a classification of complexity core lattices abstract lynch 1975 has shown that every recursive set a not in p contains an infinite polynomial complexity core a set of elements c a such that any algorithm deciding a needs more than polynomial time almost everywhere on c actually any a not in p contains infinitely many different cores the collection of which forms a lattice under inclusion we study the structure of this lattice proving that surprisingly there are only three possibilities assuming the lattice is not trivial which happens if a is in p its shape depends only on whether a is almost p immune or not it is known that the natural intractable sets usually do not have this property

the p t degrees of the recursive sets lattice embeddings extensions of embeddings and the two quantifier theory abstract ambos spies 1984a showed that the two basic nondistributive lattices can be embedded in rp t the polynomial time turing degrees of the recursive sets we introduce more general techniques to extend his results to show that every recursive lattice can be embedded in rp t in addition to lattice theoretic representation theorems we use the scheme of priority style arguments coupled with looking back techniques presented in shinoda and slaman 1988 1990 we also generalize the density type results of ladner 1975 and many others to settle the full extension of the embedding problem for rp t combined with the logical analysis of sentences with one alternation of quantifiers shore 1978 lerman 1983 these results suffice to decide the full theory of rp t they also give a strong nonhomogeneity result the p time degrees of the sets recursive in and if desired p time above two distinct sets a and b are almost never isomorphic the situation for the p time many one degrees is quite different we decide the extension of the embedding problem differently than for rp t but not the theory

orthogonal polynomials satisfying fourth order differential equations since they are rather important and quite accessible we repeat the general theoretical facts concerning weights moments and polynomials pertaining to fourth order differential equations we then briefly discuss the squares of the differential equations of the second order giving a number of easily derived examples of fourth order problems this is followed by three new orthogonal polynomial sets satisfying fourth order differential equations but which do not satisfy second order differential equations

extending lipschitz functions via random metric partitions many classical problems in geometry and analysis involve the gluing together of local information to produce a coherent global picture inevitably the difficulty of such a procedure lies at the local boundary where overlapping views of the same locality must somehow be merged it is therefore desirable that the boundaries be smooth allowing a graceful transition from one viewpoint to the next for instance one may point to whitney s use of partitions of unity in studying what is now known as the whitney extension problem 36 37 in the present work we consider what is perhaps the most basic whitney type extension problem that of extending a lipschitz function so that it remains lipschitz often such a map is extended by first producing a cover of the new domain extending the mapping locally and then gluing together the individual pieces our main observation is that in many cases if one chooses a random cover from the right distribution the boundary can be made smooth on average even when the local maps are individually quite coarse this insight leads to the unification generalization and improvement of many known results as well as to new results for many interesting spaces

oracle dependent properties of the lattice of np sets abstract we consider under the assumption p np questions concerning the structure of the lattice of np sets together with the sublattice p we show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize the two questions which we consider are whether every infinite np set contains an infinite p subset and whether there exists an np simple set we construct several oracles all of which make p np and which in addition make the above mentioned statements either true or false in particular we give a positive answer to the question raised by bennett and gill 1981 whether an oracle b exists making p b np b and such that every infinite set in np b has an infinite subset in p b the constructions of the oracles are finite injury priority arguments

on some natural complete operators abstract an operator is a mapping from integer functions to integer functions it is known from a result by baker gill and solovay 1975 that there exists an operator computable in polynomial time by a nondeterministic oracle turing machine called an np operator but not computable in polynomial time by any deterministic oracle turing machine we investigate several natural operators which share similar properties we use the concept of completeness to give a precise classification of the complexity of these operators for example the question of finding maximum values of polynomial time computable functions can be formulated as an operator complete for the class of np operators

