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Lindgren, Erikorcid.org/0000-0003-4309-9242

Open this publication in new window or tab >>Approximation of the least Rayleigh quotient for degree p homogeneous functionals### Hynd, R.

### Lindgren, Erik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_0_j_idt183_some",{id:"formSmash:j_idt179:0:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_0_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_0_j_idt183_otherAuthors",{id:"formSmash:j_idt179:0:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_0_j_idt183_otherAuthors",multiple:true}); 2017 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 12, p. 4873-4918Article in journal (Refereed) Published
##### Abstract [en]

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

Academic Press Inc., 2017
##### Keyword

Doubly nonlinear evolution, Inverse iteration, Large time behavior, Nonlinear eigenvalue problem
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-207480 (URN)10.1016/j.jfa.2017.02.024 (DOI)000400539700001 ()2-s2.0-85015702520 (Scopus ID)
#####

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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We present two novel methods for approximating minimizers of the abstract Rayleigh quotient Φ(u)/‖u‖p. Here Φ is a strictly convex functional on a Banach space with norm ‖⋅‖, and Φ is assumed to be positively homogeneous of degree p∈(1,∞). Minimizers are shown to satisfy ∂Φ(u)−λJp(u)∋0 for a certain λ∈R, where Jp is the subdifferential of 1p‖⋅‖p. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy ∂Φ(uk)−Jp(uk−1)∋0(k∈N). The second method is based on the large time behavior of solutions of the doubly nonlinear evolution Jp(v˙(t))+∂Φ(v(t))∋0(a.e.t>0) and more generally p-curves of maximal slope for Φ. We show that both schemes have the remarkable property that the Rayleigh quotient is nonincreasing along solutions and that properly scaled solutions converge to a minimizer of Φ(u)/‖u‖p. These results are new even for Hilbert spaces and their primary application is in the approximation of optimal constants and extremal functions for inequalities in Sobolev spaces.

Export Date: 22 May 2017; Article; CODEN: JFUAA; Correspondence Address: Lindgren, E.; Department of Mathematics, KTHSweden; email: eriklin@kth.se; Funding details: KVA, Royal Swedish Academy of Sciences; Funding details: 2012-3124, VR, Vetenskapsrådet; Funding text: Supported by the Swedish Research Council, grant no. 2012-3124. Partially supported by the Royal Swedish Academy of Sciences. QC 20170612

Available from: 2017-06-12 Created: 2017-06-12 Last updated: 2017-06-12Bibliographically approvedOpen this publication in new window or tab >>Approximation of the least Rayleigh quotient for degree p homogeneous funetionals### Hynd, Ryan

### Lindgren, Erik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_1_j_idt183_some",{id:"formSmash:j_idt179:1:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_1_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_1_j_idt183_otherAuthors",{id:"formSmash:j_idt179:1:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_1_j_idt183_otherAuthors",multiple:true}); 2017 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 12, p. 4873-4918Article in journal (Refereed) Published
##### Abstract [en]

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

ACADEMIC PRESS INC ELSEVIER SCIENCE, 2017
##### Keyword

Nonlinear eigenvalue problem, Doubly nonlinear evolution, Inverse iteration, Large time behavior
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-207865 (URN)10.1016/j.jfa.2017.02.024 (DOI)000400539700001 ()2-s2.0-85015702520 (Scopus ID)
#####

