The Dirichlet Laplacian generates an analytic heat semigroup

The operator $A$ is well defined with $D(A)=D(L)$ and $Au=-Lu$: by [L1], the condition $a_0(u,v)=-(f,v{)}_{L^2}$ for all $v\in H_0^1(\mathrm{\Omega })$ says exactly that the pair $(u,g)$ with $g:=-f$ satisfies $a_0(u,v)=(g,v{)}_{L^2}$ for all $v$, which is the defining identity of membership in $D(L)$ with $Lu=g$; conversely, if $u\in D(L)$ with datum $g$, then $a_0(u,v)=(g,v{)}_{L^2}=(-(-g),v{)}_{L^2}$ for all $v$, so $u\in D(A)$ with the datum $f=-g$; hence $u\in D(A)$ if and only if $u\in D(L)$ and $Au=-Lu$, the datum being unique in both definitions, so $D(A)$ is dense in $L^2(\mathrm{\Omega })$ by [L2] and $⟨Au,u⟩=-(Lu,u{)}_{L^2}=-a_0(u,u)\le 0$ for every $u\in D(A)$ again by [L2].

The slab criterion. If there are a unit vector $e$ and $a<b$ with $a<x\cdot e<b$ for every $x\in \mathrm{\Omega }$, then [L9] at $p=2$ gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le C(2)(b-a)\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$ for every $u\in H_0^1(\mathrm{\Omega })=W_0^{1,2}(\mathrm{\Omega })$ by [L1], hence $a_0(u,u)=\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}^2\ge (C(2)(b-a){)}^{-2}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2$, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}^2\le (1+C(2{)}^2(b-a{)}^2)\mathrm{\parallel }Du{\mathrm{\parallel }}_2^2$, so $a_0$ is coercive on $H_0^1$; this is the zero-extension, one-coordinate-dimension reduction performed by [L9], applied directly on $\mathrm{\Omega }$.

The bounded case under Axiom of Choice. Assume $\mathrm{\Omega }$ is bounded and the Axiom of Choice holds; then the resolvent of $L$ is compact by [L17], and [L10] supplies ${\lambda }_1>0$ with ${\lambda }_1={\min }_{0\mathrm{\ne }u\in H_0^1(\mathrm{\Omega })}\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}^2\mathrm{/}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2$ attained exactly at the nonzero first eigenfunctions and with ${\lambda }_1$ the smallest element of the eigenvalue list of an orthonormal eigenbasis $(e_j)\subseteq H_0^1(\mathrm{\Omega })$ of $L^2(\mathrm{\Omega })$, so ${\lambda }_{\ast }={\lambda }_1$ is positive and attained by a first eigenfunction; Countable Choice enters through [L10], and Axiom of Choice is used for the compact-resolvent/spectral branch through [L17].