positive relativizations for log space computability we consider some open questions about log space computability deterministic non deterministic and alternating and polynomial time computability deterministic and nondeterministic in particular the questions dl nl dl p and nl np are considered where dl nl denotes the class of sets accepted by log space bounded deterministic nondeterministic turing machines and p np denotes the class of sets accepted by polynomial time bounded deterministic nondeterministic turing machines we develop positive relativizations for the above questions in two different ways first dl and nl are relativized such that the oracle tape is not subject to the space bound and such that no other restrictions are placed on the oracle machines in this case we prove that c d iff ct dt for all tally sets t where c d is a pair of classes from dl nl p np and conp next both nl and al alternating log space are relativized with the restriction introduced by ruzzo simon and tompa j comput system sci 28 1984 216 230 we call this restriction the rst restriction and prove that c d iff carst darst for all sets a where c d is a pair of classes from dl nl conl and al and crst denotes the corresponding relativized class with the rst restriction

resource bounded immunity and simplicity revisiting the 30 years old notions of resource bounded immunity and simplicity we investigate the structural characteristics of various immunity notions strong immunity almost immunity and hyperimmunity as well as their corresponding simplicity notions we also study limited immunity and simplicity called k immunity and feasible k immunity and their simplicity notions finally we propose the k immune hypothesis as a working hypothesis that guarantees the existence of simple sets in np

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

quantitative relativizations of complexity classes consider the following open problems i text p text np ii text np text co np iii text p text pspace iv text np text pspace in this paper we study these four problems from a particular point of view to illustrate our approach consider the first problem it is known that there exist recursive sets a and b such that text p a text np a and text p b ne text np b we study restrictions r on both the deterministic and also the nondeterministic polynomial time bounded oracle machines such that the following holds text p text np if and only if for every set a text p r a text np r a the restrictions are quantitative in the sense that the size of the set of strings queried by the oracle in computations of a machine on an input is bounded by a polynomial in the length of the input we study several different ways of specifying such quantitative restrictions each of which has the desire

homotopy classes in sobolev spaces and the existence of energy minimizing maps consider the problem of finding maps f m n that are stationary for an energy functional such as fm idfl p where m and n are compact riemannian manifolds and p l such maps may be found by minimizing the functional but if we minimize among all maps from m to n then the minimum is 0 and is attained only by constant maps thus in order to find nontrivial stationary maps we would like to use the topology of m and n to define classes of maps from m to n in which we can minimize the functional for instance one could try to minimize among maps in a given homotopy class but this is not possible in general unless p dimm since a minimizing sequence of mappings in one homotopy class can converge in the appropriate weak topology to a map in another homotopy class however in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of a triangulation of m belong to a given homotopy class to state the results precisely we need to refer to certain sobolev spaces and norms we will assume without loss of generality that m and n are submanifolds of euclidean spaces r m and r respectively and we let lip m n denote the space of lipschitz maps from m to n we define li p m r to be the space of all functions fe lp m r n such that there exist functions

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

the poincare inequality is an open ended condition let p 1 and let x d be a complete metric measure space with borel and doubling that admits a 1 p poincare inequality then there exists e 0 such that x d admits a 1 q poincare inequality for every q p e quantitatively

a survey of space complexity abstract the main properties of deterministic and nondeterministic space complexity classes are given with emphasis on closure under complementation various limitations and generalizations of these classes are studied weakly space bounded classes classes defined by one way alternating or probabilistic machines and nonuniform classes in each case intrinsic properties of these classes and relationship between these classes are given then three ways of relativizing complexity classes are examined finally the space complexity of rams is defined and its relation to usual classes is given

density of smooth functions between two manifolds in sobolev spaces abstract we give some results concerning density or non density of smooth functions between two compact manifolds m n and n k in sobolev spaces w 1 p m n n k in particular we study the case n s k

