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- W3167791832 abstract "Let $$L(t) = - mathrm{div} left( A(x,t) nabla _x right) $$ for $$t in (0, tau )$$ be a uniformly elliptic operator with boundary conditions on a domain $$Omega $$ of $$mathbb {R}^d$$ and $$partial = frac{partial }{partial t}$$ . Define the parabolic operator $${{mathcal {L}}}= partial + L$$ on $$L^2(0, tau , L^2(Omega ))$$ by $$({{mathcal {L}}}u)(t) := frac{partial u(t)}{partial t} + L(t)u(t)$$ . We assume a very little of regularity for the boundary of $$Omega $$ and we assume that the coefficients A(x, t) are measurable in x and piecewise $$C^alpha $$ in t (uniformly in $$x in Omega $$ ) for some $$alpha > frac{1}{2}$$ . We prove the Kato square root property for $$sqrt{{{mathcal {L}}}}$$ and the estimate $$begin{aligned}&Vert sqrt{{{mathcal {L}}}}, u Vert _{L^2(0,tau , L^2(Omega ))} approx Vert nabla _x u Vert _{L^2(0,tau , L^2(Omega ))} + Vert u Vert _{H^{frac{1}{2}}(0,tau , L^2(Omega ))}&qquad + left( int _0^tau Vert u(t) Vert _{L^2(Omega )}^2, frac{dt}{t} right) ^{1/2}. end{aligned}$$ We also prove $$L^p$$ -versions of this result." @default.
- W3167791832 created "2021-06-22" @default.
- W3167791832 creator A5087792269 @default.
- W3167791832 date "2021-06-01" @default.
- W3167791832 modified "2023-10-13" @default.
- W3167791832 title "The Square Root of a Parabolic Operator" @default.
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- W3167791832 doi "https://doi.org/10.1007/s00041-021-09863-w" @default.
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