Unbounded domains need not attain. Fix $L>0$, put $L_k:=Lk\mathrm{/}(k+1)$, $I_k:=(k(L+1),k(L+1)+L_k)$ and $\mathrm{\Omega }:={\bigcup }_{k\ge 1}{I}_k$; since $L_k<L$, successive intervals have gap $(L+1)-L_k>1$, so the components are disjoint. Every $u\in H_0^1(\mathrm{\Omega })$ satisfies ${\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^2={\sum }_k{\int }_{I_k}\mathrm{\mid }u{\mathrm{\mid }}^2$ and ${\int }_{\mathrm{\Omega }}\mathrm{\mid }{u}^{\mathrm{\prime }}{\mathrm{\mid }}^2={\sum }_k{\int }_{I_k}\mathrm{\mid }{u}^{\mathrm{\prime }}{\mathrm{\mid }}^2$, and the restriction $u{\mathrm{\mid }}_{I_k}$ lies in $H_0^1(I_k)$, because restriction to a component carries $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ into $C_c^{\mathrm{\infty }}(I_k)$ with classical derivatives restricting to classical derivatives and is bounded for the $W^{1,2}$ norm, hence carries the closure of the test functions into the closure; on each $I_k$ the sharp inequality [L11] gives ${\int }_{I_k}\mathrm{\mid }{u}^{\mathrm{\prime }}{\mathrm{\mid }}^2\ge (\pi \mathrm{/}{L}_k{)}^2{\int }_{I_k}\mathrm{\mid }u{\mathrm{\mid }}^2\ge (\pi \mathrm{/}L{)}^2{\int }_{I_k}\mathrm{\mid }u{\mathrm{\mid }}^2$ because $L_k<L$, so $a_0(u,u)\ge (\pi \mathrm{/}L{)}^2\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2$ and ${\lambda }_{\ast }\ge (\pi \mathrm{/}L{)}^2$, while the functions ${\varphi }_k(x):=\sin (\pi (x-k(L+1))\mathrm{/}{L}_k)$ on $I_k$, extended by zero, lie in $H_0^1(\mathrm{\Omega })$ because they lie in $H_0^1(I_k)$ by [L11] and are limits of compactly supported smooth functions on $I_k$ zero-extended to $\mathrm{\Omega }$, and have Rayleigh quotient $(\pi \mathrm{/}{L}_k{)}^2$, so ${\lambda }_{\ast }\le {\inf }_k(\pi \mathrm{/}{L}_k{)}^2=(\pi \mathrm{/}L{)}^2$; hence ${\lambda }_{\ast }=(\pi \mathrm{/}L{)}^2$ is positive and $a_0$ is coercive, but the infimum is not attained: otherwise ${\sum }_k{\int }_{I_k}(\mathrm{\mid }{u}^{\mathrm{\prime }}{\mathrm{\mid }}^2-{\lambda }_{\ast }\mathrm{\mid }u{\mathrm{\mid }}^2)=0$ with every summand at least $((\pi \mathrm{/}{L}_k{)}^2-(\pi \mathrm{/}L{)}^2){\int }_{I_k}\mathrm{\mid }u{\mathrm{\mid }}^2\ge 0$ and every coefficient strictly positive, forcing ${\int }_{I_k}\mathrm{\mid }u{\mathrm{\mid }}^2=0$ for all $k$ and hence $u=0$, which is excluded from the Rayleigh quotient.

Self-adjointness of $A$. By [L3] $L=L^{\ast }$ with $D(L^{\ast })=D(L)$; since $D(A)=D(L)$ and $A=-L$ by [step 1.1], the adjoint description of [L4] gives $D(A^{\ast })=D((-L{)}^{\ast })=\{y:x\mapsto ⟨-Lx,y⟩\text{ bounded}\}=\{y:x\mapsto ⟨Lx,y⟩\text{ bounded}\}=D(L^{\ast })=D(L)=D(A)$ and $⟨x,A^{\ast }y⟩=⟨Ax,y⟩=-⟨Lx,y⟩=-⟨x,Ly⟩=⟨x,-Ly⟩$ for all $x\in D(A)$, so $A^{\ast }y=-Ly=Ay$ for every $y\in D(A)$ and $A$ is self-adjoint; with [step 1.1] $A$ is densely defined and satisfies $⟨Au,u⟩\le 0$.

Identification with the Dirichlet Laplacian. By [L8] the symbol ${\mathrm{\Delta }}_D$ denotes the $L^2$ operator associated with the Dirichlet energy form in the $u^{\mathrm{\prime }}=Au$ convention, whose defining weak identity is exactly $a_0(u,v)=-(Au,v{)}_{L^2}$ for all $v\in H_0^1(\mathrm{\Omega })$; [step 1.1] shows that $A$ has this defining identity, so $A={\mathrm{\Delta }}_D$.