rank one perturbations in a pontryagin space with one negative square let n 1 denote the class of generalized nevanfinna functions with one negative square and let n 1 n 0 be the subclass of functions q z is an element of n with the additional properties lim y infinity q i y y 0 and lim sup y infinity y im q i y i these classes form an analytic framework for studying generalized rank one perturbations a a in a pontryagin space setting many functions appearing in quantum mechanical models of point interactions either belong to the subclass n 1 n 0 or can be associated with the corresponding generalized friedrichs extension in this paper a spectral theoretical analysis of the perturbations a and the associated friedrichs extension is carried out many results such as the explicit characterizations for the critical eigenvalues of the perturbations a are based on a recent factorization result for generalized nevanlinna functions c 2002 elsevier science usa

almost every set in exponential time is p bi immune a set a is p bi immune if neither a nor its complement has an infinite subset in p we investigate here the abundance of p bi immune languages in linear exponential time e using resource bounded measure we prove that the class of p bi immune sets has measure 1 in e this implies that almost every language in e is p bi immune a bit further we show that every p random pseudorandom language is e bi immune regarding the existence of p bi immune sets in np we show that if np does not have measure 0 in e then np contains a p bi immune set another consequence is that the class of 1 tt p complete languages for e has measure 0 in e in contrast using the approach of resource bounded category it is shown that in e the class of p bi immune languages lacks the property of baire the baire category analogue of lebesgue measurability

sparse sets in np p exptime versus nexptime this paper investigates the structural properties of sets in np p and shows that the computational difficulty of lower density sets in np depends explicitly on the relations between higher deterministic and nondeterministic time bounded complexity classes the paper exploits the recently discovered upward separation method which shows for example that there exist sparse sets in np p if and only if exptime nexptime in addition the paper uses relativization techniques to determine logical possibilities limitations of these proof techniques and exhibits one of the first natural structural differences between relativized np and conp

on simple and creative sets in np abstract two structurally defined types of np sets are studied k simple sets are defined and shown to exist in np other properties of these sets are investigated k creative sets as previously defined by joseph and young 1985 are next considered a new condition is given which implies that a set is k creative several previously considered np complete sets are proved to be k creative

minimal pairs and complete problems two sets are said to form a minimal pair for polynomial many one reductions if neither set is in p and the only sets which are polynomially reducible via polynomial many one reductions to both sets are in p we consider the question of which complete sets can be the top that is the supremum of a minimal pair our main result is that any elementary recursive set which is hard for deterministic exponential time cannot be the top of a minimal pair hence in particular any complete set for an elementary recursive class containing exponential time cannot be such a supremum for np complete sets the problem remains open we show here that it is oracle dependent this is the first example of an oracle dependent property which is completely degree theoretic

density of lipschitz mappings in the class of sobolev mappings between metric spaces we prove that lipschitz mappings are dense in the newtonian sobolev classes n1 p x y of mappings from spaces x supporting p poincare inequalities into a finite lipschitz polyhedron y if and only if y is p connected 1 y 2 y p y 0 where p is the largest integer less than or equal to p

sparse sets in np p relativizations we construct an oracle relative to which bf p ne bf np and there are no sparse sets in bf np bf p the well known construction of baker gill and solovay siam j comput 4 1975 pp 431 442 gives an oracle relative to which there is a sparse set in bf np bf p together these results show that simple modifications of conventional proof techniques cannot establish whether or not sparse sets exist in bf np bf p even if one assumes bf p ne bf np

on boundary values for the laguerre operator in indefinite inner product spaces abstract it is shown that when ly xy 1 x y requires more than one boundary condition at x 0 in order to be self adjoint the more negative the more boundary conditions required

topology and sobolev spaces abstract consider the sobolev class w 1 p m n where m and n are compact manifolds we present some sufficient conditions which guarantee that w 1 p m n is path connected we also discuss cases where w 1 p m n admits more than one component there are still a number of open problems especially concerning the values of p where a change in homotopy classes occurs

