The smooth Dirichlet domain and its compact inverse

Written by GPT-6.1 Sol (OpenAI) and GPT-6 Astra (OpenAI). Self-checked by the writing AI. Original exposition: CC0.

This proves the exact bounded-region prerequisite used in AN06-U001. Let $n\geq1$ and let $\Omega\subset\mathbb R^n$ be a bounded open set with smooth boundary. Connectedness is unnecessary. The empty set gives the zero Hilbert space and all statements below trivially. No arbitrary rough boundary, higher-order boundary condition, or higher-order regularity theorem is asserted.

For related results, see John K. Hunter, Notes on Partial Differential Equations, revised 18 June 2014: Theorem 4.27, printed pp.112-113; boundary change of variables and Theorem 4.30, pp.114-116; and Proposition 4.52/Theorem 4.53, pp.124-126. The proof below supplies its own difference-quotient signs, admissible tests, compactness, energy inverse and domain identification. The higher-regularity statements following Hunter's boundary proof are not imported.

We use weak derivatives, Lebesgue integration and the unitary Fourier transform. Write $H^k(\Omega)$ for the functions whose weak derivatives of order at most $k$ are in $L^2$, and define $H_0^1(\Omega)$ to be the $H^1$-closure of $C_c^\infty(\Omega)$. Inner products are linear in the first variable.

Read Euclidean product measure, coordinate inverses and measurable change of variables, and then Fourier normalization, in that order. The approximation and convolution reading proves the translation, mollification and integer Sobolev approximation facts used below. The energy representation is proved explicitly in Section 2.

1. The zero-boundary energy space and compact embedding

Lemma 1.1. For some $C_P<\infty$, $$\|u\|_2\leq C_P\|\nabla u\|_2\qquad(u\in H_0^1(\Omega)). \tag{1}$$ Zero extension $Eu$ belongs to $H^1(\mathbb R^n)$, with derivative equal to the zero extension of each derivative of $u$. The inclusion $H_0^1(\Omega)\longrightarrow L^2(\Omega)$ is compact.

Proof. Choose a cube $Q=(-R,R)^n$ containing $\overline\Omega$ in its interior. For $u\in C_c^\infty(\Omega)$, its zero extension is smooth and, on every line parallel to the first coordinate, $$Eu(x_1,x')=\int_{-R}^{x_1}\partial_1Eu(t,x')\,dt.$$ Cauchy-Schwarz, integration over $Q$ and Fubini give (1) with $C_P=2R$. For an $H^1$-convergent sequence of such functions, their zero extensions and derivatives converge in $L^2(\mathbb R^n)$. Testing against a compactly supported smooth function identifies the derivative of the limit, proving the assertion about $E$ and extending (1). This also proves the needed completeness of the energy space: $H^1$ is complete because an $L^2$ limit of functions and their first derivatives still satisfies the weak-derivative identities, and $H_0^1$ is a closed subspace. By (1), $\|\nabla u\|_2$ is an equivalent complete norm on it.

For compactness, let $P_N$ be the whole-space Fourier multiplier $1_{\{|\xi|\leq N\}}$. Plancherel gives $$\|(I-P_N)Eu\|_2\leq N^{-1}\|\nabla Eu\|_2. \tag{2}$$ For each fixed $N$, the operator $P_N$ on functions supported in $Q$ is compact as a map $L^2(Q)\to L^2(\mathbb R^n)$. Indeed its kernel is $c_n k_N(x-y)1_Q(y)$, where $k_N=\mathcal F^{-1}1_{\{|\xi|\leq N\}}\in L^2$ and $c_n=(2\pi)^{-n/2}$. Its squared integral is $c_n^2|Q|\|k_N\|_2^2$, by translation invariance and nonnegative Fubini.

