$L^2$ Solvability of boundary value problems for divergence form parabolic equations with complex coefficients

Mardi 19 avril 2016 14:00-15:00 - Kaj Nyström - Université d'Uppsala

Résumé : We consider parabolic operators of the form
$$\partial_t+\mathcalL,\ \mathcalL=-\mboxdiv\, A(X,t)\nabla,$$
in
$\mathbb R_+^n+2 :={(X,t)=(x,x_n+1,t)\in \mathbb R^n\times \mathbb R\times \mathbb R :\ x_n+1>0}$, $n\geq 1$.
We assume that $A$ is a $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_n+1$ as well as of the time coordinate $t$. For such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on $L^2(\mathbb R^n+1,\mathbb C)=L^2(\partial\mathbb R^n+2_+,\mathbb C)$ under complex, $L^\infty$ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for $\partial_t+\mathcalL$, by way of layer potentials and with data in $L^2$, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.

Lieu : Salle 113-115 (Bâtiment 425)

$L^2$ Solvability of boundary value problems for divergence form parabolic equations with complex coefficients  Version PDF