complete distributional problems hard languages and resource bounded measure we say that a distribution is reasonable if there exists a constant s 0 such that x x n 1 ns we prove the following result which suggests that all distnp complete problems have reasonable distributions if np contains a dtime 2n bi immune set then every distnp complete set has a reasonable distribution it follows from work of mayordomo 19 that the consequent holds if the p measure of np is not zero cai and selman 6 defined a modification and extension of levin s notion of average polynomial time to arbitrary time bounds and proved that if l is p bi immune then l is distributionally hard meaning that for every polynomial time computable distribution the distributional problem l is not polynomial on the average we prove the following results which suggest that distributional hardness is closely related to more traditional notions of hardness 1 if np contains a distributionally hard set then np contains a p immune set 2 there exists a language l that is distributionally hard but not p bi immune if and only if p contains a set that is immune to all p printable sets the following corollaries follow readily 1 if the p measure of np is not zero then there exists a language l that is distributionally hard but not p bi immune 2 if the p2 measure of np is not zero then there exists a language l in np that is distributionally hard but not p bi immune

diagonalizations over polynomial time computable sets abstract a formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets the class of polynomial time computable functions respectively like e g p immunity it is shown that there are sets called p generic which have all properties enforceable by such diagonalizations we study the behaviour and the complexity of p generic sets in particular we show that the existence of p generic sets in np is oracle dependent even if we assume p np

topology of sobolev mappings ii this paper addresses some topological and analytical issues concern ing sobolev mappings between compact riemannian manifolds among the results we obtained are unified proofs of various generalizations of results obtained in a recent work of brezis and li in particular we solved two conjectures in bl we also give a topological obstruction for the weak sequential density of smooth maps in a given sobolev mapping space finally we show a necessary and suf ficient topological condition under which the smooth maps are strongly dense in the sobolev spaces the earlier result theorem 1 of b2 was shown to be not correct

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

strong separations of the polynomial hierarchy with oracles constructive separations by immune and simple sets abstract in this paper we show that the techniques introduced by furst 1984 which connected oracle separation results for the relativized polynomial time hierarchy to the problem of proving lower bounds for constant depth circuits and the subsequent probabilistic arguments introduced by yao 1985 hastad 1986 and ko 1989 in order to prove the existence of relativized polynomial time hierarchies with different structures can be adapted for resolving the main problems related to the existence of immune and simple sets in the relativized polynomial time hierarchy in particular we construct oracles which witness 1 for any k the existence of a p k immune set in p k 2 for any k the existence of a k simple set 3 for any k the existence of a p k immune set in a relativized polynomial time hierarchy for which p k p k p k 4 for any k 1 the existence of a p k 1 immune set in a relativized polynomial time hierarchy for which p k p k

reductions among polynomial isomorphism types abstract a set a is polynomial many one reducible to a set b a is karp reducible to b if there is a polynomially computable function f such that for all x x a iff f x b arbitrary sets a and b are of the same polynomial many one degree if each is polynomial many one reducible to the other a and b are polynomially isomorphic if the function f can be taken one to one onto and polynomially invertible in classical recursive function theory all many one complete sets are recursively isomorphic berman and hartmanis have observed that all known np complete sets are polynomially isomorphic berman and hartmanis have observed that all known np complete sets are polynomially isomorphic and have conjectured that all np complete sets complete under karp reducibility are isomorphic in this paper we show that not just the complete degree but every polynomial many one degree consists either of a single isomorphism type or else contains infinitely many isomorphism types densely ordered under one one size increasing polynomially invertible reductions and also contains infinitely many isomorphisms types which are incomparable under one one invertible reductions in fact we show that every countable partial ordering can be embedded in any such many one degree we also exhibit polynomial degrees which have infinitely many isomorphism types no examples are known of degrees consisting of a single isomorphism type

topology of sobolev mappings iii we establish a necessary and sufficient topological condition for maps that are in w1 p m n to be connected by continuous paths in w1 p m n to maps in w1 q m n q q 1 we also obtain a necessary and sufficient topological condition under which w1 q m n is strongly dense in w1 p m n several results concerning sequential weak density of smooth or w1 q m n maps in w1 p m n are also proven 2003 wiley periodicals inc

