The Dirichlet Laplacian generates the heat semigroup

The operator $L$. The $L^2$ operator associated with the symmetric Dirichlet form is the symmetric-case operator of The $L^2$ operator associated with a symmetric elliptic form for coefficients $a^{ij}={\delta }^{ij}$, $b\equiv 0$, $c=0$ and ellipticity constant $\theta =1$ (Uniformly elliptic divergence-form operators and their sesquilinear forms); it is densely defined, symmetric and lower bounded (The associated elliptic operator is densely defined, symmetric and lower bounded), while $a(u,u)={\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2\ge 0$. The explicit Gårding constant of this form is $\beta =1\mathrm{/}2$ (Garding's inequality for a divergence-form elliptic operator); fix ${\mu }_0:=1\ge \beta$. Then $L$ is self-adjoint and $L+{\mu }_0:D(L)\to H$ is bijective with compact inverse (The symmetric elliptic form operator is self-adjoint with compact resolvent). In particular $L$ is closed and $I+L=L+1$ is bijective.

Spectral basis and positive eigenvalues. With ${\mu }_0=1$, [F5] gives eigenvalues ${\lambda }_1\le {\lambda }_2\le \cdots \to +\mathrm{\infty }$ and an orthonormal basis $(e_j)$ with $L{e}_j={\lambda }_j{e}_j$. For an eigenvector $u\mathrm{\ne }0$ of ${\lambda }_j$, [F1] gives ${\lambda }_j\mathrm{\parallel }u{\mathrm{\parallel }}^2=(Lu,u)=a(u,u)\ge 0$. If ${\lambda }_j=0$, then $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2}=0$, and Poincaré on $H_0^1(\mathrm{\Omega })$ gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}\le C_P\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2}=0$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction); this contradicts $u\mathrm{\ne }0$. Thus $0<{\lambda }_1\le {\lambda }_2\le \cdots \to +\mathrm{\infty }$, and $A{e}_j=-{\lambda }_j{e}_j$.

Generation. The operator $A=-L$ is densely defined, closed and dissipative: closedness and density follow from [step 1.1], while for $u\in D(A)$, [F1] and [F3] give $\mathrm{Re}⟨Au,u⟩=-a(u,u)=-{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2\le 0$. Also $\mathrm{Ran}(I-A)=\mathrm{Ran}(I+L)=H$ by [step 1.1].

By [F2] with ${\lambda }_0=1$, $A$ generates a strongly continuous semigroup of contractions $T$ on $H$.

Orbit of each eigenvector. For fixed $j$, $v_j(t):=e^{-{\lambda }_{j}t}{e}_j$ belongs to $D(A)$, is $C^1$, has $v_j(0)=e_j$, and satisfies $v_j^{\mathrm{\prime }}(t)=-{\lambda }_j{v}_j(t)=A{v}_j(t)$. By uniqueness for the classical homogeneous problem in [F4], $T(t)e_j=e^{-{\lambda }_{j}t}{e}_j$. By linearity, if $f_N:={\sum }_{j\le N}(f,e_j)e_j$, then $T(t)f_N={\sum }_{j\le N}{e}^{-{\lambda }_{j}t}(f,e_j)e_j$.

Arbitrary $L^2$ data and initial trace. The finite sums $f_N$ converge to $f$ in $H$, so $c_j:=(f,e_j{)}_{L^2}$ is square-summable. Since $\mathrm{\mid }{e}^{-{\lambda }_{j}t}\mathrm{\mid }\le 1$, the spectral series converges in $H$ for each $t\ge 0$. For every $N$, contraction and orthonormality bound the distance between $T(t)f$ and this series by $\mathrm{\parallel }f-f_N{\mathrm{\parallel }}_H+({\sum }_{j>N}\mathrm{\mid }{c}_j{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, uniformly in $t\ge 0$. This tends to $0$, so the expansion holds in $H$; strong continuity also gives $T(t)f\to f$ in $H$ at $0$.

Positive-time smoothing in graph norm. Write $c_j=(f,e_j{)}_{L^2}$ and $u_N(t):={\sum }_{j\le N}{e}^{-{\lambda }_{j}t}{c}_j{e}_j$. Fix $0<\delta <T_0<\mathrm{\infty }$. For $M<N$ and $t\in [\delta ,T_0]$, orthonormality gives $\mathrm{\parallel }{u}_N(t)-u_M(t){\mathrm{\parallel }}_H^2={\sum }_{M<j\le N}{e}^{-2{\lambda }_{j}t}\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\le {\sum }_{j>M}\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\mathrm{.}$ Also $A{u}_N(t)=-{\sum }_{j\le N}{\lambda }_j{e}^{-{\lambda }_{j}t}{c}_j{e}_j$, so [F9] and continuity on bounded intervals give $\mathrm{\parallel }A(u_N(t)-u_M(t)){\mathrm{\parallel }}_H^2={\sum }_{M<j\le N}{\lambda }_j^2{e}^{-2{\lambda }_{j}t}\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\le C_{\delta }^2{\sum }_{j>M}\mathrm{\mid }{c}_j{\mathrm{\mid }}^2,C_{\delta }:={\sup }_{\lambda \ge 0}\lambda e^{-\delta \lambda }<\mathrm{\infty }\mathrm{.}$ Both tails tend to zero uniformly on $[\delta ,T_0]$. Thus $(u_N,A{u}_N)$ converges uniformly there in $H\oplus H$. By [F2] the operator $A$ is closed; its graph is closed, so the limit pair is $(u(t),Au(t))$ for $u(t)=T(t)f$. Consequently $u(t)\in D(A)$ for every $t>0$, and $t\mapsto u(t)$ is continuous on every compact positive-time interval in the graph norm [F6].

Classical evolution at positive times. Fix $0<\delta <T_0$ and put $x_{\delta }:=u(\delta )\in D(A)$ by [step 6.1]. For $t\in [\delta ,T_0]$, $u(t)=T(t-\delta )x_{\delta }$ by the semigroup law. By [F7], this orbit is differentiable for $t>\delta$ and satisfies $u^{\mathrm{\prime }}(t)=AT(t-\delta )x_{\delta }=Au(t)$; since $\delta$ can be chosen below any positive time, $u$ is a classical solution on $(0,T_0]$ (and on each closed interval bounded away from $0$). For general $f\in H$ no graph-norm trace at $0$ is asserted; if $f\in D(A)$, [F4] gives the classical solution on $[0,T_0]$.

The semigroup orbit is the unique mild solution of the homogeneous abstract Cauchy problem by [F4], and its initial value is attained in $H$ by [step 5.1]. The positive-time graph-norm and differentiability conclusions are those of [steps 6.1 and 7.1]; no spatial $H^2$ identification of $D(A)$ is made for an arbitrary bounded open $\mathrm{\Omega }$.