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Zeitschrift für Analysis und ihre Anwendungen

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Volume 15, Issue 2, 1996, pp. 419–444
DOI: 10.4171/ZAA/708

Published online: 1996-06-30

Boundary-Blow-Up Problems in a Fractal Domain

J. Matero[1]

(1) Uppsala Universitet, Sweden

Assume that $\Omega$ is a bounded domain in $\mathbb R^N$ with $N ≥ 2$, 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 $\Delta u(x) = f(u(x)) (x \in \Omega)$ with boundary blow-up, where $f(t) = e^t$ or $f(t) = t^p$ with $p \in (1,+\infty)$. The boundary estimates imply existence and uniqueness of a solution of the above problem. For $f(t) = t^p$ with $p \in (1,+\infty)$ the solution is positive. These results are used to construct a solution of the problem when $\Omega \subset \mathbb R^2$ is the von Koch snowflake domain.

Keywords: Blow-up, snowflake, fractaLe, gradient estimates

Matero J.: Boundary-Blow-Up Problems in a Fractal Domain. Z. Anal. Anwend. 15 (1996), 419-444. doi: 10.4171/ZAA/708