the jacobian the square root and the set abstract in these notes we wish to present a new approach to the problem of prescribing jacobian determinants in dimension two we restrict ourselves to the case the datum is a finite sum of dirac masses the main point is to show that we may relate this problem to the search of harmonic maps into a singular space shaped as the symbol the later problem in turn is closely linked to questions in complex analysis a large part of the paper is devoted to a presentation of these mathematical objects and their connections

immunity and pseudorandomness of context free languages we discuss the computational complexity of context free languages concentrating on two well known structural properties immunity and pseudorandomness an infinite language is reg immune resp cfl immune if it contains no infinite subset that is a regular resp context free language we prove that i there is a context free reg immune language outside reg n and ii there is a reg bi immune language that can be computed deterministically using logarithmic space we also show that iii there is a cfl simple set where a cfl simple language is an infinite context free language whose complement is cfl immune similar to the reg immunity a reg primeimmune language has no polynomially dense subsets that are also regular we further prove that iv there is a context free language that is reg n bi primeimmune concerning pseudorandomness of context free languages we show that v cfl contains reg n pseudorandom languages finally we prove that vi against reg n there exists an almost 1 1 pseudorandom generator computable in nondeterministic pushdown automata equipped with a write only output tape and vii against reg there is no almost 1 1 weakly pseudorandom generator computable deterministically in linear time by a single tape turing machine

the existence and density of generalized complexity cores if c is a class of sets and a is not in c then an infinite set h is a proper hard core for a with respect to c if h a and for every c e c such that c a c c h is finite it is shown that if c is a countable class of sets of strings that is closed under finite union and finite variation then every infinite set not in c has a proper hard core with respect to c in addition the density of such generalized complexity cores is studied

downward translations of equality in this paper we construct oracles relative to which dtime t n equals ntime t n and dtime t n does not equal ntime t n for t n sufficiently smaller than t n a stronger result than this is also obtained though for fewer t n expressedin two parts for t n 2n there is an oracle relative to which dtime t n equals ntime t n and ntime 2n contains a set not in dtime t n for any t n growing more slowly than t n for t n 22n 0 1 there is an oracle relative to which dtime t n equals ntime t n and ntime log t n contains a set not in dtime t n for t n growing more slowly than t n these results expand on those obtained by dekhtyar 1976 book et al 1982 and allender 1989

local minimizers and quasiconvexity the impact of topology the aim of this paper is to discuss the question of existence and multiplicity of strong local minimizers for a relatively large class of functionals open image in new window open image in new window from a purely topological point of view the basic assumptions on open image in new window are sequential lower semicontinuity with respect to w1 p weak convergence and w1 p weak coercivity and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds open image in new window and open image in new window

relativizing time space and time space consider the classes of formal languages specified by nondeterministic acceptors that operate simultaneously within time bounds from a set mathcal t and space bounds from a set mathcal s how large must the time bounds be in order to obtain all of the languages specified by nondeterministic acceptors that operate within space bounds from mathcal s how large must the space bounds be in order to obtain all of the languages specified by nondeterministic acceptors that operate with time bounds from mathcal t the first question is shown to be equivalent with appropriate restrictions on mathcal s and mathcal t to the question of whether it matters if the time bounds apply to all of the steps or only to the steps which query the oracle the second question is shown to be equivalent to the question of whether it matters if the space bounds apply to all of the configurations or only to the configurations in which the oracle is queried these results generalize a more specific result 6

