Bilinear embedding for divergence-form operators with negative potentials

Abstract

Let $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^d$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\hat{I}\text{c◯}$ with $L^{\infty }$ coefficients, and $V$ a locally integrable function on $\hat{I}\text{c◯}$ whose negative part is subcritical. We consider the operator $\mathcal{L}=-\mathrm{div}(A\nabla )+V$ with mixed boundary conditions on $\hat{I}\text{c◯}$. We extend the bilinear inequality of Carbonaro and Dragičević [15], originally established for nonnegative potentials, by introducing a novel condition on the coefficients that reduces to standard $p$-ellipticity when $V$ is nonnegative. As a consequence, we show that the solution to the parabolic problem $u^{\prime }(t)+\mathcal{L}u(t)=f(t)$ with $u(0)=0$ has maximal regularity on $L^p(\hat{I}\text{c◯})$, in the same spirit as [13]. Moreover, we study mapping properties of the semigroup generated by $-\mathcal{L}$ under this new condition, thereby extending classical results for the Schrödinger operator $-\hat{I}”+V$ on ${\mathbb{R}}^d$ [8,47].

59 pages