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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We present two novel methods for approximating minimizers of the abstract Rayleigh quotient Phi(u)/parallel to u parallel to(p). Here Phi is a strictly convex functional on a Banach space with norm parallel to center dot parallel to, and Phi is assumed to be positively homogeneous of degree p is an element of (1,infinity). Minimizers are shown to satisfy partial derivative Phi(u) - lambda j(p)(u) there exists 0 for a certain lambda is an element of R, where J(p) is the subdifferential of 1/p parallel to center dot parallel to(p.) The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy partial derivative Phi(u(k)) - j(p()u(k-1)) there exists 0 (k is an element of N) The second method is based on the large time behavior of solutions of the doubly nonlinear evolution j(p)((v) over circle (t)) + partial derivative Phi(v(t)) there exists 0 (a,e,t > 0) and more generally p -curves of maximal slope for Phi. We show that both schemes have the remarkable property that the Rayleigh quotient is nonincreasing along solutions and that properly scaled solutions converge to a minimizer of Phi(u)/parallel to u parallel to(p). These results are new even for Hilbert spaces and their primary application is in the approximation of optimal constants and extremal functions for inequalities in Sobolev spaces.

QC 20170530

Available from: 2017-05-30 Created: 2017-05-30 Last updated: 2017-05-30Bibliographically approvedOpen this publication in new window or tab >>Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case### Brasco, L.

### Lindgren, Erik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_2_j_idt183_some",{id:"formSmash:j_idt179:2:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_2_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_2_j_idt183_otherAuthors",{id:"formSmash:j_idt179:2:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_2_j_idt183_otherAuthors",multiple:true}); 2017 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 304, p. 300-354Article in journal (Refereed) Published
##### Abstract [en]

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

Elsevier, 2017
##### Keyword

Besov regularity, Fractional p-Laplacian, Nonlocal elliptic equations
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-195119 (URN)10.1016/j.aim.2016.03.039 (DOI)000398757500009 ()2-s2.0-84986900072 (Scopus ID)
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_2_j_idt183_j_idt360",{id:"formSmash:j_idt179:2:j_idt183:j_idt360",widgetVar:"widget_formSmash_j_idt179_2_j_idt183_j_idt360",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_2_j_idt183_j_idt366",{id:"formSmash:j_idt179:2:j_idt183:j_idt366",widgetVar:"widget_formSmash_j_idt179_2_j_idt183_j_idt366",multiple:true});
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We prove that for p≥2, solutions of equations modeled by the fractional p-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in Wloc 1,p and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation s reaches 1. © 2016 Elsevier Inc.

QC 20161121

Available from: 2016-11-21 Created: 2016-11-02 Last updated: 2017-05-30Bibliographically approvedOpen this publication in new window or tab >>Perron's Method and Wiener's Theorem for a Nonlocal Equation### Lindgren, Erik

### Lindqvist, Peter

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_3_j_idt183_some",{id:"formSmash:j_idt179:3:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_3_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_3_j_idt183_otherAuthors",{id:"formSmash:j_idt179:3:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_3_j_idt183_otherAuthors",multiple:true}); 2017 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 46, no 4, p. 705-737Article in journal (Refereed) Published
##### Abstract [en]

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

SPRINGER, 2017
##### Keyword

Fractional p-Laplacian, Non-local equation, Nonlinear equation, Degenerate equation, Singular equation, Nonlinear potential theory, Perron's method, Resolutivity
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-207667 (URN)10.1007/s11118-016-9603-9 (DOI)000399829500004 ()2-s2.0-84994430007 (Scopus ID)
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_3_j_idt183_j_idt354",{id:"formSmash:j_idt179:3:j_idt183:j_idt354",widgetVar:"widget_formSmash_j_idt179_3_j_idt183_j_idt354",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_3_j_idt183_j_idt360",{id:"formSmash:j_idt179:3:j_idt183:j_idt360",widgetVar:"widget_formSmash_j_idt179_3_j_idt183_j_idt360",multiple:true});
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We study the Dirichlet problem for non-homogeneous equations involving the fractional p-Laplacian. We apply Perron's method and prove Wiener's resolutivity theorem.