bi immunity results for cheatable sets abstract an oracle a is k cheatable if there is a polynomial time algorithm to determine the answers to 2k parallel queries to a from the answers to only k queries to some other oracle b it is known that 1 cheatable sets cannot be bi immune for p in contrast we construct 2 cheatable sets that are bi immune for arbitrary time complexity classes in addition for each k we construct a set that is k 1 cheatable but not k cheatable we show that this separation does not hold with bi immunity we show that if a recursive set a is bi immune for p then there exists a nontrivial 1 cheatable set that is polynomial time m reducible to a consequently if np contains a set that is bi immune for p then np contains a set that is not polynomial time turing equivalent to a self reducible set

on global singularities of sobolev mappings we define various invariants for sobolev mappings defined between manifolds which are stable under perturbation with respect to the strong sobolev topology we show that these invariants classify various types of global singularities of sobolev maps these invariants are used to give a simple characterization of the strong closure of the set of smooth maps in the sobolev space

lifting in besov spaces abstract let be a smooth bounded simply connected domain in r n and let u be a complex valued measurable function on such that u x 1 a e assume that u belongs to a besov space b p q s we investigate whether there exists a real valued function b p q s r such that u e i this complements the corresponding study in sobolev spaces due to bourgain brezis and the first author the microscopic parameter q turns out to play an important role in some limiting situations the analysis of this lifting problem relies on some interesting new properties of besov spaces in particular a non restriction property when q p

holonomy and skyrme s model in this paper we consider two generalizations of the skyrme model one is a variational problem for maps from a compact 3 manifold to a compact lie group the other is a variational problem for flat connections we describe the path components of the configuration spaces of smooth fields for each of the variational problems we prove that the invariants separating the path components are well defined for not necessarily smooth fields with finite skyrme energy we prove that for every possible value of these invariants there exists a minimizer of the skyrme functional throughout the paper we emphasize the importance of holonomy in the skyrme model some of the results may be useful in other contexts in particular we define the holonomy of a distributionally flat l 2 loc connection the local developing maps for such connections need not be continuous

relativizing complexity classes with sparse oracles baker gill and solovay constructed sparse sets a and b such that p a np a and np b co np b in contrast to their results we prove that p np if and only if for every tally language t p t np t and that np co np if and only if for every tally language t np t co np t we show that the polynomial hierarchy collapses if and only if there is a sparse set s such that the polynomial hierarchy relative to s collapses similar results are obtained for several other complexity classes

laguerre polynomial expansions abstract the differential operator generated by the laguerre differential equation xy 1 x y y 0 is considered for all both in the classical setting l2 0 x e x and in the sobolev space h1 0 x 1e x x e x in l2 a shifted operator exists for 1 in h1 we prove that for 0 the laguerre operator self adjoint in l2 remains self adjoint with a restricted domain for 0 a shifted operator corresponds to the case 0 just as in the l2 case

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

simplicity immunity relativizations and nondeterminism a diagonalization method is discussed by which a recursive oracle a can be constructed such that np a has a set l a which is both simple and p a immune

the gap phenomenon for variational integrals in sobolev spaces we show that a gap phenomenon occurs for general variational integrals for mappings from a domain r n into a riemannian manifold if has a non trivial topology

relativizations comparing np and exponential time the possible relationships between np and expak c 0 dtime 2cnk relative to oracles are examined it is first shown that for every oracle including the empty set and any k npa expak then it is shown that all other relationships are possible under relativization that is for each k 0 oracles a b and c are constructed such that i panpaexpak ii expbknpb and iii npc and expck are incomparable with respect to inclusion the construction of the set a is especially intricate apparently requiring a finite injury priority argument in each case in the constructions when a possible inclusion is ruled out it is done in a very strong way namely by finding a language in one of the classes which is immune with respect to the other class

bridging across the log n space frontier we believe that now is the opportune time to unify the theory of space bounded computations below log n with higher complexity classes this needs to review the basic concepts of the space complexity theory and clarify the role of the space constructibility despite of the fact that the space below log n behaves radically different from the higher bounds many important techniques do work on the both sides of this boundary in addition several important problems are closely related across the log n space bound

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we