Bilinear embedding for complex partial differential equations with first-order perturbations

In the present dissertation we further develop the L^p-theory for elliptic partial differential operators in divergence form with complex coefficients on arbitrary open subsets of \mathbb{R}^d. We extend the bilinear embedding theorem of Carbonaro and Dragičević -- originally established for homogeneous second-order divergence-form operators, under the so-called p-ellipticity condition imposed on the coefficients -- to operators that include either first-order perturbations or else negative potentials. To handle these more general situations, we introduce two new structural conditions on the coefficients that generalize p-ellipticity in an adequate manner. Our approach is based on a heat-flow method driven by a specifically designed Bellman function, which serves as the main analytic tool for proving the bilinear embedding. New notions of generalized convexity, which play the role of the convexity with respect to matrices introduced by Carbonaro and Dragičević in the unperturbed case, are essential for this method. As key consequences, we establish bounded H^\infty-functional calculus and parabolic maximal regularity in L^p for these operators. Under the new conditions on the coefficients, we further investigate the mapping properties of the associated semigroups, obtaining results on their contractivity and analyticity in L^p spaces that extend and unify previous findings. These properties play a fundamental role in modern analysis of partial differential equations and the study of evolution equations in Banach spaces. Finally, we provide a new proof of the bilinear embedding in the unperturbed case by means of a novel approximation technique. We believe that this argument might prove a powerful tool for future developments, particularly in the study of trilinear embeddings for unperturbed divergence-form operators with complex coefficients, as well as bilinear embeddings for more general classes of divergence-form operators.