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  • 1.
    Auscher, Pascal
    et al.
    Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay.
    Egert, Moritz
    Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    L2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients2016In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863Article in journal (Refereed)
    Abstract [en]

    We introduce a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space. This involves a parabolic Dirac operator at the boundary. We allow for measurable time dependence and some transversal dependence in the coefficients. We obtain layer potential representations for solutions in some classes and prove new well-posedness and perturbation results. As a byproduct, we prove for the first time a Kato estimate for the square root of parabolic operators with time dependent coefficients. This considerably extends prior results obtained by one of us under time and transversal independence. A major difficulty compared to a similar treatment of elliptic equations is the presence of non-local fractional derivatives in time.

  • 2.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Capogna, Luca
    University of Arkansas.
    Citti, Giovanna
    University of Bologna.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 257, p. 25-65Article in journal (Refereed)
    Abstract [en]

    We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype\begin{equation*}  \partial_tu= -\sum_{i=1}^{m}X_i^\ast ( |\X u|^{p-2} X_i u)\end{equation*}where $p\ge 2$, $ \ \X = (X_1,\ldots, X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $\mu$, and $X_i^*$ denotes the adjoint of $X_i$ with respect to $\mu$. Our estimates are derived assuming that (i) the control distance $d$ generated by $\X$ induces the same topology on $\M$; (ii) a doubling condition for the $\mu$-measure of $d-$metric balls and (iii) the validity of a Poincar\'e inequality involving $\X$ and $\mu$. Our results extend the recent work in \cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting including the model cases of (1) metrics generated by H\"ormander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.

  • 3.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Kuusi, Tuomo
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Boundary behavior of solutions to the parabolic p-Laplace equation2015Article in journal (Refereed)
    Abstract [en]

    We establish boundary estimates for non-negative solutions to the $p$-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and H\"older continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.

  • 4. Avelin, Benny
    et al.
    Lundström, Niklas L. P.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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)
  • 5. Avelin, Benny
    et al.
    Lundström, Niklas L.P
    Nyström, Kaj
    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 operators of p-Laplace type of the form ∇·A(x,∇u) = 0. ConcerningA we assume, for p ∈ (1,∞) fixed, an appropriate ellipticity type condition, H¨older continuityin x and that A(x, ) = ||p−1A(x, /||) whenever x ∈ Rn and ∈ Rn \ {0}. Let  ⊂ Rn be abounded domain, let D be a compact subset of . We say that ˆu = ˆup,D, is the A-capacitaryfunction for D in  if ˆu ≡ 1 on D, ˆu ≡ 0 on @ in the sense of W1,p0 () and ∇·A(x,∇ˆu) = 0 in \D in the weak sense. We extend ˆu to Rn \  by putting ˆu ≡ 0 on Rn \ . Then there existsa unique finite positive Borel measure ˆμ on Rn, with support in @, such thatZ hA(x,∇ˆu),∇i dx = −Z dˆμ whenever ∈ C∞0 (Rn \ D).In this paper we prove that if  is Reifenberg flat with vanishing constant, thenlimr→0infw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= limr→0supw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= n−1,for every , 0 < ≤ 1. In particular, we prove that ˆμ is an asymptotically optimal doublingmeasure on @.

  • 6. Avelin, Benny
    et al.
    Nyström, Kaj
    Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains2014In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 266, no 9, p. 5955-6005Article in journal (Refereed)
  • 7.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures2013In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, p. 149-159Article in journal (Refereed)
    Abstract [en]

    Let be a system of C vector fields in Rn satisfying Hörmander’s finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot–Carathéodory distance d induced by X. We prove the doubling property of certain boundary measures associated to non-negative solutions, which vanish on a portion of Ω, to the equation

    Given p, 1<p<, fixed, we impose conditions on the function A=(A1,…,Am):Rn×RmRm, which imply that the equation is a quasi-linear partial differential equation of p-Laplace type structured on vector fields satisfying the classical Hörmander condition. In the case p=2 and for linear equations, our result coincides with the doubling property of associated elliptic measures. To prove our result we establish, and this is of independent interest, a Wolff potential estimate for subelliptic equations of p-Laplace type.