Domain identification for $n\ge 2$. Assume $n\ge 2$, $\mathrm{\Omega }$ a bounded $C^2$ domain, and let $u\in D(A)=D(L)$ with datum $f=Lu\in L^2(\mathrm{\Omega })$; [L12] applied to the weak equation $Lu=f$ gives $u\in H^2(\mathrm{\Omega })$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}\le C(\mathrm{\parallel }Lu{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2})=C(\mathrm{\parallel }Au{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2})$; conversely, if $u\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$, then for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ the weak-derivative identities of [L18] applied to ${\mathrm{\partial }}_{j}u\in H^1(\mathrm{\Omega })$ with test function $\bar{\phi }$ give $a_0(u,\phi )={\sum }_j{\int }_{\mathrm{\Omega }}{\mathrm{\partial }}_{j}u\overline{{\mathrm{\partial }}_j\phi }=-{\sum }_j{\int }_{\mathrm{\Omega }}{\mathrm{\partial }}_j^{2}u\bar{\phi }=-(\mathrm{\Delta }u,\phi {)}_{L^2}$, and because $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is dense in $H_0^1(\mathrm{\Omega })$ by [L1] while $v\mapsto a_0(u,v)$ and $v\mapsto (\mathrm{\Delta }u,v{)}_{L^2}$ are both continuous on $H_0^1(\mathrm{\Omega })$, the identity $a_0(u,v)=-(\mathrm{\Delta }u,v{)}_{L^2}$ holds for every $v\in H_0^1(\mathrm{\Omega })$; hence $u\in D(L)$ with $Lu=-\mathrm{\Delta }u$, so $u\in D(A)$ with $Au=\mathrm{\Delta }u$; therefore $D(A)=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ and $Au=\mathrm{\Delta }u$ on $D(A)$, and the estimate above together with $\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{L^2}\le n\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}$ shows that the graph norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }Au{\mathrm{\parallel }}_{L^2}$ is equivalent to $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}$.

Domain identification for $n=1$. Assume $n=1$; by [L14] a bounded $C^2$ domain is then a finite disjoint union of bounded open intervals $I_k$, and the following argument applies to each component and each $u\in D(A)=D(L)$: for $\phi \in C_c^{\mathrm{\infty }}(I_k)$, extended by zero to $\mathrm{\Omega }$, the identity $a_0(u,\phi )=(f,\phi {)}_{L^2}$ with datum $f=Lu$ reads ${\int }_{I_k}{u}^{\mathrm{\prime }}\overline{{\phi }^{\mathrm{\prime }}}={\int }_{I_k}f\bar{\phi }$, and replacing $\phi$ by $\bar{\phi }$ in the bilinear weak-derivative convention of [L18] shows that $-f$ is the weak derivative of the class $u^{\mathrm{\prime }}\in L^2(I_k)$; hence $u^{\mathrm{\prime }}$ has an $L^2$ weak derivative, $u\in W^{2,2}(I_k)=H^2(I_k)$, and $Lu=f=-u^{\mathrm{\prime }\mathrm{\prime }}$; conversely, if $u\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$, then $u^{\mathrm{\prime }\mathrm{\prime }}\in L^2(\mathrm{\Omega })$ and the same weak-derivative identity gives $a_0(u,\phi )=(-u^{\mathrm{\prime }\mathrm{\prime }},\phi {)}_{L^2}$ for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, which extends to every $v\in H_0^1(\mathrm{\Omega })$ by density by [L1]; hence $u\in D(A)$ with $Au=\mathrm{\Delta }u=u^{\mathrm{\prime }\mathrm{\prime }}$; consequently $D(A)=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ and $Au=\mathrm{\Delta }u$ in dimension one as well, and the graph norm is equivalent to the $H^2$ norm, since $\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}^2=-(u^{\mathrm{\prime }\mathrm{\prime }},u{)}_{L^2}\le \mathrm{\parallel }{u}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}$ by [step 1.1] gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }{u}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}^2\le (\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }{u}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}{)}^2\le 2(\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }Au{\mathrm{\parallel }}_{L^2}{)}^2$ while $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }Au{\mathrm{\parallel }}_{L^2}\le 2\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}$.

