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  • 51.
    Jaye, Benjamin J
    et al.
    Kent State University, OH 44240 USA .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Verbitsky, Igor E
    University of Missouri, MO 65211 USA .
    Quasilinear elliptic equations and weighted Sobolev-Poincare inequalities with distributional weights2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 232, no 1, p. 513-542Article in journal (Refereed)
    Abstract [en]

    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.

  • 52.
    Jochemko, Katharina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sanyal, Raman
    Combinatorial mixed valuations2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 319, p. 630-652Article in journal (Refereed)
  • 53.
    Juhl, Andreas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Holographic formula for Q-curvature. II2011In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 4, p. 3409-3425Article in journal (Refereed)
    Abstract [en]

    We extend the holographic formula for the critical Q-curvature in Graham and Juhl (2007) [9] to all Q-curvatures. Moreover, we confirm a conjecture of Juhl (2009) [11].

  • 54. Kapulkin, Krzysztof
    et al.
    Lumsdaine, Peter LeFanu
    Stockholm University, Faculty of Science, Department of Mathematics.
    The homotopy theory of type theories2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 337, p. 1-38Article in journal (Refereed)
    Abstract [en]

    We construct a left semi-model structure on the category of intensional type theories (precisely, on CxlCat(Id,1,Sigma(,Pi ext))). This presents an infinity-category of such type theories; we show moreover that there is an infinity-functor Cl-infinity from there to the infinity-category of suitably structured quasi-categories. This allows a precise formulation of the conjectures that intensional type theory gives internal languages for higher categories, and provides a framework and toolbox for further progress on these conjectures.

  • 55.
    Kildetoft, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Parabolic projective functors in type A2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 301, p. 785-803Article in journal (Refereed)
    Abstract [en]

    We classify projective functors on the regular block of Rocha-Caridi's parabolic version of the BGG category O in type A. In fact, we show that, in type A, the restriction of an indecomposable projective functor from O to the parabolic category is either indecomposable or zero. As a consequence, we obtain that projective functors on the parabolic category O in type A are completely determined, up to isomorphism, by the linear transformations they induce on the level of the Grothendieck group, which was conjectured by Stroppel in [39].

  • 56. Klein, Patricia
    et al.
    Ma, Linquan
    Pham, Hung
    Smirnov, Ilya
    Stockholm University, Faculty of Science, Department of Mathematics.
    Yao, Yongwei
    Lech's inequality, the Stuckrad-Vogel conjecture, and uniform behavior of Koszul homology2019In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 347, p. 442-472Article in journal (Refereed)
    Abstract [en]

    Let (R, m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set {l(M/IM)/e(I, M)}(root I=m) is bounded below by 1/d!e((R) over bar) where (R) over bar = R/ Ann(M). Moreover, when (M) over cap is equidimensional, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stuckrad-Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.

  • 57.
    Koenig, Steffen
    et al.
    Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany.
    Külshammer, Julian
    Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany.
    Ovsienko, Sergiy
    Mechanics and Mathematics Faculty, Kiev National University, 01033 Kiev, Ukraine.
    Quasi-hereditary algebras, exact Borel subalgebras, A∞-categories and boxes2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 262, p. 546-592Article in journal (Refereed)
    Abstract [en]

    Highest weight categories arising in Lie theory are known to be associated with finite dimensional quasi-hereditary algebras such as Schur algebras or blocks of category O. An analogue of the PBW theorem will be shown to hold for quasi-hereditary algebras: Up to Morita equivalence each such algebra has an exact Borel subalgebra. The category F(\Delta) of modules with standard (Verma, Weyl, …) filtration, which is exact, but rarely abelian, will be shown to be equivalent to the category of representations of a directed box. This box is constructed as a quotient of a dg algebra associated with the A-infinity-structure on Ext(\Delta,\Delta). Its underlying algebra is an exact Borel subalgebra.

  • 58.
    Kozhan, Rostyslav
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Finite range perturbations of finite gap Jacobi and CMV operators2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 301, p. 204-226Article in journal (Refereed)
    Abstract [en]

    Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a Jacobi or CMV operator from a finite gap isospectral torus. The special case of eventually periodic operators solves an open problem of Simon [28, D.2.7]. We also solve the inverse resonance problem: it is shown that an operator is completely determined by the set of its eigen-values and resonances, and we provide necessary and sufficient conditions on their configuration for such an operator to exist.

  • 59.
    Kruglyak, Natan
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Milman, Mario
    Florida Atlantic University, Boca Raton.
    A distance between orbits that controls commutator estimates and invertibility of operators2004In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 182, no 1, p. 78-123Article in journal (Refereed)
    Abstract [en]

    A distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits, thus proving a version of Shneiberg's theorem for orbital methods. The same machinery is used to extend the celebrated Rochberg-Weiss commutator theorem to the setting of orbital methods. It is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element.

  • 60.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wigman, Igor
    King's College London.
    Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 330, p. 516-552Article in journal (Refereed)
    Abstract [en]

    This is a manuscript containing the full proofs of results announced in [10], together with some recent updates. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions. (C) 2018 Elsevier Inc. All rights reserved.

  • 61.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Schubert calculus and equivariant cohomology of grassmannians2008In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 217, no 4, p. 1869-1888Article in journal (Refereed)
    Abstract [en]

    We give a description of equivariant cohomology of grassmannians that places the theory into a general framework for cohomology theories of grassmannians. As a result we obtain a formalism for equivariant cohomology where the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of the general framework by a change of basis. In order to show that our formalism reflects the geometry of grassmannians we relate our theory to the treatment of equivariant cohomology of grassmannians by A. Knutson and T. Tao.

  • 62.
    Lambert, Gaultier
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Limit theorems for biorthogonal ensembles and related combinatorial identities2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 329, p. 590-648Article in journal (Refereed)
    Abstract [en]

    We study the fluctuations of certain biorthogonal ensembles for which the underlying family {P, Q} satisfies a finite term recurrence relation of the form xP(x) = JP(x). For polynomial linear statistics of such ensembles, we reformulate the cumulant method introduced in [53] in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix J. In the spirit of [12], we show that the asymptotic fluctuations are described by the right-limits of the matrix J. Moreover, whenever the right-limit is a Laurent matrix, we show that the CLT is equivalent to Soshnikov's main combinatorial lemma. We discuss several applications to unitary invariant Hermitian random matrices. In particular, we provide a general central limit theorem (CLT) and a law of large numbers in the one-cut regime. We also prove a CLT for the square singular values of the product of independent complex rectangular Ginibre matrices, as well as for the Laguerre and Jacobi biorthogonal ensembles introduced in [7], and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients. Finally, we discuss the connection with the Strong Szegto limit theorem where this combinatorial method originates.

  • 63. Lee, Ki-ahm
    et al.
    Strömqvist, Martin
    KTH, Matematik (Avd.).
    Yoo, Minha
    Highly oscillating thin obstacles2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 237, p. 286-315Article in journal (Refereed)
    Abstract [en]

    The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyper-plane Gamma in R-n and a periodic perforation T-epsilon of R-n, depending on a small parameters epsilon > 0. As epsilon -> 0, it is crucial to estimate the frequency of intersections and to determine this number locally. This is done using strong tools from uniform distribution. By employing classical estimates for the discrepancy of sequences of type {k alpha}(k=1)(infinity), alpha is an element of R, we are able to extract rather precise information about the set Gamma boolean AND T-epsilon. As epsilon -> 0, we determine the limit u of the solution u(epsilon) to the obstacle problem in the perforated domain, in terms of a limit equation it solves. We obtain the typical "strange term" behavior for the limit problem, but with a different constant taking into account the contribution of all different intersections, that we call the averaged capacity. Our result depends on the normal direction of the plane, but holds for a.e. normal on the unit sphere in R-n.

  • 64. Lee, Ki-ahm
    et al.
    Strömqvist, Martin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Yoo, Minha
    Highly oscillating thin obstacles2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 237, p. 286-315Article in journal (Refereed)
    Abstract [en]

    The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyper-plane Gamma in R-n and a periodic perforation T-epsilon of R-n, depending on a small parameters epsilon > 0. As epsilon -> 0, it is crucial to estimate the frequency of intersections and to determine this number locally. This is done using strong tools from uniform distribution. By employing classical estimates for the discrepancy of sequences of type {k alpha}(k=1)(infinity), alpha is an element of R, we are able to extract rather precise information about the set Gamma boolean AND T-epsilon. As epsilon -> 0, we determine the limit u of the solution u(epsilon) to the obstacle problem in the perforated domain, in terms of a limit equation it solves. We obtain the typical "strange term" behavior for the limit problem, but with a different constant taking into account the contribution of all different intersections, that we call the averaged capacity. Our result depends on the normal direction of the plane, but holds for a.e. normal on the unit sphere in R-n.

