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  • 1.
    Avelin, Benny
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Lundström, Niklas L.P.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Boundary estimates for solutions to operators of p-Laplace type with lower order terms2011In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, no 1, p. 264-291Article in journal (Refereed)
    Abstract [en]

    In this paper we study the boundary behavior of solutions to equations of the form∇⋅A(x,∇u)+B(x,∇u)=0, in a domain ΩRn, assuming that Ω is a δ-Reifenberg flat domain for δ sufficiently small. The function A is assumed to be of p-Laplace character. Concerning B, we assume that |∇ηB(x,η)|⩽c|η|p−2, |B(x,η)|⩽c|η|p−1, for some constant c, and that B(x,η)=|η|p−1B(x,η/|η|), whenever xRn, ηRn∖{0}. In particular, we generalize the results proved in J. Lewis et al. (2008) [12] concerning the equation ∇⋅A(x,∇u)=0, to equations including lower order terms.

  • 2.
    Avelin, Benny
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Lundström, Niklas L.P.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Optimal doubling, reifenberg flatness and operators of p-laplace type2011In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 17, p. 5943-5955Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider equations of p-Laplace type of the form A(x,u)=0. Concerning A we assume, for p∈(1,) fixed, an appropriate ellipticity type condition, Hölder continuity in x and that A(x,η)=|η|p−1A(x,η/|η|) whenever xRn and ηRn∖{0}. Let ΩRn be a bounded domain, let D be a compact subset of Ω. We say that is the A-capacitary function for D in Ω if on D, on Ω in the sense of and in ΩD in the weak sense. We extend to RnΩ by putting on RnΩ. Then there exists a unique finite positive Borel measure on Rn, with support in Ω, such that In this paper, we prove that if Ω is Reifenberg flat with vanishing constant, then for every τ, 0<τ≤1. In particular, we prove that is an asymptotically optimal doubling measure on Ω.

  • 3. Cinti, Chiara
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Polidoro, Sergio
    A boundary estimate for non-negative solutions to Kolmogorov operators in non-divergence form2012In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 191, no 1, p. 1-23Article in journal (Refereed)
    Abstract [en]

    We consider non-negative solutions to a class of second-order degenerate Kolmogorov operators of the form L=Sigma(m)(i,j=1)a(i,j)(z)partial derivative(xixj)+Sigma(m)(i=1)a(i)(z)partial derivative(xi)+Sigma(N)(i,j=1)b(i,j)x(i)partial derivative(xj)-partial derivative(t), where z = (x, t) belongs to an open set Omega subset of R-N x R, and 1 <= m <= N. Let (z) over bar is an element of Omega, let K be a compact subset of (Omega) over bar, and let Sigma subset of partial derivative Omega be such that K boolean AND partial derivative Omega subset of Sigma. give sufficient geometric conditions for the validity of the following Carleson type estimate. There exists a positive constant C-K, depending only on Omega, Sigma, K, (z) over tilde and on L, such that sup(K) u <= C(K)u((z) over tilde), for every non-negative solution u of Lu = 0 in Omega such that u|(Sigma) = 0.

  • 4. Cinti, Chiara
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Polidoro, Sergio
    A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators2010In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 33, no 4, p. 341-354Article in journal (Refereed)
    Abstract [en]

    In this paper we are concerned with Harnack inequalities for non-negative solutions u:Ω→ℝ to a class of second order hypoelliptic ultraparabolic partial differential equations in the form Lu:=∑j=1mX2ju+X0u−∂tu=0 where Ω is any open subset of ℝN + 1, and the vector fields X1, ..., Xm and X0t are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x0, t0) ∈ Ω we give a geometric sufficient condition on the compact sets K⊆Ω for which the Harnack inequality supK u≤CK u(x0, t0) holds for all non-negative solutions u to the equation Lu=0. We also compare our result with an abstract Harnack inequality from potential theory.

  • 5.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Garofalo, Nicola
    Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
    Götmark, Elin
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Munive, Isidro
    Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions2012In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 11, no 2, p. 437-474Article in journal (Refereed)
    Abstract [en]

    In a cylinder Omega(T) = Omega x (0, T) subset of R-+(n+1) we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form

    H u = Sigma(m)(i,j=1) a(ij)(x, t)XiX (j)u - partial derivative(t)u = 0, (x, t) is an element of R-+(n+1),

    where X = {X-l, . . . , X-m} is a system of C-infinity vector fields inR(n) satisfying Hormander's rank condition (1.2), and Omega is a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. Concerning the matrix-valued function A = {a(ij)}, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries a(ij) are Holder continuous with respect to the parabolic distance associated with d. Our main results are: I) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Holder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator H (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20, 39]. With one proviso: in those papers the authors assume that the coefficients a(ij) be only bounded and measurable, whereas we assume Holder continuity with respect to the intrinsic parabolic distance.