  • 8.
    Azzam, Jonas
    et al.
    University of Washington, Seattle, USA.
    Hofmann, Steve
    University of Missouri, Columbia, USA.
    Martell, Jose Maria
    Instituto de Ciencias Matematicas, Madrid, Spain.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Toro, Tatiana
    University of Washington, Seattle, USA.
    A new characterization of chord-arc domains2015In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863Article in journal (Refereed)
    Abstract [en]

    We show that if $\om \subset \ree$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $\om$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $\om$ is a chord-arc domain,i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew conditions, and an interior Harnack Chain condition.  We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain free boundary problems.

  • 9.
    Castro, Alejandro
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Sande, Olow
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients2016In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 5, article id 124Article in journal (Refereed)
    Abstract [en]

    We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is an $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. We prove that the boundedness of associated single layer potentials, with data in $L^2$, can be reduced to two crucial estimates (Theorem \ref{th0}), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two  crucial estimates in the case of real, symmetric operators (Theorem \ref{th2}). Our results are crucial when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator $\partial_t+\mathcal{L}$ in $\mathbb R_+^{n+2}$, with $L^2$-data on $\mathbb R^{n+1}=\partial\mathbb R_+^{n+2}$, and by way of layer potentials.

  • 10. Cinti, Chiara
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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 \fancyscriptL=∑i,j=1mai,j(z)∂xixj+∑i=1mai(z)∂xi+∑i,j=1Nbi,jxi∂xj−∂t, where z = (x, t) belongs to an open set Ω⊂RN×R , and 1 ≤ m ≤ N. Let z˜∈Ω , let K be a compact subset of Ω−− , and let Σ⊂∂Ω be such that K∩∂Ω⊂Σ . We give sufficient geometric conditions for the validity of the following Carleson type estimate. There exists a positive constant C K , depending only on Ω,Σ,K,z˜ and on \fancyscriptL , such that supKu≤CKu(z˜), for every non-negative solution u of \fancyscriptLu=0 in Ω such that u∣Σ=0 .

  • 11. Cinti, Chiara
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Polidoro, Sergio
    A Carleson-type estimate in Lipschitz type domains for non-negative solutions to Kolmogorov equations2013In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 12, no 2, p. 439-465Article in journal (Refereed)
    Abstract [en]

    We prove a Carleson type estimate, in Lipschitz type domains, for non-negative solutions to a class of second order degenerate differential operators of Kolmogorov type of the form L = Sigma(m)(i,j=1)a(i,j)(z)partial derivative x(i)x(j) + 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) is an element of RN+1, 1 <= m <= N. Our estimate is scale-invariant and generalizes previous results valid for second order uniformly parabolic equations to the class of operators considered.

  • 12. Cinti, Chiara
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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)
  • 13.
    Colesanti, Andrea
    et al.
    U.Dini, Firenze, Italy.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Salani, Paolo
    U.Dini, Firenze, Italy.
    Xiao, Jie
    Memorial University of Newfoundland, St. John's, Canada.
    Yang, Deane
    Polytechnic Institute of New York University, New York, USA.
    Zhang, Gaoyong
    Polytechnic Institute of New York University, New York, USA.
    The Hadamard variational formula and the Minkowski problem for p-Capacity2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 285, p. 1511-1585Article in journal (Refereed)
    Abstract [en]

    A Hadamard variational formula for p-capacity of convex bodies in R-n is established when 1 < p < n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge Ampere type equation. zkiniqueness for the Minkowski problem for p-capacity is established when 1 < p < n and existence and regularity when 1 < p < 2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p = 2).

  • 14. Frentz, Marie
    et al.
    Garofalo, Nicola
    Götmark, Elin
    Munive, Isidro
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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.

