The analytic Dirichlet heat semigroup

Diagonal action and domain characterisation. Since $a(e_j,v)={\lambda }_j(e_j,v{)}_{L^2}$ for all $v\in H_0^1(\mathrm{\Omega })$, the defining identity gives $e_j\in D(A)$ with $A{e}_j=-{\lambda }_j{e}_j$; for $u\in L^2(\mathrm{\Omega })$ with coefficients $c_j=(u,e_j{)}_{L^2}$ one has $u\in D(A)$ if and only if ${\sum }_j{\lambda }_j^2\mathrm{\mid }{c}_j{\mathrm{\mid }}^2<\mathrm{\infty }$, and then $Au=-{\sum }_j{\lambda }_j{c}_j{e}_j$: if $u\in D(A)$ then $(Au,e_j{)}_{L^2}=-a(u,e_j)=-{\lambda }_j(u,e_j{)}_{L^2}$ by symmetry and [L1] gives ${\sum }_j{\lambda }_j^2\mathrm{\mid }{c}_j{\mathrm{\mid }}^2=\mathrm{\parallel }Au{\mathrm{\parallel }}_{L^2}^2<\mathrm{\infty }$ together with $Au=-{\sum }_j{\lambda }_j{c}_j{e}_j$, while conversely ${\sum }_j{\lambda }_j^2\mathrm{\mid }{c}_j{\mathrm{\mid }}^2<\mathrm{\infty }$ makes $u_N:={\sum }_{j\le N}{c}_j{e}_j$ Cauchy in the form norm by [L1], hence convergent in $H_0^1(\mathrm{\Omega })$ to a class equal to $u$ in $L^2(\mathrm{\Omega })$, and for $v\in H_0^1(\mathrm{\Omega })$ continuity of $a$ in the $H^1$ norm gives $a(u,v)={\lim }_{N}a(u_N,v)={\sum }_j{\lambda }_j{c}_j\overline{(v,e_j{)}_{L^2}}$, the scalar series converging absolutely by Cauchy-Schwarz and [L1], so $u\in D(A)$ with $Au=-{\sum }_j{\lambda }_j{c}_j{e}_j$.

The spectral series defines the semigroup. For $t>0$ put $S(t)f:={\sum }_j{e}^{-{\lambda }_{j}t}{c}_j{e}_j$ and $S(0)f:=f$: the series converges in $L^2(\mathrm{\Omega })$ because the weights $e^{-{\lambda }_{j}t}$ are bounded in $j$ and ${\sum }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2=\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}^2$ by [L1], the family is linear with $\mathrm{\parallel }S(t)\mathrm{\parallel }\le {\sup }_j{e}^{-{\lambda }_{j}t}\le e^{-{\lambda }_{1}t}$ when ${\lambda }_1>0$ by [L7], $S(t)S(s)=S(t+s)$ is coefficientwise, and $\mathrm{\parallel }S(t)f-f{\mathrm{\parallel }}_{L^2}^2={\sum }_j(e^{-{\lambda }_{j}t}-1{)}^2\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\to 0$ as $t\downarrow 0$ by dominated convergence; each $S(t)f$ lies in $D(A)$ with $AS(t)f=-{\sum }_j{\lambda }_j{e}^{-{\lambda }_{j}t}{c}_j{e}_j$ by the criterion of [step 1.1], since ${\sum }_j{\lambda }_j^2{e}^{-2{\lambda }_{j}t}\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\le {\sup }_j\left({\lambda }_j^2{e}^{-2{\lambda }_{j}t}\right)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}^2<\mathrm{\infty }$.

Powers and smoothing constants. For $m\ge 1$ the criterion of [step 1.1] applied inductively along the diagonal action gives $A^{m}S(t)f={\sum }_j(-{\lambda }_j{)}^m{e}^{-{\lambda }_{j}t}{c}_j{e}_j$ with $S(t)f\in D(A^m)$, and ${\lambda }_j\ge {\lambda }_1>0$ by [L7] gives $\mathrm{\parallel }{A}^{m}S(t)\mathrm{\parallel }\le {\sup }_{s\ge 0}{s}^m{e}^{-st}=(m\mathrm{/}(et){)}^m$, the supremum being at $s=m\mathrm{/}t$ by one-variable calculus; the same identities hold for $T$ once $T=S$ is established.

