Change search
Refine search result
1 - 21 of 21
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Rows per page
  • 5
  • 10
  • 20
  • 50
  • 100
  • 250
Sort
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
Select
The maximal number of hits you can export is 250. When you want to export more records please use the Create feeds function.
  • 1. Allen, Mark
    et al.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Petrosyan, Arshak
    THE TWO-PHASE FRACTIONAL OBSTACLE PROBLEM2015In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 47, no 3, p. 1879-1905Article in journal (Refereed)
    Abstract [en]

    We study minimizers of the functional integral(+)(B1) vertical bar del u vertical bar(2)x(n)(a) dx + 2 integral(')(B1)(lambda + u(+) + lambda-u(-)) dx' for a is an element of (- 1, 1). The problem arises in connection with heat flow with control on the boundary. It can also be seen as a nonlocal analogue of the, by now well studied, two-phase obstacle problem. Moreover, when u does not change signs this is equivalent to the fractional obstacle problem. Our main results are the optimal regularity of the minimizer and the separation of the two free boundaries Gamma(+) = partial derivative'{u(center dot, 0) > 0} and Gamma(-) = partial derivative' {u(center dot, 0) < 0} when a >= 0.

  • 2.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal regularity for the obstacle problem for the p-Laplacian2015In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, no 6, p. 2167-2179Article in journal (Refereed)
    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.

  • 3.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal regularity for the parabolic no-sign obstacle type problem2013In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 15, no 4, p. 477-499Article in journal (Refereed)
    Abstract [en]

    We study the parabolic free boundary problem of obstacle type Delta u - partial derivative u/partial derivative t = f chi({u not equal 0}). Under the condition that f = H nu for some function nu with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution u. Both the regularity and the assumptions are optimal. Using this result and assuming that f is Dini continuous, we prove that the free boundary is, near so called low energy points, a C-1 graph. Our result completes the theory for this type of problems for the heat operator.

  • 4. Brasco, L.
    et al.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 304, p. 300-354Article in journal (Refereed)
    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.

  • 5. Brasco, L.
    et al.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Parini, E.
    The fractional Cheeger problem2014In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 16, no 3, p. 419-458Article in journal (Refereed)
    Abstract [en]

    Given an open and bounded set Omega subset of R-N, we consider the problem of minimizing the ratio between the s-perimeter and the N-dimensional Lebesgue measure among subsets of Omega. This is the non-local version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and non-local eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.

  • 6.
    Edquist, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A two-phase obstacle-type problem for the p-Laplacian2009In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 35, no 4, p. 421-433Article in journal (Refereed)
    Abstract [en]

    We study the so-called two-phase obstacle-type problem for the p-Laplacian when p is close to 2. We introduce a new method to obtain the optimal growth of the function from branch points, i.e. two-phase points in the free boundary where the gradient vanishes. As a by-product we can locally estimate the (n - 1)-Hausdorff-measure of the free boundary for the special case when p > 2.

  • 7.
    Edquist, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Regularity of a parabolic free boundary problem with Hölder continuous coefficientsManuscript (preprint) (Other academic)
  • 8. Hynd, Ryan
    et al.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A doubly nonlinear evolution for the optimal Poincare inequality2016In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 4, article id 100Article in journal (Refereed)
    Abstract [en]

    We study the large time behavior of solutions of the PDE vertical bar v(t)vertical bar(p-2)v(t) = Delta(p)v. A special property of this equation is that the Rayleigh quotient integral(Omega) vertical bar Dv(x,t)vertical bar(p) dx/integral(Omega) vertical bar v(x,t)vertical bar(p) dx is nonincreasing in time along solutions. As t tends to infinity, this ratio converges to the optimal constant in Poincare's inequality. Moreover, appropriately scaled solutions converge to a function for which equality holds in this inequality. An interesting limiting equation also arises when p tends to infinity, which provides a new approach to approximating ground states of the infinity Laplacian.

  • 9. Hynd, Ryan
    et al.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Approximation of the least Rayleigh quotient for degree p homogeneous funetionals2017In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 12, p. 4873-4918Article in journal (Refereed)
    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.

  • 10. Hynd, Ryan
    et al.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution2016In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 9, no 6, p. 1447-1482Article in journal (Refereed)
    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.

  • 11. Hynd, Ryan
    et al.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    INVERSE ITERATION FOR p-GROUND STATES2016In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, no 5, p. 2121-2131Article in journal (Refereed)
    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.

