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Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains
Moshe Marcus1, Laurent Veron2

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.
1:  TECHNION - Department of Mathematics
2:  LMPT - Laboratoire de Mathématiques et Physique Théorique
Harmonic measure – Lischitz domains – Harnack inequality – singular integrals – Besov spaces – Interpolation – polyedron – Borel measures – Bessel capacities – Removable sets