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Pré-Publication, Document De Travail Année : 2009

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

Moshe Marcus
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Résumé

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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Dates et versions

hal-00402084 , version 1 (06-07-2009)
hal-00402084 , version 2 (07-07-2009)
hal-00402084 , version 3 (16-07-2009)

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Moshe Marcus, Laurent Veron. Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains. 2009. ⟨hal-00402084v3⟩
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