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Nyström, Kaj

Open this publication in new window or tab >>Boundary behavior of solutions to the parabolic p-Laplace equation### Avelin, Benny

### Kuusi, Tuomo

### Nyström, Kaj

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##### Abstract [en]

##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-265500 (URN)10.2140/apde.2019.12.1 (DOI)000446595400001 ()
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##### Funder

Swedish Research Council, 637-2014-6822
Available from: 2015-10-30 Created: 2015-10-30 Last updated: 2018-11-28Bibliographically approved

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

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

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 Hölder 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.

Open this publication in new window or tab >>A unified deep artificial neural network approach to partial differential equations in complex geometries### Berg, Jens

### Nyström, Kaj

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_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}); 2018 (English)In: Neurocomputing, ISSN 0925-2312, E-ISSN 1872-8286, Vol. 317, p. 28-41Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Computational Mathematics
##### Research subject

Mathematics with specialization in Applied Mathematics
##### Identifiers

urn:nbn:se:uu:diva-362369 (URN)10.1016/j.neucom.2018.06.056 (DOI)000444237900003 ()
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##### Funder

Göran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2018-08-23 Created: 2018-10-04 Last updated: 2018-11-14Bibliographically approved

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

In this paper, we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator. The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method. We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.

Open this publication in new window or tab >>Quasi-linear PDEs and low-dimensional sets### Lewis, John L.

### Nyström, Kaj

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_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: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 20, no 7, p. 1689-1746Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-260772 (URN)10.4171/JEMS/797 (DOI)000433038100005 ()
#####

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##### Funder

Göran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2015-08-24 Created: 2015-08-24 Last updated: 2018-08-17Bibliographically approved

University of Kentucky, Lexington, KY, USA.

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.

Open this publication in new window or tab >>A new characterization of chord-arc domains### Azzam, Jonas

### Hofmann, Steve

### Martell, Jose Maria

### Nyström, Kaj

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_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); ### Toro, Tatiana

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}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_3_j_idt188_j_idt202",{id:"formSmash:j_idt184:3:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2017 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, no 4, p. 967-981Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Chord-arc domains, NTA domains, 1-sided NTA domains, uniform domains, uniform rectifiability, Carleson measures, harmonic measure, A∞ Muckenhoupt weights
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-226720 (URN)10.4171/JEMS/685 (DOI)000402474800002 ()
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##### Funder

EU, FP7, Seventh Framework Programme, 615112 HAPDEGMTSwedish Research Council
Available from: 2014-06-19 Created: 2014-06-19 Last updated: 2018-09-24Bibliographically approved

University of Washington, Seattle, USA.

University of Missouri, Columbia, USA.

Instituto de Ciencias Matematicas, Madrid, Spain.

University of Washington, Seattle, USA.

We show that if Ω⊂Rn^{+1}, n≥1, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of Ω implies the existence of exterior corkscrew points at all scales, so that in fact, Ω 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.

Open this publication in new window or tab >>*L*^{2} Solvability of boundary value problems for divergence form parabolic equations with complex coefficients### Nyström, Kaj

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}); 2017 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 262, no 3, p. 2808-2939Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-268050 (URN)10.1016/j.jde.2016.11.011 (DOI)000392463200026 ()
#####

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

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 a $(n+1)\times (n+1)$-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 $t$. For such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on $L^2(\mathbb R^{n+1},\mathbb C)=L^2(\partial\mathbb R^{n+2}_+,\mathbb C)$ under complex, $L^\infty$ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for $\partial_t+\mathcal{L}$, by way of layer potentials and with data in $L^2$, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.

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_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}); 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 >>The A∞-Property of the Kolmogorov Measure### Nyström, Kaj

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}); 2017 (English)In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 10, no 7, p. 1709-1756Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Kolmogorov equation, ultraparabolic, hypoelliptic, Lipschitz domain, doubling measure, parabolic measure, Kolmogorov measure, A(infinity)
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-335534 (URN)10.2140/apde.2017.10.1709 (DOI)000409094100005 ()
#####

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##### Funder

Göran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2017-12-08 Created: 2017-12-08 Last updated: 2017-12-08Bibliographically approved

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

We consider the Kolmogorov-Fokker-Planck operator K := Sigma(m)(i=1) partial derivative(xixi) + Sigma(m)(i=1) x(i)partial derivative(yi) - partial derivative(t) in unbounded domains of the form Omega={(x,x(m),y,y(m),t) RN+1 vertical bar x(m)>psi(x,y,t)}. Concerning and psi, we assume that Omega is what we call an (unbounded) admissible Lip(K)-domain: psi satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator K, as well as an additional regularity condition formulated in terms of a Carleson measure. We prove that in admissible Lip(K)-domains the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon-Nikodym derivative defines an A(infinity) weight with respect to this surface measure. Our result is sharp.

Open this publication in new window or tab >>The weak-A∞ PROPERTY OF HARMONIC AND p-HARMONIC MEASURES IMPLIES UNIFORM RECTIFIABILITY### Hofmann, Steve

### Le, Phi

### Maria Martell, Jose

### Nyström, Kaj

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_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}); 2017 (English)In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 10, no 3, p. 513-558Article in journal (Refereed) Published
##### Abstract [en]

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

MATHEMATICAL SCIENCE PUBL, 2017
##### Keywords

harmonic measure and p-harmonic measure, Poisson kernel, uniform rectifiability, Carleson measures, Green function, weak-A(infinity)
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-328276 (URN)10.2140/apde.2017.10.513 (DOI)000403096000001 ()
#####

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Available from: 2017-11-13 Created: 2017-11-13 Last updated: 2017-11-13Bibliographically approved

Univ Missouri, Dept Math, Columbia, MO 65211 USA..

Syracuse Univ, Math Dept, 215 Carnegie Bldg, Syracuse, NY 13244 USA..

CSIC, CSIC, Inst Ciencias Matemat, UAM,UC3M,UCM, C Nicolas Cabrera 13-15, E-28049 Madrid, Spain..

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.

Open this publication in new window or tab >>Tug-of-war, market manipulation, and option pricing### Nyström, Kaj

### Parviainen, Mikko

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}); 2017 (English)In: Mathematical Finance, ISSN 0960-1627, E-ISSN 1467-9965, Vol. 27, no 2, p. 279-312Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

infinity Laplace, nonlinear parabolic partial differential equation, option pricing, stochastic differential game, tug-of-war
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-214209 (URN)10.1111/mafi.12090 (DOI)000397560600001 ()
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Available from: 2014-01-08 Created: 2014-01-08 Last updated: 2017-05-16Bibliographically approved

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

University of Jyväskylä, Jyväskylä, Finland.

We develop an option pricing model based on a tug-of-war game. This two-playerzero-sum stochastic differential game is formulated in the context of a multidimen-sional ﬁnancial market. The issuer and the holder try to manipulate asset price pro-cesses in order to minimize and maximize the expected discounted reward. We provethat the game has a value and that the value function is the unique viscosity solutionto a terminal value problem for a parabolic partial differential equation involving thenonlinear and completely degenerate inﬁnity Laplace operator.

Open this publication in new window or tab >>Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients### Castro, Alejandro

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.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}); 2016 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 5, article id 124Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-266835 (URN)10.1007/s00526-016-1058-8 (DOI)000386708700016 ()
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Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2017-12-01Bibliographically approved

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.