EL

Erik Lindgren

  • Analysis of PDEs
  • ArticleHigher Sobolev regularity for the fractional $p$-Laplace equation in the superquadratic case


    Advances in Mathematics 304, 300-354, 2017

    Posted by: Erik Lindgren

    DOIMSC 2010: 35R11 35B65 35J70 35R09

  • ArticlePerron's method and {W}iener's theorem for a nonlocal equation


    Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis 46 (4), 705-737, 2017

    Posted by: Erik Lindgren

    DOIMSC 2010: 35R11 31B35 35J60 35J70 35R09

  • ArticleRegularity of the $p$-Poisson equation in the plane


    Journal d'Analyse Mathématique 132, 217-228, 2017

    Posted by: Erik Lindgren

    DOIMSC 2010: 35J92 35B65

  • ArticleApproximation of the least Rayleigh quotient for degree p homogeneous functionals


    Journal of Functional Analysis 272 (12), 2017

    • Nonlinear eigenvalue problem
    • doubly nonlinear evolution
    • inverse iteration
    • large time behavior

    Posted by: Erik Lindgren

    arXivfulltext

    We present two novel methods for approximating minimizers of the abstract Rayleigh quotient $\Phi(u)/ \|u\|^p$. Here $\Phi$ is a strictly convex functional on a Banach space with norm $\|\cdot\|$, and $\Phi$ is assumed to be positively homogeneous of degree $p\in (1,\infty)$. Minimizers are shown to satisfy $\partial \Phi(u)- \lambda\mathcal{J}_p(u)\ni 0$ for a certain $\lambda\in \mathbb{R}$, where $\mathcal{J}_p$ is the subdifferential of $\frac{1}{p}\|\cdot\|^p$. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy $$ \partial \Phi(u_k)- \mathcal{J}_p(u_{k-1})\ni 0 \quad (k\in \mathbb{N}). $$ The second method is based on the large time behavior of solutions of the doubly nonlinear evolution $$ \mathcal{J}_p(\dot v(t))+\partial\Phi(v(t))\ni 0 \quad(a.e.\;t>0) $$ 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 $\Phi(u)/ \|u\|^p$. 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.

  • ArticleInverse iteration for $p$-ground states


    Proceedings of the American Mathematical Society 144 (5), 2121-2131, 2016

    Posted by: Erik Lindgren

    DOIMSC 2010: 35J62 35J20 35J70 35J92 35P30

  • ArticleA doubly nonlinear evolution for the optimal {P}oincaré inequality


    Calculus of Variations and Partial Differential Equations 55 (4), Art. 100, 22, 2016

    Posted by: Erik Lindgren

    DOIMSC 2010: 35K55 35K15 35P30 39B62 47J10

  • ArticleHölder estimates for viscosity solutions of equations of fractional $p$-Laplace type


    NoDEA : Nonlinear Differential Equations and Applications 23 (5), Art. 55, 18, 2016

    Posted by: Erik Lindgren

    DOIMSC 2010: 35R11 35B65 35D40 35J70 35J92 35R09

  • ArticleHölder estimates and large time behavior for a nonlocal doubly nonlinear evolution


    Analysis & PDE 9 (6), 1447-1482, 2016

    Posted by: Erik Lindgren

    DOIMSC 2010: 35R11 35A15 35B65 35J60 35J70 35R09 47J35

  • ArticleThe two-phase fractional obstacle problem


    SIAM Journal on Mathematical Analysis 47 (3), 1879-1905, 2015

    Posted by: Erik Lindgren

    DOIMSC 2010: 35R35 35B65 35J20 35J70

  • ArticleOptimal regularity for the obstacle problem for the $p$-Laplacian


    Journal of Differential Equations 259 (6), 2167-2179, 2015

    Posted by: Erik Lindgren

    DOIMSC 2010: 35J87 35B65 35K86

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