Boundary-Blow-Up Problems in a Fractal Domain

  • J. Matero

    Uppsala Universitet, Sweden


Assume that Ω\Omega is a bounded domain in RN\mathbb R^N with N2N ≥ 2, which satisfies a uniform interior and exterior cone condition. We determine uniform a priori lower and upper bounds for the growth of solutions and their gradients, of the problem Δu(x)=f(u(x))(xΩ)\Delta u(x) = f(u(x)) (x \in \Omega) with boundary blow-up, where f(t)=etf(t) = e^t or f(t)=tpf(t) = t^p with p(1,+)p \in (1,+\infty). The boundary estimates imply existence and uniqueness of a solution of the above problem. For f(t)=tpf(t) = t^p with p(1,+)p \in (1,+\infty) the solution is positive. These results are used to construct a solution of the problem when ΩR2\Omega \subset \mathbb R^2 is the von Koch snowflake domain.

Cite this article

J. Matero, Boundary-Blow-Up Problems in a Fractal Domain. Z. Anal. Anwend. 15 (1996), no. 2, pp. 419–444

DOI 10.4171/ZAA/708