  • 15. Frentz, Marie
    et al.
    Götmark, Elin
    Nyström, Kaj
    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. 5002-5041Article 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 RnRn, X={X1,…,Xq}X={X1,…,Xq}, q⩽nq⩽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.

  • 16. Frentz, Marie
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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)
  • 17. Frentz, Marie
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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)
  • 18. Götmark, Elin
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Boundary behaviour 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.

  • 19.
    Hofmann, Steve
    et al.
    Univ Missouri, Dept Math, Columbia, MO 65211 USA..
    Le, Phi
    Syracuse Univ, Math Dept, 215 Carnegie Bldg, Syracuse, NY 13244 USA..
    Maria Martell, Jose
    CSIC, CSIC, Inst Ciencias Matemat, UAM,UC3M,UCM, C Nicolas Cabrera 13-15, E-28049 Madrid, Spain..
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The weak-A∞ PROPERTY OF HARMONIC AND p-HARMONIC MEASURES IMPLIES UNIFORM RECTIFIABILITY2017In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 10, no 3, p. 513-558Article in journal (Refereed)
    Abstract [en]

    Let E subset of Rn+1, n >= 2, be an Ahlfors-David regular set of dimension n. We show that the weak- A 1 property of harmonic measure, for the open set Omega: =Rn+1 \ E, implies uniform rectifiability of E. More generally, we establish a similar result for the Riesz measure, p-harmonic measure, associated to the p-Laplace operator, 1 < p < infinity.

  • 20.
    Hofmann, Steve
    et al.
    University of Missouri, Columbia, USA.
    Le, Phi
    University of Missouri, Columbia, USA.
    Martell, Jose Maria
    Instituto de Ciencias Matematicas, Madrid, Spain.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The weak-A property of harmonic and p-harmonic measures implies uniform rectifiability2015In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206XArticle in journal (Refereed)
    Abstract [en]

    Let $E\subset \ree$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set$\Omega:= \ree\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure,associated to the $p$-Laplace operator, $1<p<\infty$.

  • 21. Hofmann, Steve
    et al.
    Lewis, John L.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Caloric measure in parabolic flat domains2004In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 122, p. 281-346Article in journal (Refereed)
  • 22. Hofmann, Steve
    et al.
    Lewis, John L.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Existence of big pieces of graphs for parabolic problems2003In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 28, no 2, p. 355-384Article in journal (Refereed)
  • 23. Hofmann, Steve
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Dirichlet problems for a nonstationary linearized system of Navier-Stokes equations in non-cylindrical domains2002In: Methods and Applications of Analysis, ISSN 1073-2772, E-ISSN 1945-0001, Vol. 9, p. 13-98Article in journal (Refereed)
  • 24. Kukavica, Igor
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Unique continuation on the boundary for Dini domains1998In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 126, p. 441-446Article in journal (Refereed)
  • 25. Kuusi, Tuomo
    et al.
    Mingione, Giuseppe
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    A Boundary Harnack Inequality for Singular Equations of p-Parabolic Type2014In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 142, no 8, p. 2705-2719Article in journal (Refereed)
    Abstract [en]

    We prove a boundary Harnack type inequality for nonnegative solutions to singular equations of p-parabolic type, 2n/(n+1)<p<2, in a time-independent cylinder whose base is C1,1-regular. Simple examples show, using the corresponding estimates valid for the heat equation as a point of reference, that this type of inequality cannot, in general, be expected to hold in the degenerate case ( 2<p<∞).

  • 26. Kuusi, Tuomo
    et al.
    Mingione, Giuseppe
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets2014In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 101, no 2, p. 119-151Article in journal (Other academic)
    Abstract [en]

    In this paper we prove, by showing that solutions have exactly the same degree of regularity as the obstacle, optimal regularity results for obstacle problems involving evolutionary p-Laplace type operators. A main ingredient, of independent interest, is a new intrinsic interpolative geometry allowing for optimal linearization principles via blow-up analysis at contact points. This also opens the way to the proof of a removability theorem for solutions to evolutionary p-Laplace type equations. A basic feature of the paper is that no differentiability in time is assumed on the obstacle; this is in line with the corresponding linear results.

  • 27. Lewis, John L.
    et al.
    Lundström, Niklas
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Boundary Harnack inequalities for operators of p-Laplace type in Reifenberg flat domains2008In: Proceedings of Symposia in Pure Mathematics, Vol. 79, p. 229-266Article in journal (Refereed)
  • 28. 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.
    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)
  • 29. 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.
    Boundary behaviour for $p$ harmonic functions in Lipschitz and starlike Lipschitz ring domains2007In: Annales Scientifiques de l'Ecole Normale Supérieure, ISSN 0012-9593, E-ISSN 1873-2151, Vol. 40, p. 765-813Article in journal (Refereed)
  • 30. 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.
    Boundary behaviour of $p$-harmonic functions in domains beyond Lipschitz domains2008In: Advances in Calculus of Variations, ISSN 1864-8258, E-ISSN 1864-8266, Vol. 1, p. 133-170Article in journal (Refereed)
  • 31. 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.
    New results for $p$ harmonic functions2011In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 7, p. 345-363Article in journal (Refereed)
  • 32. 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.
    On a parabolic symmetry problem2007In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 23, p. 513-536Article in journal (Refereed)
  • 33. 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 and free boundary regularity for the $p$ Laplacian in Lipschitz and $C^1$ domains2008In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 33, p. 523-548Article in journal (Refereed)
  • 34. 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 and free boundary regularity for the p-Laplace operator in Reifenberg flat and Ahlfors regular domains2012In: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 25, no 3, p. 827-862Article in journal (Refereed)
  • 35. 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 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 Nyström (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.

  • 36. 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)
  • 37. 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.
    The boundary Harnack inequality for infinity harmonic functions in the plane2008In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 136, p. 1311-1323Article in journal (Refereed)
  • 38. 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.
    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)
  • 39. 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.
    Vogel, Andrew
    On the dimension of p-harmonic measure in space2013In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 15, no 6, p. 2197-2256Article in journal (Refereed)
    Abstract [en]

    Let Omega subset of R-n, n >= 3, and let p, 1 < p < infinity, p not equal D 2, be given. In this paper we study the dimension of p-harmonic measures that arise from nonnegative solutions to the p-Laplace equation, vanishing on a portion of partial derivative Omega, in the setting of delta-Reifenberg flat domains. We prove, for p >= n, that there exists (delta) over tilde = (delta) over tilde (p, n) > 0 small such that if Omega is a delta-Reifenberg flat domain with delta < <(delta)over tilde>, then p-harmonic measure is concentrated on a set of sigma-finite Hn-1-measure. We prove, for p >= n, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p-harmonic measure is always less than n - 1. We also prove that if 2 < p < n, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is less than n - 1, while if 1 < p < 2, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is larger than n - 1. Furthermore, perturbing off the case p = 2; we derive estimates for the Hausdorff dimension of p-harmonic measure when p is near 2.

  • 40.
    Lewis, John
    et al.
    University of Kentucky, Lexington, KY, USA.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Quasi-linear PDEs and low-dimensional sets2015In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863Article in journal (Refereed)
    Abstract [en]

    In this paper we establish new results concerning boundary Harnack inequalities and the Martin boundary problem, for non-negative solutions to equations of $p$-Laplace type with variable coefficients. The key novelty is that we consider solutions which vanish only on a low-dimensional set $\Sigma$ in $\mathbb R^n$ and this is different compared to the more traditional setting of boundary value problems set in the geometrical situation of a bounded domain in $\mathbb R^n$ having a boundary with (Hausdorff) dimension in the range $[n-1,n)$. We establish our  quantitative and scale-invariant estimates in the context of low-dimensional Reifenberg flat sets.

