 n q 1 n o n with its usual t action and kms state for p q rational in lowest terms q o n n q p 1 with uhf fixed point algebra of type p q for any n 1 q o n for infinitely many with distinct kms states and uhf fixed point algebras for any 0 1 q o for irrational the fixed point algebras are not af and the q are usually not cuntz algebras for transcendental k 1 q k 0 q z so that q is cuntz q n cuntz 2008 16 if and 1 are both algebraic integers the only o n which appear are those for which n 3 mod 4 for each the representation of q defined by the kms state generates a type iii factor these algebras fit into the framework of modular index theory twisted cyclic theory of carey et al 2010 8 carey et al 2009 12 carey et al in press 5

semigroup c algebras and amenability of semigroups we construct reduced and full semigroup c algebras for left cancellative semigroups our new construction covers particular cases already considered by a nica and also toeplitz algebras attached to rings of integers in number fields due to j cuntz moreover we show how left amenability of semigroups can be expressed in terms of these semigroup c algebras in analogy to the group case