$L^p$ and endpoint solvability results for divergence form elliptic equations with complex $L^\infty$ coefficients

Lundi 28 janvier 2013 14:00-15:00 - Mihalis Mourgoglou - Université Paris-Sud

Résumé : We consider divergence form elliptic equations Lu :=\nabla\cdot(A\nabla u)=0 in the half space \mathbb{R}^{n+1}_+ :=\{(x,t)\in\mathbb{R}^n\times(0,\infty)\}, whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x,t)-A(x,0) satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we obtain solvability of the Dirichlet problem for L, with data in L^p(\mathbb{R}^n), 2-\varepsilon< p <\infty, or in \Lambda_\alpha(\mathbb{R}^n) (which is defined to be <span class="caps">BMO</span>(\mathbb{R}^n) when \alpha=0 and C^\alpha(\mathbb{R}^n) when \alpha \in (0,1)), for \alpha<\alpha_0, where \alpha_0 is the De Giorgi-Nash exponent, and solvability of the Neumann and Regularity problems, with data in the spaces L^p(\mathbb{R}^n)/H^p(\mathbb{R}^n) and L^p_1(\mathbb{R}^n)/H^{1,p}(\mathbb{R}^n) respectively, for p \in (\frac{n}{n+\alpha_0},2+\varepsilon), assuming that we have bounded Layer Potentials in L^2(\mathbb{R}^n), and invertible Layer Potentials in \Lambda_\alpha(\mathbb{R}^n) and L^p(\mathbb{R}^n)/H^p(\mathbb{R}^n) for the t-independent operator L_0 := -\nabla\cdot(A(\cdot,0)\nabla).

Lieu : bât. 425 - 113-115

$L^p$ and endpoint solvability results for divergence form elliptic equations with complex $L^\infty$ coefficients  Version PDF