Generation and maximal angle. Applying [L5] to the self-adjoint operator $A$ of [step 2.1] with its quadratic bound of [step 1.1] yields that $A$ is sectorial of angle $\pi \mathrm{/}2$ with vertex $0$ in the $e^{tA}$ convention and generates a bounded analytic semigroup $(T(z){)}_{z\in {\mathrm{\Sigma }}_{\pi \mathrm{/}2}\cup \{0\}}$ of angle $\pi \mathrm{/}2$ with generator $A$, contractive on $[0,\mathrm{\infty })$; by the definition of the analytic-semigroup angle in [L6] the admissible angles are the $\delta \in (0,\pi \mathrm{/}2]$ for which such a family exists, so the exhibited family realizes the maximal allowed angle $\pi \mathrm{/}2$, and $\mathrm{\parallel }T(t)\mathrm{\parallel }\le 1$ for $t\ge 0$.

Higher-order domains for $n\ge 2$. Assume $n\ge 2$ and $\mathrm{\Omega }$ a bounded $C^{2m}$ domain, and let $E_m:=\{u\in H^{2m}(\mathrm{\Omega }):{\mathrm{\Delta }}^{j}u\in H_0^1(\mathrm{\Omega }),0\le j<m\}$; the claim is $D(A^m)=E_m$ for every $m\ge 1$, and $E_1=D(A)$ with $Au=\mathrm{\Delta }u$ is [step 2.3]; if $u\in D(A^m)$ with $m\ge 2$, then $u\in D(A^{m-1})$ and $Au\in D(A^{m-1})$, so the induction hypothesis gives $u\in H^{2m-2}(\mathrm{\Omega })$ with ${\mathrm{\Delta }}^{j}u\in H_0^1(\mathrm{\Omega })$ for $j<m-1$, and gives $Au\in H^{2m-2}(\mathrm{\Omega })$ with ${\mathrm{\Delta }}^j(Au)\in H_0^1(\mathrm{\Omega })$ for $j<m-1$; since $Au=\mathrm{\Delta }u$ by [step 2.3], the datum $f=Lu=-Au$ lies in $H^{2m-2}(\mathrm{\Omega })$, so [L13] with $k=2m-2$ applied to the weak solution $u\in H_0^1(\mathrm{\Omega })$ upgrades $u$ to $H^{2m}(\mathrm{\Omega })$, and ${\mathrm{\Delta }}^{j}u\in H_0^1(\mathrm{\Omega })$ for $j<m$ holds by the induction hypothesis and ${\mathrm{\Delta }}^j(Au)\in H_0^1(\mathrm{\Omega })$; hence $D(A^m)\subseteq E_m$; conversely, if $u\in E_m$ with $m\ge 2$, then $u\in E_1=D(A)$ with $Au=\mathrm{\Delta }u$, and $\mathrm{\Delta }u\in H^{2m-2}(\mathrm{\Omega })$ has ${\mathrm{\Delta }}^j(\mathrm{\Delta }u)={\mathrm{\Delta }}^{j+1}u\in H_0^1(\mathrm{\Omega })$ for $j<m-1$, so $\mathrm{\Delta }u\in E_{m-1}=D(A^{m-1})$ by the induction hypothesis, and $u\in D(A)$ with $Au=\mathrm{\Delta }u\in D(A^{m-1})$ gives $u\in D(A^m)$ by the recursive definition; hence $E_m\subseteq D(A^m)$ and $D(A^m)=E_m$ for all $m\ge 1$.

