Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains

Details Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains

2009-07-06

arxiv.org/abs/0907.1006v3

Mathematics

Analysis of PDEs

120 pages

Scientific paper

We study the generalized boundary value problem for nonnegative solutions of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Gw$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Gw$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $\Gw$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Gw$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace. We obtain sharp results involving Besov spaces with negative index on k-dimensional edges and apply our results to the characterization of removable sets and good measures on the boundary of a polyhedron.

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