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Strömqvist, Martin

Open this publication in new window or tab >>Harnack's Inequality for Parabolic Nonlocal Equations### Strömqvist, Martin

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##### Abstract [en]

##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-346641 (URN)10.1016/j.anihpc.2019.03.003 (DOI)000488142500007 ()
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Available from: 2018-03-20 Created: 2018-03-20 Last updated: 2019-10-25Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.

Open this publication in new window or tab >>Local Boundedness of Solutions to Non-Local Parabolic Equations Modeled on the Fractional P−Laplacian### Strömqvist, Martin

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}); 2019 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 266, no 12, p. 7948-7959Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematical Analysis
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-346640 (URN)10.1016/j.jde.2018.12.021 (DOI)000461785900007 ()
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Available from: 2018-03-20 Created: 2018-03-20 Last updated: 2019-04-04Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the form

partial derivative(t)u(x, t) + P.V integral(Rn) K(x, y, t)vertical bar u(x, t) - u(y, t)vertical bar(p-2)(u(x, t) - u(y, t)) dy = 0,

(x, t) is an element of R-n x R, where P.V. means in the principle value sense, p is an element of (1, infinity) and the kernel obeys K (x,y,t) approximate to vertical bar x-y vertical bar(n+ps) for some s is an element of (0, 1), uniformly in (x, y, t) is an element of R-n x R-n x R.

Open this publication in new window or tab >>Homogenization of a parabolic Dirichlet problem by a method of Dahlberg### Castro, Alejandro J.

### Strömqvist, Martin

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}); 2018 (English)In: Publicacions matemàtiques, ISSN 0214-1493, E-ISSN 2014-4350, Vol. 62, no 2, p. 439-473Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-345204 (URN)10.5565/PUBLMAT6221805 (DOI)000435637100005 ()
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Available from: 2018-03-08 Created: 2018-03-08 Last updated: 2018-09-10Bibliographically approved

Nazarbayev Univ, Dept Math, Astana 010000, Kazakhstan.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Consider the linear parabolic operator in divergence form: Hu := partial derivative(t)u (X, t) - div(A (X) del u (X, t)). We employ a method of Dahlberg to show that the Dirichlet problem for H in the upper half plane is well-posed for boundary data in L-p, for any elliptic matrix of coef-ficients A which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation partial derivative(t)u(epsilon) (X, t) - div(A (X/epsilon) del u(epsilon) (X, t)) in Lipschitz domains with L-p-boundary data.

Open this publication in new window or tab >>On the parabolic Lipschitz approximation of parabolic uniform rectifiable sets### Nyström, Kaj

### Strömqvist, Martin

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}); 2017 (English)In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 33, no 4, p. 1397-1422Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Parabolic Lipschitz graph, parabolic uniform rectifiability, big pieces, parabolic measure, caloric measure
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-348942 (URN)10.4171/RMI/976 (DOI)000425897100012 ()
#####

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Available from: 2018-04-19 Created: 2018-04-19 Last updated: 2018-04-19Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

We prove the existence of big pieces of regular parabolic Lip-schitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized two cube condition. An application to the fine properties of parabolic measure is given.

Open this publication in new window or tab >>Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization### Karakhanyan, Aram L.

### Strömqvist, Martin

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.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}); 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}); 2016 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 6Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-313544 (URN)10.1007/s00526-016-1088-2 (DOI)000390043500009 ()
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Available from: 2017-02-01 Created: 2017-01-20 Last updated: 2017-11-29Bibliographically approved

Univ Edinburgh, Sch Math, Edinburgh, Midlothian, Scotland..

We consider the intersection of a convex surface Gamma with a periodic perforation of R-d, which looks like a sieve, given by T epsilon = boolean OR(d)(k is an element of Z) {epsilon k + a epsilon T} where T is a given compact set and a epsilon << epsilon is the size of the perforation in the epsilon-cell (0, epsilon)(d) subset of R-d. When epsilon tends to zero we establish uniform estimates for p- capacity, 1 < p < d, of the set Gamma n T-epsilon. Additionally, we prove that the intersections Gamma boolean AND {epsilon k + a(epsilon)T}(k) are uniformly distributed over Gamma and give estimates for the discrepancy of the distribution. As an application we show that the thin obstacle problem with the obstacle defined on the intersection of Gamma and the perforations, in a given bounded domain, is homogenizable when p < 1+ d/4. This result is new even for the classical Laplace operator.

Open this publication in new window or tab >>On the parabolic Lipschitz approximation of parabolic uniform rectifiable sets### Nyström, Kaj

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Strömqvist, Martin

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.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}); 2015 (English)In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616Article in journal (Refereed) Accepted
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-263916 (URN)
#####

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Available from: 2015-10-04 Created: 2015-10-04 Last updated: 2017-12-01Bibliographically approved

We prove the existence of big pieces of regular parabolic Lipschitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized in time two cube condition. An application to the fine properties of parabolic measure is given.

Open this publication in new window or tab >>Application of Uniform Distribution to Homogenization of a Thin Obstacle Problem with p-Laplacian### Karakhanyan, Aram

### Strömqvist, Martin

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}); 2014 (English)In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 39, no 10, p. 1870-1897Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Capacity, Free boundary, Homogenization, p-Laplacian, Perforated domains, Quasiuniform convergence, Thin obstacle, Uniform distributions
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-268041 (URN)10.1080/03605302.2014.895013 (DOI)000341003700004 ()2-s2.0-84906490732 (Scopus ID)
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Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved

Edinburgh University.

KTH, Matematik (Avd.).

In this paper we study the homogenization of p-Laplacian with thin obstacle in a perforated domain. The obstacle is defined on the intersection between a hyperplane and a periodic perforation. We construct the family of correctors for this problem and show that the solutions for the epsilon-problem converge to a solution of a minimization problem of similar form but with an extra term involving the mean capacity of the obstacle. The novelty of our approach is based on the employment of quasi-uniform convergence. As an application we obtain Poincare's inequality for perforated domains.

Open this publication in new window or tab >>Homogenization in Perforated Domains### Strömqvist, Martin

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}); 2014 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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

Stockholm: KTH Royal Institute of Technology, 2014. p. vii, 22
##### Series

TRITA-MAT-A ; 2014:11
##### National Category

Natural Sciences
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-268044 (URN)978-91-7595-213-0 (ISBN)
##### Public defence

2014-09-05, F3, Lindstedtsvägen 25, KTH, 13:00 (English)
##### Opponent

### Armstrong, Scott

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##### Supervisors

### Shahgholian, Henrik

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##### Note

KTH, Matematik (Avd.).

Homogenization theory is the study of the asymptotic behaviour of solutionsto partial differential equations where high frequency oscillations occur.In the case of a perforated domain the oscillations are due to variations in thedomain of the equation. The four articles that constitute this thesis are devotedto obstacle problems in perforated domains. Paper A treats an optimalcontrol problem where the objective is to control the solution to the obstacleproblem by the choice of obstacle. The optimal obstacle in the perforated domain,as well as its homogenized limit, are characterized in terms of certainauxiliary problems they solve. In papers B,C and D the authors solve homogenizationproblems in a perforated domain where the perforation is definedas the intersection between a periodic perforation and a hyper plane. Thetheory of uniform distribution is an indespensible tool in the analysis of theseproblems. Paper B treats the obstacle problem for the Laplace operator andthe authors use correctors to derive a homogenized equation. Paper D is ageneralization of paper B to the p-Laplacian. The authors employ capacitytechniques which are well adapted to the problem. In Paper C the obstaclevaries on the same scale as the perforations. In this setting the authorsemploy the theory of Gamma-convergence to prove a homogenization result.

CEREMADE Université Paris-Dauphine, Paris.

KTH, Matematik (Avd.).

QC 20140703

Available from: 2015-12-04 Created: 2015-12-01 Last updated: 2015-12-04Bibliographically approvedOpen this publication in new window or tab >>Gamma-convergence of Oscillating Thin Obstacles### Strömqvist, Martin

### Koroleva, Yulia

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}); 2013 (English)In: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 4, p. 88-100Article in journal (Refereed) Published
##### Keywords

obstacle problem, homogenization theory, Γ-convergence
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-268046 (URN)
#####

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Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved

KTH, Matematik (Avd.).

Lomonosov Moscow State University.

Open this publication in new window or tab >>Highly oscillating thin obstacles### Lee, Ki-ahm

### Strömqvist, Martin

### Yoo, Minha

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}); 2013 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 237, p. 286-315Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Homogenization, Thin obstacle, Ergodicity, Discrepancy, Corrector
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-268042 (URN)10.1016/j.aim.2013.01.007 (DOI)000316512500008 ()2-s2.0-84874437735 (Scopus ID)
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Available from: 2013-04-22 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved

KTH, Matematik (Avd.).

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