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  • 1.
    Aronsson, Gunnar
    Luleå tekniska universitet.
    Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞1986In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 56, no 2, p. 135-158Article in journal (Refereed)
    Abstract [en]

    Here, all solutions of the form u=rkf(φ) to the p-harmonic equation, div(|∇u|p-2∇u)=0, (p>2) in the plane are determined. One main result is a representation formula for such solutions. Further, solutions with an isolated singularity at the origin are constructed (Theorem 1). Graphical illustrations are given at the end of the paper. Finally, all solutions u=rkf(φ) of the limit equation for p=∞, ux2uxx+2uxuyuxy+uy2uyy=2, are constructed, some of which have a "strong" singularity at the origin (Theorem 2).

  • 2.
    Aronsson, Gunnar
    Luleå tekniska universitet.
    On certain singular solutions of the partial differential equation ux2uxx+2uxuyuxy+uy2uyy=01984In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 47, no 1-3, p. 133-151Article in journal (Refereed)
    Abstract [en]

    The partial differential equation treated here is the formal limit of the p-harmonic equation in R2, for p →∞. Questions related to the smoothness of solutions and the possible existence of stationary points are central for the theory. The singular solutions constructed here bring new light on these questions.

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