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Nyström, Kaj

Open this publication in new window or tab >>Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness### Lundström, Niklas

### Nyström, Kaj

### Olofsson, Marcus

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##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-65955 (URN)10.1007/s00229-014-0683-9 (DOI)000343881600008 ()
#####

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##### Note

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Uppsala Univ, Dept Math.

Uppsala Univ, Dept Math.

In this paper we study viscosity solutions to the systemwhere . Concerning , we assume that where is a linear, possibly degenerate, parabolic operator of second order and 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 the dynamics of the underlying state variables is described by an N-dimensional Levy process. 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 the continuous functions , c (i,j) , and g (i) . Using the comparison principle we prove the existence of a unique viscosity solution (u (1), . . . , u (d) ) to the system by Perron's method. Our main contribution 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.

Available from: 2013-02-13 Created: 2013-02-13 Last updated: 2018-06-08Bibliographically approvedOpen this publication in new window or tab >>Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type### Lundström, Niklas L P

### Nyström, Kaj

### Olofsson, Marcus

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2014 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 193, no 4, p. 1213-1247Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2014
##### Keywords

System, Variational inequality, Existence, Viscosity solution, Obstacle problem, Regularity, Kolmogorov equation, Ultraparabolic, Hypoelliptic, Backward stochastic differential equation, Reflected backward stochastic differential equation, Optimal switching problem
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-65956 (URN)10.1007/s10231-013-0325-y (DOI)000339962000017 ()
#####

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Available from: 2013-02-13 Created: 2013-02-13 Last updated: 2018-06-08Bibliographically approved

Department of Mathematics, Uppsala University, Sweden.

Department of Mathematics, Uppsala University, Sweden.

Department of Mathematics, Uppsala University, Sweden.

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.

Open this publication in new window or tab >>Boundary behavior of non-negative solutions to degenerate sub-elliptic equations### Götmark, Elin

### Nyström, Kaj

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2013 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 254, no 8, p. 3431-3460Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Hormander condition, Boundary Harnack inequality, Elliptic measure, Sub-elliptic PDEs, Muckenhoupt weights, Quasi-linear equations p-Laplace
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-67954 (URN)10.1016/j.jde.2013.01.030 (DOI)000315831000011 ()
#####

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Available from: 2013-04-15 Created: 2013-04-09 Last updated: 2018-06-08Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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.

Open this publication in new window or tab >>A boundary estimate for non-negative solutions to Kolmogorov operators in non-divergence form### Cinti, Chiara

### Nyström, Kaj

### Polidoro, Sergio

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2012 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 191, no 1, p. 1-23Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-53272 (URN)10.1007/s10231-010-0172-z (DOI)000298648300001 ()
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Available from: 2012-03-23 Created: 2012-03-19 Last updated: 2018-06-08Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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.

Open this publication in new window or tab >>Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions### Frentz, Marie

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Garofalo, Nicola

### Götmark, Elin

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Munive, Isidro

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); ### Nyström, Kaj

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_4_j_idt188_j_idt202",{id:"formSmash:j_idt184:4:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2012 (English)In: 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) Published
##### Abstract [en]

##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:umu:diva-47921 (URN)000309320600009 ()
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Available from: 2011-10-07 Created: 2011-10-03 Last updated: 2018-06-08Bibliographically approved

Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.

Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.

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.

Open this publication in new window or tab >>On an inverse type problem for the heat equation in parabolic regular graph domains### Nyström, Kaj

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2012 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 270, no 1-2, p. 197-222Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

New York, USA: Springer, 2012
##### Keywords

Caloric measure, Heat equation, Absolute continuity, Poisson kernel, One-phase free boundary problem, Two-phase free boundary problem, Inverse problem, Blow-up
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-52167 (URN)10.1007/s00209-010-0793-3 (DOI)000299125000011 ()
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Available from: 2012-02-23 Created: 2012-02-13 Last updated: 2018-06-08Bibliographically approved

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.

Open this publication in new window or tab >>Regularity of flat free boundaries in two-phase problems for the p-Laplace operator### Lewis, John L.

### Nyström, Kaj

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2012 (English)In: 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) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-53271 (URN)10.1016/j.anihpc.2011.09.002 (DOI)000300603500005 ()
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Available from: 2012-03-23 Created: 2012-03-19 Last updated: 2018-06-08Bibliographically approved

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.

Open this publication in new window or tab >>The obstacle problem for parabolic non-divergence form operators of Hörmander type### Frentz, Marie

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.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2012 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 252, no 9, p. 5002-2041Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2012
##### Keywords

obstacle problem, parabolic equations, Hormander condition, hypo-elliptic, embedding theorem, a priori estimates
##### National Category

Mathematical Analysis
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-51517 (URN)10.1016/j.jde.2012.01.032 (DOI)000301090200014 ()
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##### Note

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={X_{1},…,X_{q}}X={X_{1},…,X_{q}}, 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.

Originally published in thesis in manuscript form.

Available from: 2012-01-25 Created: 2012-01-24 Last updated: 2018-06-08Bibliographically approvedOpen this publication in new window or tab >>Boundary estimates for solutions to operators of *p*-Laplace type with lower order terms### Avelin, Benny

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.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2011 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, no 1, p. 264-291Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Boundary Harnack inequality, p-harmonic function, A-harmonic function, (A, B)-harmonic function, Variable coefficients, Operators with lower order terms, Reifenberg flat domain, Martin boundary
##### National Category

Mathematics Probability Theory and Statistics
##### Research subject

Mathematical Statistics; Mathematics
##### Identifiers

urn:nbn:se:umu:diva-40224 (URN)10.1016/j.jde.2010.09.011 (DOI)000284919600013 ()
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_j_idt371",{id:"formSmash:j_idt184:8:j_idt188:j_idt371",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_j_idt371",multiple:true});
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Available from: 2011-02-17 Created: 2011-02-17 Last updated: 2018-06-08Bibliographically approved

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 *Ω*⊂*R**n*, 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 |∇*η**B*(*x*,*η*)|⩽*c*|*η*|^{p−2}, |*B*(*x*,*η*)|⩽*c*|*η*|^{p−1}, for some constant *c*, and that *B*(*x*,*η*)=|*η*|^{p−1}*B*(*x*,*η*/|*η*|), whenever *x*∈*R**n*, *η*∈*R**n*∖{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.

Open this publication in new window or tab >>New results for p harmonic functions### Lewis, John L

### Nyström, Kaj

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2011 (English)In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 7, no 2, p. 345-363Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

International Press of Boston, 2011
##### Keywords

boundary Harnack inequality, p harmonic function, Lipschitz domain, Martin boundary, free boundary
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-104579 (URN)10.4310/PAMQ.2011.v7.n2.a4 (DOI)000274659200004 ()
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Available from: 2015-06-12 Created: 2015-06-11 Last updated: 2018-06-07Bibliographically approved

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.