  • 65. Lewis, John L.
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Regularity of Lipschitz free boundaries in two-phase problems for the $p$-Laplace operator2010In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 225, no 5, p. 2565-2597Article in journal (Refereed)
  • 66. Lewis, John L
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator2010In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 225, no 5, p. 2565-2597Article in journal (Refereed)
    Abstract [en]

    In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C(1,gamma)-smooth for some gamma is an element of (0, 1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.

  • 67.
    Lindahl, Karl-Olof
    Universidad de Santiago de Chile.
    The size of quadratic p-adic linearization disks2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 248, p. 872-894Article in journal (Refereed)
    Abstract [en]

    We find the exact radius of linearization disks at indifferentfixed points ofquadratic maps in Cp. We also show thatthe radius is invariant under power series perturbations.Localizing all periodic orbits of these quadratic-like maps wethen show that periodic points are not the only obstruction for linearization. In so doing, we provide the first known examples in the dynamics ofpolynomials over Cp where the boundary of the linearization disk does not contain any periodic point.

  • 68.
    Mazorchuk, Volodymyr
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Stroppel, Catharina
    Categorification of (induced) cell modules and the rough structure of generalised Verma modules2008In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 219, no 4, p. 1363-1426Article in journal (Refereed)
    Abstract [en]

    This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. fit type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.

  • 69.
    Mazorchuk, Volodymyr
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Stroppel, Catharina
    Cuspidal sl(n)-modules and deformations of certain Brauer tree algebras2011In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 228, no 2, p. 1008-1042Article in journal (Refereed)
    Abstract [en]

    We show that the algebras describing blocks of the category of cuspidal weight (resp. generalized weight) sl(n) -modules are one-parameter (resp. multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley-Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities.

  • 70.
    Nadirashvili, Nikolai
    et al.
    Centre de Mathématiques et Informatique.
    Tkachev, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Vlăduţc, Serge
    Institut de Mathématiques de Luminy .
    A non-classical solution to a Hessian equation from Cartan isoparametric cubic2012In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 231, no 3, p. 1589-1597Article in journal (Refereed)
    Abstract [en]

    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.

  • 71.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Square function estimates and the Kato problem for second order parabolic operators in Rn+12016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 293, p. 1-36Article in journal (Refereed)
    Abstract [en]

    In [4] the Kato square root problem for second order complex uniformly elliptic operators of the form L = -div(A del f), with only bounded and measurable coefficients, was solved. The solution is a consequence of a square function estimate for the operator (1 + lambda L-2)(-1)lambda L. This and related square function estimates have recently spurred a wave of new and ground breaking results in the area of elliptic PDEs. In this paper we establish similar square function estimates for second order parabolic operators in Rn+1 of the form at partial derivative(t) + L paving the way for important developments in the area of parabolic PDEs.

  • 72.
    Pagani, Nicola
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Chen-Ruan cohomology of moduli of curves of genus 2 with marked points2012In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 229, no 3, p. 1643-1687Article in journal (Refereed)
    Abstract [en]

    In this work we describe the Chen-Ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points. In the first half of the paper we compute the additive structure of the Chen-Ruan cohomology ring for the moduli stack of stable n-pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen-Ruan cohomology ring as an algebra on the ordinary cohomology.

  • 73.
    Rodriguez-Lopez, Salvador
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Rule, David
    Staubach, Wolfgang
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    A Seeger-Sogge-Stein theorem for bilinear Fourier integral operators2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 264, p. 1-54Article in journal (Refereed)
    Abstract [en]

    We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in S-1,0(m) (n, 2) and non-degenerate phase functions, from L-P x L-q -> L-r under the assumptions that m <= -(n - 1)(vertical bar 1/p - 1/2 vertical bar + vertical bar 1/q - 1/2 vertical bar) and 1/p + 1/q = 1/r. 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.

  • 74.
    Rodríguez-López, Salvador
    et al.
    Uppsala University, Sweden.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Staubach, Wolfgang
    Uppsala University, Sweden.
    A Seeger-Sogge-Stein theorem for bilinear Fourier integral operators2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 264, p. 1-54Article in journal (Refereed)
    Abstract [en]

    We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in

    and non-degenerate phase functions, from Lp×Lq→Lr under the assumptions that

     and . This is a bilinear version of the classical theorem  of Seeger–Sogge–Stein concerning the Lp 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.

  • 75.
    Shahgholian, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Uraltseva, Nina
    Weiss, Georg S.
    A parabolic two-phase obstacle-like equation2009In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 221, no 3, p. 861-881Article in journal (Refereed)
    Abstract [en]

    For the parabolic obstacle-problem-like equation Delta u - partial derivative(t)u = lambda(+chi{u>0})-lambda(-chi{u<0}), where lambda(+) and lambda(-) are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary delta {u > 0} boolean OR partial derivative{u < 0} is in a neighborhood of each "branch point" the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper [Henrik Shahgholian, Nina Uraltseva, Georg S. Weiss, The two-phase membrane problem-regularity in higher dimensions, Int. Math. Res. Not. (8) (2007)] to the parabolic case. There are substantial difficulties in the parabolic case clue to the fact that the time derivative of the solution is in general not a continuous function. Our result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.

  • 76.
    Shahgholian, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    The two-phase membrane problem - An intersection-comparison approach to the regularity at branch points2006In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 205, no 2, p. 487-503Article in journal (Refereed)
    Abstract [en]

    For the two-phase membrane problem Delta u = (lambda)+/2 chi{u>0} - (lambda)-/2 chi{u<0}, where lambda(+) > 0 and lambda(-) > 0, we prove in two dimensions that the free boundary is in a neighborhood of each branch point the union of two C-1-graphs. We also obtain a stability result with respect to perturbations of the boundary data. Our analysis uses an intersection-comparison approach based on the Aleksandrov reflection.

  • 77. Stenflo, Örjan
    V-variable fractals: Fractals with partial self similarity2008In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, ISSN 0001-8708, Vol. 218, no 6, p. 2051-2088Article in journal (Refereed)
  • 78.
    Strömbergsson, Andreas
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Södergren, Anders
    Univ Copenhagen, Dept Math Sci, Univ Pk 5, DK-2100 Copenhagen O, Denmark;Chalmers Univ Technol, Dept Math Sci, SE-41296 Gothenburg, Sweden;Univ Gothenburg, SE-41296 Gothenburg, Sweden.
    On the generalized circle problem for a random lattice in large dimension2019In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 345, p. 1042-1074Article in journal (Refereed)
    Abstract [en]

    In this note we study the error term Rn,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f : Z+ → R+ satisfying limn→∞ f (n) = ∞ and f (n) = Oε(eεn) for every ε > 0. Then, the random function

    t |→ 1/root 2f (n) Rn,L (t f(n))

    on the interval [0, 1] converges in distribution to one-dimensional Brownian motion as n → ∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers' mean value formula from [18]. For the individual kth moment of the variable (2f (n))-1/2 Rn,L (f (n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f (n) = O(ecn) for any fixed c ∈ (0, ck), where ck is a constant depending on k whose optimal value we determine.

  • 79.
    Toft, Joachim
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Nabizadeh Morsalfard, Elmira
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Periodic distributions and periodic elements in modulation spaces2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 323, p. 193-225Article in journal (Refereed)
    Abstract [en]

    We characterize periodic elements in Gevrey classes, Gelfand-Shilov distribution spaces and modulation spaces, in terms of estimates of involved Fourier coefficients, and by estimates of their short-time Fourier transforms. If q is an element of [1, infinity), omega is a suitable weight and (epsilon(E)(0))' is the set of all E -periodic elements, then we prove that the dual of M-(omega)(infinity,q) boolean AND (epsilon(E)(0))' equals M-(1/omega)(infinity,q)' boolean AND (epsilon(E)(0))' by suitable extensions of Bessel's identity. (C) 2017 Elsevier Inc. All rights reserved.

  • 80.
    Tonkonog, Dmitry
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. Univ Calif Berkeley, Berkeley, CA 94720 USA;HSE Univ, Moscow, Russia.
    From symplectic cohomology to Lagrangian enumerative geometry2019In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 352, p. 717-776Article in journal (Refereed)
    Abstract [en]

    We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials. We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains. 

  • 81.
    Åhag, Per
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cegrell, Urban
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Kolodziej, Slawomir
    Phạm, Hoàng Hiệp
    Zeriahi, Ahmed
    Partial pluricomplex energy and integrability exponents of plurisubharmonic functions2009In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 222, no 6, p. 2036-2058Article in journal (Refereed)
    Abstract [en]

    We first prove a quantitative estimate of the volume of the sublevel sets of a plurisubharmonic function in a hyperconvex domain with boundary values 0 (in a quite general sense) in terms of its Monge–Ampère mass in the domain. Then we deduce a sharp sufficient condition on the Monge–Ampère mass of such a plurisubharmonic function φ for exp(−2φ) to be globally integrable as well as locally integrable.

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