Special partial matchings (SPMs) are a generalisation of Brentis special matchings. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Mariettis zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic W-sets constitute pircons. Together, these results extend a result of Can, Cherniaysky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.
In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Caratheodory and Nakki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.
We consider divergence form elliptic operators of the form L = -div A (x)del, defined in Rn+1 = {(x, t) is an element of R-n x R}, n andgt;= 2, where the L-infinity coefficient matrix A is (n + 1) x (n + 1), uniformly elliptic, complex and t-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L-2 (R-n) = L-2(partial derivative R-+(n+1)) is stable under complex, L-infinity perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L-2(R-n) whenever A (x) is real and symmetric (and thus, by our stability result, also when A is complex, parallel to A - A(0)parallel to(infinity) is small enough and A(0) is real, symmetric, L-infinity and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L-2 (resp. (L) over dot(1)(2)) data, for small complex perturbations of a real symmetric matrix. Previously, L-2 solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients A (j,n+1)= 0 = A(n+1,j), 1 andlt;= j andlt;= n, which corresponds to the Kato square root problem.
We obtain a sufficient condition for boundary regularity of quasiminimirers of the p-energy integral in terms of a Wiener type sum. The exponent in the sum is independent of the dimension and is explicitly expressed in terms of p and the quasiminimizing constant. We also show by an example that the exponent is sharp in a certain sense. (C) 2016 Elsevier Inc. All rights reserved.
A theory of Sobolev inequalities in arbitrary open sets in R-n is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities exhibit the same critical exponents as in the classical framework. Moreover, they involve constants independent of the geometry of the domain, and hence yield genuinely new results even in the case when just smooth domains are considered. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set. (C) 2016 Elsevier Inc. All rights reserved.
Let G be a connected reductive linear algebraic group over C with an involution theta. Denote by K the subgroup of fixed points. In certain cases, the K-orbits in the flag variety G/B are indexed by the twisted identities t = {theta(omega(-1))omega | omega is an element of W} in the Weyl group W. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph" whose vertices form a subset of t. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on t is rank symmetric. less thanbrgreater than less thanbrgreater thanIn the special case K = Sp(2n) (C), G = SL(2n) (C), we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one", needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.
We introduce a class of weak solutions to the quasilinear equation -Delta(p)u = sigma vertical bar u vertical bar(P-2)u in an open set Omega subset of R-n with p andgt; 1, where Delta(p)u = del. (vertical bar del u vertical bar(p-2)del u) is the p-Laplacian operator. Our notion of solution is tailored to general distributional coefficients sigma which satisfy the inequality less thanbrgreater than less thanbrgreater than-Lambda integral(Omega) vertical bar del h vertical bar(p)dx andlt;= andlt;vertical bar h vertical bar(p), sigma andgt; andlt;= lambda integral(Omega) vertical bar del h vertical bar(p)dx, less thanbrgreater than less thanbrgreater thanfor all h is an element of C-0(infinity)(Omega). Here 0 andlt; Lambda andlt; +infinity-, and less thanbrgreater than less thanbrgreater than0 andlt; lambda andlt; (p - 1)(2-p) if p andgt;= 2, or 0 andlt; lambda andlt; 1 if 0 andlt; p andlt; 2. less thanbrgreater than less thanbrgreater thanAs we shall demonstrate, these conditions on lambda are natural for the existence of positive solutions, and cannot be relaxed in general. Furthermore, our class of solutions possesses the optimal local Sobolev regularity available under such a mild restriction on sigma. less thanbrgreater than less thanbrgreater thanWe also study weak solutions of the closely related equation -Delta p nu = (p - 1)vertical bar del nu vertical bar(p) +sigma, under the same conditions on . Our results for this latter equation will allow us to characterize the class of sigma satisfying the above inequality for positive lambda and Lambda. thereby extending earlier results on the form boundedness problem for the Schrodinger operator to p not equal 2.
We show how to construct a non-smooth solution to a Hessian fully nonlinear second-order uniformly elliptic equation using the Cartan isoparametric cubic in 5 dimensions.
We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in
and non-degenerate phase functions, from L^{p}×L^{q}→L^{r }under the assumptions that
and . This is a bilinear version of the classical theorem of Seeger–Sogge–Stein concerning the L^{p }boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of quasi-Banach target spaces.