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

Poggio, Andrea (author), Carbonaro, Andrea Bruno (mentor), Dragičević, Oliver (mentor)

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 Dragićević - 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 Dragićević 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.

• Language: English
• Keywords: Elliptic operators, first-order perturbations, subcritical potentials, bilinear embedding, Bellman function
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Year: 2026
• PID: 20.500.12556/RUL-181139
• UDC: 517.9
• COBISS.SI-ID: 274381827
• Publication date in RUL: 26.03.2026

Secondary language

• Language: Slovenian
• Title: Bilinearna vložitev za kompleksne parcialne diferencialne enačbe s perturbacijami prvega reda

Abstract

V doktorski disertaciji nadaljujemo z razvojem razvojem $L^p$-teorije za parcialne diferencialne operatorje v divergenčni obliki s kompleksnimi koeficienti na poljubnih odprtih podmnožicah ${\mathbb R}^d$. Razširimo bilinearni vložitveni izrek Carbonara in Dragićevića - ki je bil sprva dokazan za homogene operatorje drugega reda v divergenčni obliki, pod tako imenovanim pogojem $p$-eliptičnosti na koeficiente - na operatorje, ki vključujejo bodisi perturbacije prvega reda bodisi negativne potenciale. Za obravnavo teh splošnejših situacij uvedemo dva nova strukturna pogoja na koeficiente, ki na ustrezen način posplošita $p$-eliptičnost. Naš pristop temelji na metodi toplotnega toka za posebej zasnovano Bellmanovo funkcijo; to je naše glavno analitično orodje za dokaz bilinearne vložitve. Novi koncepti generalizirane konveksnosti, ki igrajo vlogo konveksnosti glede na matrike, vpeljane s strani Carbonara in Dragićevića v neperturbiranem primeru, so ključni za uspešno uporabo te metode. Med temeljnimi posledicami izpeljemo omejen $H^\infty$-funkcijski račun ter parabolično maksimalno regularnost v $L^p$-prostorih za omenjene operatorje. Pod novimi pogoji na koeficiente nadalje preučujemo lastnosti pripadajočih polgrup, pri čemer pridobimo rezultate o njihovi kontraktivnosti in analitičnosti v $L^p$-prostorih, ki razširjajo in povezujejo prejšnje ugotovitve. Te lastnosti igrajo temeljno vlogo v sodobni analizi parcialnih diferencialnih enačb in pri preučevanju evolucijskih enačb v Banachovih prostorih. Na koncu podamo nov dokaz bilinearne vložitve v neperturbiranem primeru s pomočjo nove, izvirne aproksimacijske tehnike. Menimo, da bi ta argument lahko predstavljal močno orodje za prihodnje raziskave, zlasti pri preučevanju trilinearnih vložitev za neperturbirane operatorje v divergenčni obliki s kompleksnimi koeficienti ter bilinearnih vložitev za širše razrede operatorjev v divergenčni obliki.

• Keywords: Eliptični operatorji, perturbacije prvega reda, subkritični potenciali, bilinearna vložitev, Bellmanova funkcija.