  • 6.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Götmark, Elin
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The obstacle problem for parabolic non-divergence form operators of Hörmander type2012In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 252, no 9, p. 5002-2041Article in journal (Refereed)
    Abstract [en]

    In this paper we establish the existence and uniqueness of strong solutions to the obstacle problem for a class of parabolic sub-elliptic operators in non-divergence form structured on a set of smooth vector fields in Rn, X={X1,…,Xq}X={X1,…,Xq}, q⩽n, satisfying Hörmanderʼs finite rank condition. We furthermore prove that any strong solution belongs to a suitable class of Hölder continuous functions. As part of our argument, and this is of independent interest, we prove a Sobolev type embedding theorem, as well as certain a priori interior estimates, valid in the context of Sobolev spaces defined in terms of the system of vector fields.

  • 7.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type2010In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, no 1, p. 146-164Article in journal (Refereed)
    Abstract [en]

    Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.

  • 8.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Pascucci, Andrea
    Polidoro, Sergio
    Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options2010In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 347, no 4, p. 805-838Article in journal (Refereed)
    Abstract [en]

    In this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstacles as well as non-smooth obstacles. All our proofs follow the same line of thought and are based on blow-ups, compactness, barriers and arguments by contradiction. The problem considered arises in financial mathematics, when considering path-dependent derivative contracts with early exercise feature.

  • 9.
    Götmark, Elin
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Boundary behavior of non-negative solutions to degenerate sub-elliptic equations2013In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 254, no 8, p. 3431-3460Article in journal (Refereed)
    Abstract [en]

    Let X = {X-1, ..., X-m} be a system of C-infinity vector fields in R-n satisfying Hormander's finite rank condition and let Omega be a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. We study the boundary behavior of non-negative solutions to the equation Lu = Sigma(i, j -1) X-i*(a(ij)X(j)u) = Sigma X-i, j=1(i)*(x)(aij(x)X-j(x)u(x)) = 0 for some constant beta >= 1 and for some non-negative and real-valued function lambda = lambda(x). Concerning kappa we assume that lambda defines an A(2)-weight with respect to the metric introduced by the system of vector fields X =, {X-1,..., X-m}. Our main results include a proof of the doubling property of the associated elliptic measure and the Holder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes et al. (1982, 1983) [18-20] (m = n, {X-(1), ..., X-m} = {partial derivative(x1), ...., partial derivative x(n)}, A is an A(2)-weight) and Capogna and Garofalo (1998) [6] (X = {X-1,..., X-m} satisfies Hormander's finite rank condition and X(x) equivalent to lambda A for some constant lambda). One motivation for this study is the ambition to generalize, as far as possible, the results in Lewis and Nystrom (2007, 2010, 2008) [35-38], Lewis et al. (2008) [34] concerning the boundary behavior of non-negative solutions to (Euclidean) quasi-linear equations of p-Laplace type, to non-negative solutions, to certain sub-elliptic quasi-linear equations of p-Laplace type. (C) 2013 Elsevier Inc. All rights reserved.

  • 10. Lewis, John L.
    et al.
    Lundström, Niklas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Boundary Harnack inequalities for operators of p-Laplace type in Reifenberg flat domains2008In: Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz'ya's 70th Birthday / [ed] Dorina Mitrea and Marius Mitrea, American Mathematical Society (AMS), 2008, Vol. 79, p. 229-266Chapter in book (Refereed)
    Abstract [en]

    In this paper we highlight a set of techniques that recently have been used to establish boundary Harnack inequalities for p-harmonic functions vanishing on a portion of the boundary of a domain which is ‘flat’ in the sense that its boundary is well-approximated by hyperplanes. Moreover, we use these techniques to establish new results concerning boundary Harnack inequalities and the Martin boundary problem for operators of p-Laplace type with variable coefficients in Reifenberg flat domains.