  • 41.
    Li, Kai
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Olofsson, Marcus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Optimal switching problems under partial information2015In: Monte Carlo Methods and Applications, ISSN 1569-3961, Vol. 21, no 2, p. 91-120Article in journal (Refereed)
    Abstract [en]

    In this paper we formulate and study an optimal switching problem under partial information. In our model the agent/manager/investor attempts to maximize the expected reward by switching between different states/investments. However, he is not fully aware of his environment and only an observation process, which contains partial information about the environment/underlying, is accessible. It is based on the partial information carried by this observation process that all decisions must be made. We propose a probabilistic numerical algorithm based on dynamic programming, regression Monte Carlo methods, and stochastic filtering theory to compute the value function. In this paper, the approximation of the value function and the corresponding convergence result are obtained when the underlying and observation processes satisfy the linear Kalman-Bucy setting. A numerical example is included to show some specifc features of partial information.

  • 42. Lundström, Niklas L. P.
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    On a two-phase free boundary condition for $p$-harmonic measures2009In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 129, p. 231-249Article in journal (Refereed)
  • 43. Lundström, Niklas L. P.
    et al.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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)
  • 44.
    Lundström, Niklas
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Olofsson, Marcus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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)
    Abstract [en]

    In this paper we study viscosity solutions to the system  \begin{eqnarray*}&&\min\biggl\{-\mathcal{H}u_i(x,t)-\psi_i(x,t),u_i(x,t)-\max_{j\neq i}(-c_{i,j}(x,t)+u_j(x,t))\biggr\}=0,\notag\\&&u_i(x,T)=g_i(x),\ i\in\{1,\dots,d\},\end{eqnarray*}where $(x,t)\in\mathbb R^{N}\times [0,T]$. Concerning $\mathcal{H}$ we assume that $\mathcal{H}=\mathcal{L}+\mathcal{I}$ where$\mathcal{L}$ is a linear, possibly degenerate, parabolic operator of second order and $\mathcal{I}$ is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when thedynamics of the underlying state variables is described by $N$-dimensional Levy processes.  We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth andstructural assumptions on the data, i.e., on the operator $\mathcal{H}$ and on continuous functions $\psi_i$, $c_{i,j}$, and$g_i$.   Using the comparison principle we establish the existence of a unique viscosity solution $(u_1,\dots,u_d)$  to the system by using Perron's method. Our contribution, compared to the existing literature, is that we establish existence and uniqueness of viscosity solutions in the setting of Levy processes and non-local operators with no sign assumption on the switching costs $\{c_{i,j}\}$ and allowing $c_{i,j}$  to depend on $x$ as well as $t$.

  • 45.
    Lundström, Niklas
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Olofsson, Marcus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    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)
  • 46.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    A backward in time Harnack inequality for non-negative solutions to fully non-linear parabolic equations2014In: Revista di Matematica della Universita Di Parma, ISSN 0035-6298, Vol. 5, no 1, p. 1-14Article in journal (Refereed)
    Abstract [en]

    We consider fully non-linear parabolic equations of the form\[Hu =F(D^2u(x,t),Du(x,t),x,t)-\partial_tu = 0\]in bounded space-time domains $D\subset\mathbb R^{n+1}$, assuming only $F(0,0,x,t)=0$ and a uniform parbolicity condition on $F$. For domains of the form $\Omega_T=\Omega\times (0,T)$, where $\Omega\subset\mathbb R^n$ is a bounded Lipschitz and $T>0$, we establish a scale-invariant backward in time Harnackinequality for non-negative solutions vanishing on the lateral boundary. Our argument rests on the comparison principle, the Harnack inequality and local H{\"o}lder continuity estimates.

  • 47.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    An estimate of a polynomial capacity1998In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 9, no 3, p. 217-227Article in journal (Refereed)
  • 48.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Boundary value problems and duality between $L^p$ Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains2006In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 57, p. 93-119Article in journal (Refereed)
  • 49.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Boundary value problems for parabolic Lamé systems in time-varying domains1999In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 48, p. 1285-1355Article in journal (Refereed)
  • 50.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Caloric measure and Reifenberg flatness2006In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 31, p. 405-436Article in journal (Refereed)
12 1 - 50 of 87
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