Here are the kernel details. Every $L^2$ function on a Euclidean product is approximated by finite-measure simple functions, and then by finite sums of rectangle indicators, using the proved rectangle approximation. A rectangle in $\mathbb R^n\times\mathbb R^n$ is the product of two rectangles; intersecting its second factor with $Q$ keeps it a product. Thus finite sums $\sum_{j=1}^m a_j(x)b_j(y)$, with both factors in their respective $L^2$ spaces, are dense in $L^2(\mathbb R^n\times Q)$. A square-integrable kernel $G$ defines a bounded operator: pointwise Cauchy–Schwarz in $y$, followed by Fubini in $x$, gives $\|T_Gv\|_2\leq\|G\|_2\|v\|_2$. The same bound applies to kernel differences. The approximating product kernels have range in the finite span of the $a_j$, so their operators have finite rank. Bounded subsets of such a finite-dimensional range have compact closure by the finite-dimensional compactness proof.

To verify the displayed kernel represents the Fourier cutoff, choose $m_j\in C_c^\infty(\mathbb R^n)$ tending to $1_{\{|\xi|\leq N\}}$ in $L^2$, and set $k_j=\mathcal F^{-1}m_j$. For $v\in C_c^\infty(Q)$, the Fourier formula and absolute Fubini give $c_n k_j*v=\mathcal F^{-1}(m_j\mathcal Fv)$. The left side converges in $L^2$ to the stated kernel operator by the preceding kernel bound and $k_j\to k_N$ in $L^2$. The right side converges to $P_Nv$, since $\mathcal Fv$ is bounded and Plancherel applies. For $v\in L^2(Q)$, extend by zero, truncate to the points whose distance from the complement of $Q$ exceeds $1/j$, and then mollify with radius less than $1/(2j)$. Dominated convergence gives the truncation limit. The mollification theorem permits each radius to be chosen still smaller so that its $L^2$ error tends to zero; the convolutions belong to $C_c^\infty(Q)$. This proves the required density and extends the kernel identity to all of $L^2(Q)$.

Consequently $P_N$ on functions supported in $Q$ is a norm limit of finite-rank operators. Equation (2) shows that $E:H_0^1\to L^2(\mathbb R^n)$ is a norm limit of compact operators. Such a limit is compact: a finite net for the image of a unit ball under an approximant is a slightly enlarged net for its image under the limit. Total boundedness in the complete target gives compact closure, as proved in the same finite-dimensional-tools reading. Restriction to $\Omega$ proves the claimed inclusion compact. $\square$

2. The energy inverse

Put $a(u,v)=\int_\Omega\nabla u\cdot\overline{\nabla v}$. For $f\in L^2(\Omega)$ there is a unique $u\in H_0^1(\Omega)$ satisfying $$a(u,v)=(f,v)\qquad(v\in H_0^1(\Omega)). \tag{3}$$ Here is the Hilbert representation step explicitly. A continuous conjugate-linear functional $F$ on a Hilbert space has closed kernel. Projection onto that kernel exists: a minimizing sequence for distance is Cauchy by the parallelogram identity and converges by completeness; variation along the kernel shows its remainder is orthogonal. If $F\ne0$, choose a unit vector $e$ in the orthogonal complement of its kernel. Every vector is a kernel vector plus $ce$, since subtracting the multiple chosen by its $F$-value puts it in the kernel. Hence $F(v)=(F(e)e,v)$. The zero functional is represented by zero. Apply this argument to the complete energy inner product $a$ and $F(v)=(f,v)$, bounded by $C_P\|f\|_2\|\nabla v\|_2$.

Define $Kf=u$. Uniqueness makes $K$ linear, and testing (3) with $u$ gives $$\|\nabla Kf\|_2\leq C_P\|f\|_2,\qquad \|Kf\|_2\leq C_P^2\|f\|_2. \tag{4}$$ Thus $K:L^2\to H_0^1$ is bounded and, by Lemma 1.1, $K:L^2\to L^2$ is compact. Furthermore $$(f,Kg)=a(Kf,Kg)=(Kf,g),\qquad (Kf,f)=\|\nabla Kf\|_2^2\geq0. \tag{5}$$ Consequently $K$ is self-adjoint and positive. If $Kf=0$, (3) gives $(f,v)=0$ for all $v\in C_c^\infty(\Omega)$, so $f=0$ by their $L^2$ density. The density follows by first truncating away from the complement of $\Omega$ and then mollifying; dominated convergence and continuity of translations in $L^2$ give the two approximations. Thus $K$ is injective and has dense range: a vector orthogonal to its range is killed by $K^*=K$.

3. The full second-order boundary estimate

Theorem 3.1. If $u\in H_0^1(\Omega)$ and $-\Delta u=f\in L^2(\Omega)$ in distributions, then $$u\in H^2(\Omega),\qquad \|u\|_{H^2(\Omega)}\leq C_\Omega(\|f\|_2+\|u\|_2). \tag{6}$$

Proof. The distribution equation extends to (3) by $H^1$ approximation of tests. In particular $$\|\nabla u\|_2^2=(f,u)\leq\|f\|_2\|u\|_2, \qquad \|u\|_{H^1}\leq C(\|f\|_2+\|u\|_2). \tag{7}$$

We first give the two difference-quotient facts used below. Set $\delta_h^k q(y)=(q(y+he_k)-q(y))/h$. Translations commute with weak derivatives: substitute $y'=y+he_k$ in the integral against a compact smooth test, use the original weak-derivative identity with the oppositely translated test, and substitute back. Taking the difference and dividing by $h$ gives $\partial_i\delta_h^kq=\delta_h^k\partial_iq$ wherever those derivatives are defined. Where supports and shifts fit in a larger coordinate patch, change of variable gives $$\int q\,\overline{\delta_{-h}^k r} =-\int\delta_h^kq\,\overline r, \qquad \delta_h^k(bq)=b(y+he_k)\delta_h^kq+(\delta_h^kb)q. \tag{8}$$ For $q\in H^1$, the fundamental theorem of calculus along lines, followed by Cauchy-Schwarz and Fubini, gives $$\|\delta_h^kq\|_{L^2(U)}\leq\|\partial_kq\|_{L^2(U_h)}, \tag{9}$$ where $U_h$ contains all intervening segments. This identity first holds for smooth functions and then for the $H^1$ limits used here. Conversely, if the difference quotients of an $L^2$ function are bounded by $M$ on $U$ for all sufficiently small $h$, then it has weak $k$th derivative there of norm at most $M$. For $\phi\in C_c^\infty(U)$, (8) yields $$\left|\int q\,\overline{\partial_k\phi}\right| =\lim_{h\to0}\left|\int\delta_h^kq\,\overline\phi\right| \leq M\|\phi\|_2.$$ The Hilbert representation argument in Section 2 represents the functional $\phi\mapsto-\int q\,\overline{\partial_k\phi}$ by an $L^2$ function. That is exactly the weak derivative, including its sign. No assertion of differentiability is made before this argument.

Near a boundary point, write the smooth boundary as a graph after rotating coordinates, and use the graph map to flatten it. On nested, relatively compact coordinate patches it is a smooth diffeomorphism $x=\psi(y)$ with $y_n>0$ inside $\Omega$, bounded derivatives and bounded inverse derivatives. Let $w=u\circ\psi$, $J=|\det D\psi|$, and $B=D\psi^{-1}$, evaluated at $x=\psi(y)$. Changing variables in the weak formulation gives $$\int A_{ij}\partial_iw\,\overline{\partial_jv} =\int F\overline v,\qquad A=JBB^T,\quad F=J(f\circ\psi). \tag{10}$$ Repeated indices are summed here. The matrix is real symmetric, its entries and first derivatives are bounded, and $\xi^TA(y)\xi\geq\theta|\xi|^2$ for a common $\theta>0$ on the chosen patch. These statements follow from the determinant being bounded above and below and the inverse matrix being bounded. All Sobolev change-of-variable identities in (10) follow by taking $u_j\in C_c^\infty(\Omega)$ converging to $u$ in $H^1$, using the ordinary chain rule for $u_j$, and passing to the limit. Test functions supported away from the patch's artificial edges and belonging to the zero-boundary energy space pull back to admissible tests in (3).

Choose $\eta$ smooth with $0\leq\eta\leq1$, equal to one on the smaller patch and supported away from its artificial edges. It may meet $y_n=0$. For each tangential index $k<n$ and small $h$, use $$v=-\delta_{-h}^k(\eta^2\delta_h^kw) \tag{11}$$ in (10). This is an admissible $H_0^1$ test. To justify it without importing a trace theorem, multiply the transformed $u_j$ by a cutoff equal to one around all supports in (11). They are compactly supported smooth functions in the half-space and converge in $H^1$. Tangential translations preserve the half-space; for fixed $h$, differences and multiplication by $\eta$ are bounded operations on $H^1$. Their corresponding tests therefore converge to (11) in $H^1$. No normal translation across the boundary is used. At an interior patch the same test is admissible for every $k$ by choosing its support a positive distance from the patch boundary.