  • 11. Lewis, John L
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    New results for p harmonic functions2011In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 7, no 2, p. 345-363Article in journal (Refereed)
    Abstract [en]

    In this paper we first discuss new results of the authors concerning a boundary Harnack inequality and Holder continuity up to the boundary for the ratio of two positive pharmonic functions, 1 < p < infinity, which vanish on a portion of a Lipschitz domain. Second we discuss applications of these results to the Martin boundary problem for p harmonic functions and to certain boundary regularity-free boundary problems.

  • 12. Lewis, John L.
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Regularity of flat free boundaries in two-phase problems for the p-Laplace operator2012In: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430, Vol. 29, no 1, p. 83-108Article in journal (Refereed)
    Abstract [en]

    In this paper we continue the study in Lewis and Nystrom (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.

  • 13. 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.

  • 14. Lewis, John L.
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Poggi-Corradini, Pietro
    p Harmonic Measure in Simply Connected Domains2011In: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 61, no 2, p. 689-715Article in journal (Refereed)
    Abstract [en]

    Let Omega be a bounded simply connected domain in the complex plane, C. Let N be a neighborhood of partial derivative Omega, let p be fixed, 1 < p < infinity, and let (u) over cap be a positive weak solution to the p Laplace equation in Omega boolean AND N. Assume that (u) over cap has zero boundary values on partial derivative Omega in the Sobolev sense and extend (u) over cap to N \ Omega by putting 11 E 0 on N Then there exists a positive finite Borel measure (mu) over cap on C with support contained in partial derivative Omega and such that integral vertical bar del(u) over cap vertical bar(p-2) <del(u) over cap, del phi > dA = - integral phi d (mu) over cap whenever phi is an element of C(0)(infinity)(N). If p = 2 and if (u) over cap is the Green function for Omega with pole at x is an element of Omega\(N) over bar then the measure (mu) over cap coincides with harmonic measure at x, omega = omega(x), associated to the Laplace equation. In this paper we continue the studies initiated by the first author by establishing new results, in simply connected domains, concerning the Hausdorff dimension of the support of the measure (mu) over cap. In particular, we prove results, for 1 < p < infinity, p not equal 2, reminiscent of the famous result of Makarov concerning the Hausdorff dimension of the support of harmonic measure in simply connected domains.

  • 15. Lewis, John
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Boundary behavior and the Martin boundary problem for p harmonic functions in Lipschitz domains2010In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 172, no 3, p. 1907-1948Article in journal (Refereed)
    Abstract [en]

    In a previous article, we proved a boundary Harnack inequality for the ratio of two positive p harmonic functions, vanishing on a portion of the boundary of a Lipschitz domain. In the current paper we continue our study by showing that this ratio is Holder continuous up to the boundary. We also consider the Martin boundary of certain domains and the corresponding question of when a minimal positive p harmonic function (with respect to a given boundary point w) is unique up to constant multiples. In particular we show that the Martin boundary can be identified with the topological boundary in domains that are convex or C(1). Minimal positive p harmonic functions relative to a boundary point w in a Lipschitz domain are shown to be unique, up to constant multiples, provided the boundary is sufficiently flat at w.

  • 16.
    Lundström, Niklas L P
    et al.
    Department of Mathematics, Uppsala University, Sweden.
    Nyström, Kaj
    Department of Mathematics, Uppsala University, Sweden.
    Olofsson, Marcus
    Department of Mathematics, Uppsala University, Sweden.
    Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type2014In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 193, no 4, p. 1213-1247Article in journal (Refereed)
    Abstract [en]

    In this paper we study the system where . A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth, and structural assumptions on the data, i.e., on the operator and on continuous functions , , and . A key aspect is that we make no sign assumption on the switching costs and that is allowed to depend on as well as . Using the comparison principle, the existence of a unique viscosity solution to the system is constructed as the limit of an increasing sequence of solutions to associated obstacle problems. Having settled the existence and uniqueness, we subsequently focus on regularity of beyond continuity. In this context, in particular, we assume that belongs to a class of second-order differential operators of Kolmogorov type of the form: where . The matrix is assumed to be symmetric and uniformly positive definite in . In particular, uniform ellipticity is only assumed in the first coordinate directions, and hence, may be degenerate.