QC 20170602

Available from: 2017-06-02 Created: 2017-06-02 Last updated: 2017-06-02Bibliographically approvedOpen this publication in new window or tab >>Regularity of the p-Poisson equation in the plane### Lindgren, Erik

### Lindqvist, P.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_4_j_idt183_some",{id:"formSmash:j_idt179:4:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_4_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_4_j_idt183_otherAuthors",{id:"formSmash:j_idt179:4:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_4_j_idt183_otherAuthors",multiple:true}); 2017 (English)In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 132, no 1, p. 217-228Article in journal (Refereed) Published
##### Abstract [en]

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

Springer New York LLC, 2017
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-216478 (URN)10.1007/s11854-017-0019-2 (DOI)000404532200008 ()2-s2.0-85021306620 (Scopus ID)
#####

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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We study the regularity of the p-Poisson equation Δpu=h, h∈Lq, in the plane. In the case p > 2 and 2 < q < ∞, we obtain the sharp Hölder exponent for the gradient. In the other cases, we come arbitrarily close to the sharp exponent.

QC 20171201

Available from: 2017-12-01 Created: 2017-12-01 Last updated: 2017-12-01Bibliographically approvedOpen this publication in new window or tab >>Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution### Hynd, Ryan

### Lindgren, Erik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_5_j_idt183_some",{id:"formSmash:j_idt179:5:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_5_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_5_j_idt183_otherAuthors",{id:"formSmash:j_idt179:5:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_5_j_idt183_otherAuthors",multiple:true}); 2016 (English)In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 9, no 6, p. 1447-1482Article in journal (Refereed) Published
##### Abstract [en]

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

Mathematical Sciences Publishers, 2016
##### Keyword

doubly nonlinear evolution, Holder estimates, eigenvalue problem, fractional p-Laplacian, nonlocal equation
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-200043 (URN)10.2140/apde.2016.9.1447 (DOI)000389831600006 ()2-s2.0-84992724068 (Scopus ID)
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##### Funder

Swedish Research Council, 2012-3124
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

The nonlinear and nonlocal PDE vertical bar nu(t)vertical bar(p-2) nu(t) + (-Delta p)(s)nu = 0, where (-Delta(p))(s) v(x,t) = 2 P.V. integral(Rn) vertical bar nu(x,t) - nu(x+y,t)vertical bar(p-2)nu(x,t)-nu(x,y,t))/vertical bar y vertical bar(n+sp) dy; has the interesting feature that an associated Rayleigh quotient is nonincreasing in time along solutions. We prove the existence of a weak solution of the corresponding initial value problem which is also unique as a viscosity solution. Moreover, we provide Hlder estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional p-Laplacian.

QC 20170126

Available from: 2017-01-26 Created: 2017-01-20 Last updated: 2017-11-29Bibliographically approvedOpen this publication in new window or tab >>Hölder estimates for viscosity solutions of equations of fractional p-Laplace type### Lindgren, Erik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_6_j_idt183_some",{id:"formSmash:j_idt179:6:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_6_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_6_j_idt183_otherAuthors",{id:"formSmash:j_idt179:6:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_6_j_idt183_otherAuthors",multiple:true}); 2016 (English)In: NoDEA. Nonlinear differential equations and applications (Printed ed.), ISSN 1021-9722, E-ISSN 1420-9004, Vol. 23, no 5, article id 55Article in journal (Refereed) Published
##### Abstract [en]

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

Birkhauser Verlag AG, 2016
##### Keyword

35B65, 35D40, 35J60, 35J70, 35R09
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-195276 (URN)10.1007/s00030-016-0406-x (DOI)000385149000005 ()2-s2.0-84988358251 (Scopus ID)
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##### Funder

Swedish Research Council, 2012-3124The Royal Swedish Academy of Sciences
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We prove Hölder estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional p-Laplace equation (Formula presented.), where s∈ (0 , 1) and p> 2 or 1 / (1 - s) < p< 2. Our results also apply for inhomogeneous equations with more general kernels, when p and s are allowed to vary with x, without any regularity assumption on p and s. This complements and extends some of the recently obtained Hölder estimates for weak solutions.