Higher-order domains for $n=1$. If $n=1$, then $\mathrm{\Omega }$ is a finite disjoint union of bounded open intervals by [L14], and the weak-derivative argument of [step 2.4] gives, by induction on $m$, the description $D(A^m)=\{u\in H^{2m}(\mathrm{\Omega }):{\mathrm{\Delta }}^{j}u\in H_0^1(\mathrm{\Omega }),0\le j<m\}$: the case $m=1$ is [step 2.4]; if $u\in D(A^m)$ with $m\ge 2$, then $u\in D(A^{m-1})$ and $Au\in D(A^{m-1})$, so the induction hypothesis applied to $u$ gives $u\in H^{2m-2}(\mathrm{\Omega })$ with $u^{(2j)}\in H_0^1(\mathrm{\Omega })$ for $j<m-1$ and applied to $Au=u^{\mathrm{\prime }\mathrm{\prime }}$ gives $u^{\mathrm{\prime }\mathrm{\prime }}\in H^{2m-2}(\mathrm{\Omega })$ with $u^{(2j+2)}\in H_0^1(\mathrm{\Omega })$ for $j<m-1$, hence $u\in H^{2m}(\mathrm{\Omega })$ and all $u^{(2j)}$, $0\le j<m$, lie in $H_0^1(\mathrm{\Omega })$; conversely, if $u\in H^{2m}(\mathrm{\Omega })$ has $u^{(2j)}\in H_0^1(\mathrm{\Omega })$ for $j<m$, then $u\in D(A)$ with $Au=u^{\mathrm{\prime }\mathrm{\prime }}$ by [step 2.4] and $u^{\mathrm{\prime }\mathrm{\prime }}\in H^{2m-2}(\mathrm{\Omega })$ satisfies $(u^{\mathrm{\prime }\mathrm{\prime }}{)}^{(2j)}=u^{(2j+2)}\in H_0^1(\mathrm{\Omega })$ for $j<m-1$, so $u^{\mathrm{\prime }\mathrm{\prime }}\in D(A^{m-1})$ by the induction hypothesis and $u\in D(A^m)$; in dimension one no boundary-regularity hypothesis is needed, because the weak-derivative identity of [L18] is available on every open set.

Exponential decay under coercivity. If $a_0(u,u)\ge \alpha \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2$ for some $\alpha >0$ and all $u\in H_0^1(\mathrm{\Omega })$, then ${\lambda }_{\ast }\ge \alpha >0$, and [step 1.1] gives $⟨Au,u⟩=-a_0(u,u)\le -{\lambda }_{\ast }\mathrm{\parallel }u{\mathrm{\parallel }}^2$ for every $u\in D(A)$, so the ${\lambda }_{\ast }>0$ branch of [L7] produces a bounded analytic semigroup of angle $\pi \mathrm{/}2$ generated by $A$ with $\mathrm{\parallel }T(z)\mathrm{\parallel }\le e^{-{\lambda }_{\ast }\mathrm{Re}z}$ on ${\mathrm{\Sigma }}_{\pi \mathrm{/}2}$; by [L15] the contour semigroup is the unique exponentially bounded strongly continuous semigroup with generator $A$, and both the family of [L7] and the family $T$ of [step 3.1] are exponentially bounded and have generator $A$, so they coincide and $\mathrm{\parallel }T(t)\mathrm{\parallel }\le e^{-{\lambda }_{\ast }t}$ for $t\ge 0$.

Positive-time smoothing. By [L15] applied to the sectorial operator $A$ of [step 3.1], the semigroup $T$ generated by $A$ satisfies $T(t)L^2(\mathrm{\Omega })\subseteq D(A^m)$ for every $t>0$ and $m\ge 1$, where the graph domains are the recursive domains of the statement: $D(A^m)=\{u\in D(A^{m-1}):Au\in D(A^{m-1})\}$ is equivalent to $\{u\in D(A^{m-1}):A^{m-1}u\in D(A)\}$ by induction on $m$, and $D(A^0)=L^2(\mathrm{\Omega })$.

The boundary-compatibility caveat. The spatial identifications in steps 3.2 and 3.3 retain ${\mathrm{\Delta }}^{j}u\in H_0^1(\mathrm{\Omega })$ for every $0\le j<m$: these conditions follow from the recursive operator domain and cannot be discarded merely because $u\in H^{2m}(\mathrm{\Omega })$. For example, on a nonempty bounded smooth domain the constant function $1$ belongs to every $H^{2m}$ but not to $H_0^1$, since [L9] would give $\mathrm{\parallel }1{\mathrm{\parallel }}_2\le C\mathrm{\parallel }\mathrm{\nabla }1{\mathrm{\parallel }}_2=0$ if it did. Moreover [L16] shows that the regularity suppliers for $n\ge 2$ apply only under their boundary and data hypotheses. Thus the abstract smoothing of step 4.2 supplies the recursive graph domain on general open sets; the stated spatial conclusions require their additional hypotheses. This reasoning uses neither a trace lifting nor any additional choice assumption.