  • 17.
    Lundström, Niklas L.P.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On a two-phase free boundary condition for p-harmonic measures2009In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 129, no 2, p. 231-249Article in journal (Refereed)
    Abstract [en]

    Let Ωi⊂Rn,i∈{1,2} , be two (δ, r 0)-Reifenberg flat domains, for some 0<δ<δ^ and r 0 > 0, assume Ω1∩Ω2=∅ and that, for some w∈Rn and some 0 < r, w∈∂Ω1∩∂Ω2,∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . Let p, 1 < p < ∞, be given and let u i , i∈{1,2} , denote a non-negative p-harmonic function in Ω i , assume that u i , i∈{1,2}, is continuous in Ω¯i∩B(w,2r) and that u i = 0 on ∂Ωi∩B(w,2r) . Extend u i to B(w, 2r) by defining ui≡0 on B(w,2r)∖Ωi. Then there exists a unique finite positive Borel measure μ i , i∈{1,2} , on R n , with support in ∂Ωi∩B(w,2r) , such that if ϕ∈C∞0(B(w,2r)) , then∫Rn|∇ui|p−2⟨∇ui,∇ϕ⟩dx=−∫Rnϕdμi.Let Δ(w,2r)=∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . The main result proved in this paper is the following. Assume that μ 2 is absolutely continuous with respect to μ 1 on Δ(w, 2r), d μ 2 = kd μ 1 for μ 1-almost every point in Δ(w, 2r) and that logk∈VMO(Δ(w,r),μ1) . Then there exists δ~=δ~(p,n)>0 , δ~<δ^ , such that if δ≤δ~ , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).

  • 18.
    Lundström, Niklas L.P.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The Boundary Harnack Inequality for Solutions to Equations of Aronsson type in the Plane2011In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 36, p. 261-278Article in journal (Refereed)
    Abstract [en]

    In this paper we prove a boundary Harnack inequality for positive functions which vanish continuously on a portion of the boundary of a bounded domain \Omega \subset R2 and which are solutions to a general equation of p-Laplace type, 1 < p < \infty. We also establish the same type of result for solutions to the Aronsson type equation \nabla (F(x,\nabla u)) \cdot F\eta(x,\nabla u) = 0. Concerning \Omega we only assume that \partial\Omega is a quasicircle. In particular, our results generalize the boundary Harnack inequalities in [BL] and [LN2] to operators with variable coefficients.

  • 19.
    Lundström, Niklas
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Uppsala Univ, Dept Math.
    Olofsson, Marcus
    Uppsala Univ, Dept Math.
    Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness2014In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 145, no 3-4, p. 407-432Article in journal (Refereed)
  • 20.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    A framework for the valuation and risk management of non-maturing liabilities2005Report (Other academic)
  • 21.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    An estimate of a polynomial capacity1998In: Potential Analysis, ISSN 0926-2601, Vol. 9, no 3, p. 217-227Article in journal (Refereed)
  • 22.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains2006In: Collectanea Mathematica, ISSN 0010-0757, Vol. 57, no 1, p. 93-119Article in journal (Refereed)
  • 23.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Boundary value problems for parabolic Lamé systems in time-varying domains1999In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 48, no 4, p. 1285-1355Article in journal (Refereed)
  • 24.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Caloric measure and Reifenberg flatness2006In: Annales Academiae Scientiarium Fennicae. Mathematica, ISSN 1239-629X, Vol. 31, p. 405-436Article in journal (Refereed)
  • 25.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Free boundary regularity of multi-dimensional American options through blow-ups and global solutions2007In: Journal of Computational Mathematics and Optimization, Vol. 3, no 1, p. 39-76Article in journal (Refereed)
  • 26.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Harmonic analysís, quadratic forms and asymptotic expansions of risk measures2008In: Applied Mathematical Sciences, Vol. 2, no 21-24, p. 1023-1052Article in journal (Refereed)
  • 27.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Integrability of Green potentials in fractal domains1996In: Arkiv för Matematik, ISSN 0004-2080, Vol. 34, p. 335-381Article in journal (Refereed)
  • 28.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On an inverse type problem for the heat equation in parabolic regular graph domains2012In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 270, no 1-2, p. 197-222Article in journal (Refereed)
    Abstract [en]

    In this paper we prove some results concerning inverse/free boundary type problems, below the continuous threshold, for the heat equation in the setting of parabolic regular graph domains.