  • 12.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hölder estimates for viscosity solutions of equations of fractional p-Laplace type2016In: 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)
    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.

  • 13.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On the penalized obstacle problem in the unit half ball2010In: Electronic Journal of Differential Equations, ISSN 1550-6150, E-ISSN 1072-6691, no 9, p. 1-12Article in journal (Refereed)
    Abstract [en]

    We study the penalized obstacle problem in the unit half ball,i.e. an approximation of the obstacle problem in the unit half ball. The mainresult states that when the approximation parameter is small enough and whencertain level sets are sufficiently close to the hyperplane {x1 = 0}, then theselevel sets are uniformly C1 regular graphs. As a by-product, we also recoversome regularity of the free boundary for the limiting problem, i.e., for theobstacle problem.

  • 14.
    Lindgren, Erik
    Department of Mathematical Sciences, Norwegian University of Science and Technology, Alfred Getz vei 1, NO-7491 Trondheim, Norway .
    On the regularity of solutions of the inhomogeneous infinity laplace equation2014In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 142, no 1, p. 277-288Article in journal (Refereed)
    Abstract [en]

    We study the inhomogeneous infinity Laplace equation and prove that for bounded and continuous inhomogeneities, any blow-up is linear but not necessarily unique. If, in addition, the inhomogeneity is assumed to be C-1, then we prove that any solution is differentiable, i.e., that any blow-up is unique.

  • 15.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Regularity properties of two-phase free boundary problems2009Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four papers which are all related to the regularity properties of free boundary problems. The problems considered have in common that they have some sort of two-phase behaviour.In papers I-III we study the interior regularity of different two-phase free boundary problems. Paper I is mainly concerned with the regularity properties of the free boundary, while in papers II and III we devote our study to the regularity of the function, but as a by-product we obtain some partial regularity of the free boundary.The problem considered in paper IV has a somewhat different nature. Here we are interested in certain approximations of the obstacle problem. Two major differences are that we study regularity properties close to the fixed boundary and that the problem converges to a one-phase free boundary problem.

  • 16.
    Lindgren, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lindqvist, P.
    Regularity of the p-Poisson equation in the plane2017In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 132, no 1, p. 217-228Article in journal (Refereed)
    Abstract [en]

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

  • 17.
    Lindgren, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lindqvist, Peter
    Perron's Method and Wiener's Theorem for a Nonlocal Equation2017In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 46, no 4, p. 705-737Article in journal (Refereed)
    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.

  • 18.
    Lindgren, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Monneau, Regis
    Pointwise regularity of the free boundary for the parabolic obstacle problem2015In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 54, no 1, p. 299-347Article in journal (Refereed)
    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.

  • 19.
    Lindgren, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Petrosyan, Arshak
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Regularity of the free boundary in a two-phase semilinear problem in two dimensions2008In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 57, no 7, p. 3397-3418Article in journal (Refereed)
    Abstract [en]

    We study minimizers of the energy functional ∫D (|∇u|2 + 2(λ+(u+)p + λ-(u-)p)) dx for p ∈ (0, 1) without any sign restriction on the function u. The main result states that in dimension two the free boundaries Γ+ = ∂{u > 0} ∩ D and Γ- = ∂{u < 0} n D are C1 regular.

  • 20.
    Lindgren, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Privat, Yannick
    A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition2007In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 67, no 8, p. 2497-2505Article in journal (Refereed)
    Abstract [en]

    We study a free boundary problem for the Laplace operator, where we impose a Bernoulli-type boundary condition. We show that there exists a solution to this problem. We use A. Beurling's technique, by defining two classes of sub- and super-solutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian. We extend these results in different directions.

  • 21.
    Lindgren, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Edquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the two-phase membrane problem with coefficients below the Lipschitz threshold2009In: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430, Vol. 26, no 6, p. 2359-2372Article in journal (Refereed)
    Abstract [en]

    We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C-1,C-1-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C-1-graphs. In our case, the same monotonicity formula does not apply in the same way. In the absence of a monotonicity formula, we use a specific scaling argument combined with the classification of certain global solutions to obtain C-1,C-1-estimates. Then we exploit some stability properties with respect to the coefficients to prove that the free boundary is the union of two Reifenberg vanishing sets near so-called branching points.

1 - 21 of 21
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf