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Hynd, R. & Lindgren, E. (2017). Approximation of the least Rayleigh quotient for degree p homogeneous functionals. Journal of Functional Analysis, 272(12), 4873-4918
Open this publication in new window or tab >>Approximation of the least Rayleigh quotient for degree p homogeneous functionals
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]

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.

Place, publisher, year, edition, pages
Academic Press Inc., 2017
Keywords
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)
Note

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 approved
Hynd, R. & Lindgren, E. (2017). Approximation of the least Rayleigh quotient for degree p homogeneous funetionals. Journal of Functional Analysis, 272(12), 4873-4918
Open this publication in new window or tab >>Approximation of the least Rayleigh quotient for degree p homogeneous funetionals
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]

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.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2017
Keywords
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)
Note

QC 20170530

Available from: 2017-05-30 Created: 2017-05-30 Last updated: 2017-05-30Bibliographically approved
Brasco, L. & Lindgren, E. (2017). Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case. Advances in Mathematics, 304, 300-354
Open this publication in new window or tab >>Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case
2017 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 304, p. 300-354Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
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)
Note

QC 20161121

Available from: 2016-11-21 Created: 2016-11-02 Last updated: 2017-05-30Bibliographically approved
Lindgren, E. & Lindqvist, P. (2017). Perron's Method and Wiener's Theorem for a Nonlocal Equation. Potential Analysis, 46(4), 705-737
Open this publication in new window or tab >>Perron's Method and Wiener's Theorem for a Nonlocal Equation
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]

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.

Place, publisher, year, edition, pages
SPRINGER, 2017
Keywords
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)
Note

QC 20170602

Available from: 2017-06-02 Created: 2017-06-02 Last updated: 2017-06-02Bibliographically approved
Lindgren, E. & Lindqvist, P. (2017). Regularity of the p-Poisson equation in the plane. Journal d'Analyse Mathematique, 132(1), 217-228
Open this publication in new window or tab >>Regularity of the p-Poisson equation in the plane
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]

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.

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)
Note

QC 20171201

Available from: 2017-12-01 Created: 2017-12-01 Last updated: 2017-12-01Bibliographically approved
Hynd, R. & Lindgren, E. (2016). Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution. Analysis & PDE, 9(6), 1447-1482
Open this publication in new window or tab >>Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution
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]

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.

Place, publisher, year, edition, pages
Mathematical Sciences Publishers, 2016
Keywords
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)
Funder
Swedish Research Council, 2012-3124
Note

QC 20170126

Available from: 2017-01-26 Created: 2017-01-20 Last updated: 2017-11-29Bibliographically approved
Lindgren, E. (2016). Hölder estimates for viscosity solutions of equations of fractional p-Laplace type. NoDEA. Nonlinear differential equations and applications (Printed ed.), 23(5), Article ID 55.
Open this publication in new window or tab >>Hölder estimates for viscosity solutions of equations of fractional p-Laplace type
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]

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.

Place, publisher, year, edition, pages
Birkhauser Verlag AG, 2016
Keywords
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)
Funder
Swedish Research Council, 2012-3124The Royal Swedish Academy of Sciences
Note

QC 20161111

Available from: 2016-11-11 Created: 2016-11-02 Last updated: 2017-11-29Bibliographically approved
Hynd, R. & Lindgren, E. (2016). INVERSE ITERATION FOR p-GROUND STATES. Proceedings of the American Mathematical Society, 144(5), 2121-2131
Open this publication in new window or tab >>INVERSE ITERATION FOR p-GROUND STATES
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]

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.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2016
Keywords
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)
Note

QC 20160330

Available from: 2016-03-30 Created: 2016-03-22 Last updated: 2017-11-30Bibliographically approved
Andersson, J., Lindgren, E. & Shahgholian, H. (2015). Optimal regularity for the obstacle problem for the p-Laplacian. Journal of Differential Equations, 259(6), 2167-2179
Open this publication in new window or tab >>Optimal regularity for the obstacle problem for the p-Laplacian
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]

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.

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)
Funder
Swedish Research Council, 2012-3124
Note

QC 20150630. QC 20160216

Available from: 2015-06-30 Created: 2015-06-29 Last updated: 2017-12-04Bibliographically approved
Lindgren, E. & Monneau, R. (2015). Pointwise regularity of the free boundary for the parabolic obstacle problem. Calculus of Variations and Partial Differential Equations, 54(1), 299-347
Open this publication in new window or tab >>Pointwise regularity of the free boundary for the parabolic obstacle problem
2015 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 54, no 1, p. 299-347Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
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)
Note

QC 20150915

Available from: 2015-09-15 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-4309-9242

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