QC 20161111

Available from: 2016-11-11 Created: 2016-11-02 Last updated: 2017-11-29Bibliographically approvedOpen this publication in new window or tab >>INVERSE ITERATION FOR p-GROUND STATES### Hynd, Ryan

### Lindgren, Erik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_7_j_idt183_some",{id:"formSmash:j_idt179:7:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_7_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_7_j_idt183_otherAuthors",{id:"formSmash:j_idt179:7:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_7_j_idt183_otherAuthors",multiple:true}); 2016 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, no 5, p. 2121-2131Article in journal (Refereed) Published
##### Abstract [en]

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

American Mathematical Society (AMS), 2016
##### Keyword

Nonlinear eigenvalue problem, p-Laplacian, inverse iteration, power method
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-184009 (URN)10.1090/proc/12860 (DOI)000370723200026 ()2-s2.0-84958549975 (Scopus ID)
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_7_j_idt183_j_idt354",{id:"formSmash:j_idt179:7:j_idt183:j_idt354",widgetVar:"widget_formSmash_j_idt179_7_j_idt183_j_idt354",multiple:true});
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We adapt the inverse iteration method for symmetric matrices to some nonlinear PDE eigenvalue problems. In particular, for p is an element of (1, infinity) and a given domain Omega subset of R-n, we analyze a scheme that allows us to approximate the smallest value the ratio integral(Omega)vertical bar D psi vertical bar(p)dx/ integral(Omega)vertical bar psi vertical bar(p)dx can assume for functions psi that vanish on partial derivative Omega. The scheme in question also provides a natural way to approximate minimizing psi. Our analysis also extends in the limit as p -> infinity and thereby fashions a new approximation method for ground states of the infinity Laplacian.

QC 20160330

Available from: 2016-03-30 Created: 2016-03-22 Last updated: 2017-11-30Bibliographically approvedOpen this publication in new window or tab >>Optimal regularity for the obstacle problem for the p-Laplacian### Andersson, John

### Lindgren, Erik

### Shahgholian, Henrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_8_j_idt183_some",{id:"formSmash:j_idt179:8:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_8_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_8_j_idt183_otherAuthors",{id:"formSmash:j_idt179:8:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_8_j_idt183_otherAuthors",multiple:true}); 2015 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, no 6, p. 2167-2179Article in journal (Refereed) Published
##### Abstract [en]

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

Elsevier, 2015
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-170288 (URN)10.1016/j.jde.2015.03.019 (DOI)000368466500003 ()2-s2.0-84930540839 (Scopus ID)
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##### Funder

Swedish Research Council, 2012-3124
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In this paper we discuss the obstacle problem for the p-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the p-obstacle problem: Find the smallest u such thatdiv(|∇u|p-2∇u)≤0,u≥ϕ,in B1, with ϕ∈C1,1(B1) and given boundary datum on ∂B1. We prove that the solution is uniformly C1,1 at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to L∞. When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is C1,1p-1 in the spatial variables.

QC 20150630. QC 20160216

Available from: 2015-06-30 Created: 2015-06-29 Last updated: 2017-12-04Bibliographically approvedOpen this publication in new window or tab >>Pointwise regularity of the free boundary for the parabolic obstacle problem### Lindgren, Erik

### Monneau, Regis

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##### Abstract [en]

##### Keyword

VARIABLE-COEFFICIENTS
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-173421 (URN)10.1007/s00526-014-0787-9 (DOI)000359941200012 ()2-s2.0-84939471926 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We study the parabolic obstacle problem Delta u - u(t) = f chi((u>0)), u >= 0, f is an element of L-p with f(0) = 1 and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition). Moreover, under the assumption that f is Dini continuous, we prove that the set of regular points is locally a (parabolic) C-1-surface and that the set of singular points is locally contained in a union of (parabolic) C-1 manifolds.

QC 20150915

Available from: 2015-09-15 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved