Form-generated sectorial elliptic semigroups

The form-generated semigroup. By [L1], applied to the given closed sectorial form with constants $M,\theta$, the associated operator $A$ is closed and densely defined, is sectorial with vertex $M$ and every exponent $\delta <\pi \mathrm{/}2-\theta$, and $B=A-MI$ generates a bounded analytic semigroup $S$ on every ${\mathrm{\Sigma }}_{\delta }$ with $\delta <\pi \mathrm{/}2-\theta$; moreover $B$ is sectorial with vertex $0$ and the same exponents, because $\lambda I-B=(\lambda +M)I-A$ for every $\lambda$, so $\lambda \in \rho (B)$ exactly when $\lambda +M\in \rho (A)$ with $R(\lambda ,B)=R(\lambda +M,A)$ and the defining bound $\mathrm{\parallel }R(\lambda +M,A)\mathrm{\parallel }\le K_{\epsilon }\mathrm{/}\mathrm{\mid }\lambda +M-M\mathrm{\mid }$ becomes $\mathrm{\parallel }R(\lambda ,B)\mathrm{\parallel }\le K_{\epsilon }\mathrm{/}\mathrm{\mid }\lambda \mathrm{\mid }$; finally $T(z)=e^{Mz}S(z)$ is the analytic semigroup generated by $A$ with $\mathrm{\parallel }T(z)\mathrm{\parallel }\le C_{{\delta }^{\mathrm{\prime }}}{e}^{M\mathrm{Re}z}$ on every smaller sector ${\mathrm{\Sigma }}_{{\delta }^{\mathrm{\prime }}}$, ${\delta }^{\mathrm{\prime }}<\delta$, by [L1] and the definition of the analytic-semigroup angle in [L3].

The coercive case. If $\mathrm{Re}a(u,u)\ge \alpha \mathrm{\parallel }u{\mathrm{\parallel }}_V^2$ with $\alpha >0$, then [L6] gives $\mathrm{\parallel }T(t)\mathrm{\parallel }\le e^{-\alpha t\mathrm{/}{\kappa }^2}$ for every $t\ge 0$ using the chosen positive embedding bound. If $H=\{0\}$, density forces $V=\{0\}$ and the semigroup has norm $0$, so the same estimate holds directly.

The angle caveat. The assertion is that for every $\delta <\pi \mathrm{/}2-\theta$ the shifted family $S$ is analytic and bounded on the sector ${\mathrm{\Sigma }}_{\delta }$, and on each strictly smaller ${\mathrm{\Sigma }}_{{\delta }^{\mathrm{\prime }}}$ the bound carries the factor $e^{M\mathrm{Re}z}$; the definition [L3] asserts no family on ${\mathrm{\Sigma }}_{\pi \mathrm{/}2-\theta }$ itself and makes no maximality claim, and when $M>0$ the factor $e^{M\mathrm{Re}z}$ is unbounded on any sector, so the unshifted semigroup $T$ is not asserted to be bounded; only the shifted semigroup $S$ is bounded, and no symmetry of the form is assumed, so none of the stronger conclusions of [L4] applies to a general $a$.

The Dirichlet specialisation. For a nonempty open $\mathrm{\Omega }$, the principal form $a_0(u,v)={\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \overline{\mathrm{\nabla }v}dx$ is a closed sectorial form on $V=H_0^1(\mathrm{\Omega })\subseteq H=L^2(\mathrm{\Omega })$ with $M=0$ and $\theta =0$: it is bounded on $V$ by Cauchy-Schwarz, its real part is $\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}^2\ge 0$ with vanishing imaginary part, and the shifted form norm is the complete $H^1$ norm; the associated operator of [L2] is exactly the operator $A$ of [L4], since both are defined by $a_0(u,v)=-(Au,v{)}_{L^2}$; the abstract construction of [step 1.1] therefore produces a bounded analytic semigroup generated by this $A$, and [L4] identifies $A={\mathrm{\Delta }}_D$ and exhibits the contraction analytic heat semigroup generated by it; by [L5] the semigroup constructed here and the heat semigroup of [L4] are the same exponentially bounded semigroup with generator $A$, so the abstract heat flow is recovered, and no elliptic regularity or domain identification beyond [L4] is used.