  • 29.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On area integral estimates for solutions to parabolic systems in time-varying and non-smooth cylinders2008In: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 360, no 6, p. 2987-3017Article in journal (Refereed)
  • 30.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On blow-ups and the classification of global solutions to parabolic free boundary problems2005Report (Other academic)
  • 31.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On blow-ups and the classification of global solutions to parabolic free boundary problems2006In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 55, no 4, p. 1233-1290Article in journal (Refereed)
  • 32.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On deposit volumes and the valuation of non-maturing liabilities2008In: Journal of Economic Dynamics & Control, ISSN 0165-1889, Vol. 32, no 3, p. 709-756Article in journal (Refereed)
  • 33.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On the behaviour near expiration for multidimensional American options2006Report (Other academic)
  • 34.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On the behaviour near expiry for multi-dimensional American options2008In: Journal of Mathmatical Analysis and Applications, ISSN 0022-247X, Vol. 339, no 1, p. 644-654Article in journal (Refereed)
  • 35.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On the rating and pricing of mortgage portfolios through structured finance2007In: Journal of Risk Model Validation, Vol. 1, no 4, p. 1-61Article in journal (Refereed)
  • 36.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Regularity below the continuous threshold in a two-phase parabolic free boundary problem2006In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 99, no 2, p. 257-286Article in journal (Refereed)
  • 37.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Smoothness properties of solutions to Dirichlet problems in domains with a fractal boundary1994Doctoral thesis, monograph (Other academic)
  • 38.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Structuring, credit quality and pricing of cash flow CDOS2006Report (Other academic)
  • 39.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    The Dirichlet problem for second order parabolic operators1997In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 46, no 1, p. 183-245Article in journal (Refereed)
  • 40.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Hoffman, S.
    Dirichlet problems for a non-stationary linearized system of Navier-Stokes equations in non-cylindrical domains.2002In: Journal of Methods and Applications of Analysis, Vol. 9, no 1, p. 13-98Article in journal (Refereed)
  • 41.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Hoffman, S.
    Lewis, J.
    Caloric measure in parabolic flat domains2004In: Duke Mathematical Journal, ISSN 0012-7094, Vol. 122, no 2, p. 281-346Article in journal (Refereed)
  • 42.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Hoffman, S.
    Lewis, J.
    Existence of big pieces of graphs for parabolic problems2003In: Annales Academiae Scientiarium Fennicae. Mathematica, ISSN 1239-629X, Vol. 28, p. 355-384Article in journal (Refereed)
  • 43.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Kukavica, Igor
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Unique continuation on the boundary for Dini domains1998In: Proceedings of the American Mathematical Society, ISSN 0002-9939, Vol. 126, no 2, p. 441-446Article in journal (Refereed)
  • 44.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Lewis, J.
    Boundary behaviour for p-harmonic functions in Lipschitz and starlike Lipschitz ring domains2007In: Annales Scientifigues de L'Ecole Normale Superieure, ISSN 0012-9593, Vol. 40, no 5, p. 765-813Article in journal (Refereed)
  • 45.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Lewis, J.
    Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains2008In: Advances in the Calculus of Variations, Vol. 1, p. 1-38Article in journal (Refereed)
  • 46.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Lewis, J.
    On a parabolic symmetry problem2007In: Revista Matemática Iberoamericana, ISSN 0213-2230, Vol. 23, no 2, p. 513-536Article in journal (Refereed)
  • 47.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Lewis, J.
    Regularity and free boundary regularity for the p-Laplacian in Lipschitz and C1-domains2008In: Annales Academiae Scientiarium Fennicae, ISSN 1239-629X, Vol. 33, p. 1-26Article in journal (Refereed)
  • 48.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Lewis, J.
    The boundary Harnack inequality for infinity harmonic functions in the plane2008In: Proceedings of the American Mathematical Society, ISSN 0002-9939, Vol. 136, p. 1311-1323Article in journal (Refereed)
  • 49.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Pascucci, Andrea
    Polidoro, Sergio
    Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators2010In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 249, no 8, p. 2044-2060Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to a proof of regularity, near the initial state, for solutions to the Cauchy-Dirichlet and obstacle problem for a class of second order differential operators of Kolmogorov type. The approach used here is general enough to allow us to consider smooth obstacles as well as non-smooth obstacles.

  • 50.
    Nyström, Kaj
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Skoglund, Jimmy
    A credit risk model for large dimensional portfolios with application to economic capital2005Report (Other academic)
12 1 - 50 of 60
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