The generator is $A$, so $S=T$. For $\lambda >0$ and $g\in L^2(\mathrm{\Omega })$ with coefficients $g_j$, the series $u:={\sum }_j{g}_j(\lambda +{\lambda }_j{)}^{-1}{e}_j$ satisfies ${\sum }_j{\lambda }_j^2\mathrm{\mid }{u}_j{\mathrm{\mid }}^2\le {\sum }_j\mathrm{\mid }{g}_j{\mathrm{\mid }}^2<\mathrm{\infty }$, so $u\in D(A)$ with $(\lambda I-A)u={\sum }_j(\lambda +{\lambda }_j)u_j{e}_j=g$ by [step 1.1], while $(\lambda I-A)u=0$ forces $(\lambda +{\lambda }_j)(u,e_j{)}_{L^2}=0$ for all $j$ and hence $u=0$; therefore $R(\lambda ,A)g={\sum }_j(\lambda +{\lambda }_j{)}^{-1}{g}_j{e}_j$ with $\mathrm{\parallel }R(\lambda ,A)\mathrm{\parallel }\le 1\mathrm{/}(\lambda +{\lambda }_1)\le 1\mathrm{/}\lambda$. To compare with the Laplace transform of $S$, for $N\ge 1$ put $S_N(t)g:={\sum }_{j\le N}{e}^{-{\lambda }_{j}t}{g}_j{e}_j$. Parseval and ${\lambda }_j\ge 0$ give $\mathrm{\parallel }{S}_N(t)g{\mathrm{\parallel }}_{L^2}^2={\sum }_{j\le N}{e}^{-2{\lambda }_{j}t}\mathrm{\mid }{g}_j{\mathrm{\mid }}^2\le \mathrm{\parallel }g{\mathrm{\parallel }}_{L^2}^2$; by [L1], $S_N(t)g\to S(t)g$ in $L^2$ for every $t\ge 0$. Thus the functions $t\mapsto e^{-\lambda t}{S}_N(t)g$ converge pointwise in norm and are dominated by the integrable scalar function $t\mapsto e^{-\lambda t}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2}$, so [L10] gives ${\int }_0^{\mathrm{\infty }}{e}^{-\lambda t}S(t)gdt={\lim }_{N\to \mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{e}^{-\lambda t}{S}_N(t)gdt={\lim }_{N\to \mathrm{\infty }}{\sum }_{j\le N}(\lambda +{\lambda }_j{)}^{-1}{g}_j{e}_j=R(\lambda ,A)g,$ where the last limit holds in $L^2$ since ${\sum }_j\mathrm{\mid }{g}_j{\mathrm{\mid }}^2\mathrm{/}(\lambda +{\lambda }_j{)}^2<\mathrm{\infty }$. By [L4] the left side is $R(\lambda ,G_S)g$, where $G_S$ is the generator of $S$. Since $G_S$ is closed as a generator by [L5] and $A$ is closed as the generator of $T$ by [L3, L5], their common everywhere-defined inverse $R(\lambda ,A)=R(\lambda ,G_S)$ gives $D(G_S)=R(\lambda ,A)L^2=D(A)$ and $G_S=A$; finally [L6] supplies the unique exponentially bounded semigroup with generator $A$, and both $S$ and the heat semigroup $T$ of [L3] are exponentially bounded strongly continuous semigroups with generator $A$, so $S=T$ and the spectral series represents the heat semigroup.

Abstract smoothing and the spatial reading. By [L6] the semigroup $T$ generated by the sectorial operator $A$ of [L3] satisfies $T(t)L^2\subseteq D(A^m)$ for every $t>0$ and $m\ge 1$, which is the abstract content of the identities of [step 3.1]. For $n\ge 2$ and a bounded $C^{2m}$ domain, put $v:=T(t)f$ and $w_r:=A^{r}v$ for $0\le r\le m$. Since $v\in D(A^m)$, each $w_r\in D(A)\subseteq H_0^1(\mathrm{\Omega })$ for $r<m$, $w_m\in L^2$, and $A{w}_r=w_{r+1}$. Thus $-\mathrm{\Delta }{w}_{m-1}=-w_m\in L^2$, so [L8] gives $w_{m-1}\in H^2$. Descending from $r=m-2$ to $r=0$, if $w_{r+1}\in H^{2(m-r-1)}$, then [L9] with $k=2(m-r-1)$ applies to $-\mathrm{\Delta }{w}_r=-w_{r+1}$; its domain requirement is $C^{2(m-r)}$, supplied by $C^{2m}$, and its coefficient requirements hold for the Laplacian. It follows that $w_r\in H^{2(m-r)}$, in particular $v\in H^{2m}(\mathrm{\Omega })$. For $n=1$ and any bounded open $\mathrm{\Omega }$, $A={\mathrm{\partial }}_x^2$ distributionally. Each $w_r\in D(A)\subseteq H_0^1(\mathrm{\Omega })$ for $r<m$ satisfies $w_r^{\mathrm{\prime }\mathrm{\prime }}=w_{r+1}\in L^2$; induction gives $w_r=v^{(2r)}$ distributionally, while $w_r^{\mathrm{\prime }}\in L^2$ for $r<m$ and $w_m\in L^2$. Hence all weak derivatives of $v$ through order $2m$ lie in $L^2$, so $v\in H^{2m}(\mathrm{\Omega })$ without boundary regularity. The abstract $D(A^m)$ conclusion remains valid in every dimension for arbitrary bounded open $\mathrm{\Omega }$.