Bilinear embedding for divergence-form operators with complex coefficients on irregular domains
Carbonaro, Andrea ; Dragičević, Oliver
Naj bo ▫$\Omega\subseteq \mathbb{R}^{d}$▫ odprta množica in ▫$A$▫ kompleksna, enakomerno strogo akretivna ▫$d\times d$▫ matrična funkcija na ▫$\Omega$▫ z ▫$L^{\infty}$▫ koeficienti. Imejmo operator v ...divergenčni formi ▫${\mathscr L}^{A}=-{\rm div}(A\nabla)$▫ z mešanimi robnimi pogoji na ▫$\Omega$▫. V članku razširimo bilinearno neenakost, ki smo jo v [17: Carbonaro, A., Dragičević, O.: Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients. J. Eur. Math. Soc. (sprejeto v objavo)] dokazali v posebnem primeru, ko je ▫$\Omega=\mathbb{R}^{d}$▫. Kot posledico dobimo, da ima rešitev paraboličnega problema ▫$u^{\prime}(t)+{\mathscr L}^{A}u(t)=f(t)$▫, ▫$u(0)=0$▫, maksimalno regularnost v ▫$L^{p}(\Omega)$▫ za vse ▫$p>1$▫, za katere ▫$A$▫ zadošča pogoju ▫$p$▫-eliptičnosti, ki sva ga vpeljala v [17]. Ta razpon eksponentov je za razred operatorjev, ki ga obravnavamo, optimalen. Na ▫$\Omega$▫ ne privzemamo nobenih pogojev, v posebnem, ne privzemamo regularnosti ▫$\partial\Omega$▫ niti obstoja vložitev Soboljeva. Metode iz [17] ne moremo direktno uporabiti na našem primeru, zato potrebujemo drugačen pristop.
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