For clarity, put $X_h=\|\eta\delta_h^k\nabla w\|_2$ and $W=\|w\|_{H^1}$ on a fixed larger patch. Substituting (11), using (8), and expanding the derivative of $\eta^2\delta_h^kw$ leaves the principal term $$\int\eta^2 A_{ij}(y+he_k) \delta_h^k\partial_iw\, \overline{\delta_h^k\partial_jw} \geq\theta X_h^2.$$ The remaining terms contain either $\delta_h^kA_{ij}\,\partial_iw$ or $2\eta\partial_j\eta\,\delta_h^kw$. The coefficient differences are uniformly bounded by their first derivatives, and (9) bounds $\|\delta_h^kw\|_2$ by $W$ on the larger patch. Cauchy-Schwarz therefore bounds these terms by $C(WX_h+W^2)$. The right side of (10) is bounded by $$\|F\|_2\|\delta_{-h}^k(\eta^2\delta_h^kw)\|_2 \leq C\|F\|_2(X_h+W).$$ For the last inequality apply (9) to $\eta^2\delta_h^kw$ and expand its $k$th derivative; its derivative term is bounded by $X_h$ because $0\leq\eta\leq1$. Hence $$\theta X_h^2\leq C(W+\|F\|_2)X_h +C(W^2+\|F\|_2W).$$ The inequality $ab\leq\varepsilon a^2+(4\varepsilon)^{-1}b^2$, with fixed small $\varepsilon$, absorbs the terms linear in $X_h$ and proves $$X_h\leq C(W+\|F\|_2), \tag{12}$$ uniformly in $h$. The converse to (9), applied to every $\partial_iw$ where $\eta=1$, now gives $\partial_k\partial_iw\in L^2$. In the interior all second derivatives are obtained this way. At the boundary it gives every second derivative except $\partial_n^2w$; commutation of distribution derivatives includes those with the tangential derivative written second.

Equation (10) implies, in distributions on the smaller half-patch, $$A_{nn}\partial_n^2w =-F-\sum_{(i,j)\ne(n,n)}A_{ij}\partial_j\partial_iw -\sum_{i,j}(\partial_jA_{ij})\partial_iw. \tag{13}$$ The right side is in $L^2$ by (12). Since $A_{nn}\geq\theta$, multiplication by $1/A_{nn}$ identifies the remaining distribution derivative with an $L^2$ function of the same controlled norm. When $n=1$ the first sum in (13) is empty and this step gives the entire second-derivative conclusion directly.

To transform back explicitly, put $\phi=\psi^{-1}$ and $c_{ir}=\partial_{x_i}\phi_r$. On each compact subpatch in the interior of the half-patch, mollification approximates $w$ in $H^2$: derivatives commute with convolution there, and continuity of translations in $L^2$ proves convergence of each derivative. The ordinary chain rule for these smooth approximants, followed by change of variables and passage in $L^2$, gives $$\partial_{x_j}\partial_{x_i}u =\sum_{r,s}c_{ir}c_{js} (\partial_{y_s}\partial_{y_r}w)\circ\phi +\sum_r(\partial_{x_j}c_{ir}) (\partial_{y_r}w)\circ\phi. \tag{14}$$ The coefficients, their first derivatives and the Jacobians are uniformly bounded on the full smaller patch. The right side therefore belongs to $L^2$ on that full patch with the asserted estimate. Exhausting its interior compact subpatches identifies it with the global weak second derivative, without requiring any $H^2$ approximation across the boundary. A finite collection of these boundary patches and interior patches covers $\overline\Omega$, which is compact. Local weak derivatives agree on overlaps since they represent the same distributions. Adding their estimates proves $$\|u\|_{H^2(\Omega)}\leq C_\Omega(\|f\|_2+\|u\|_{H^1(\Omega)}).$$ Equation (7) yields (6). $\square$

The zero-boundary condition is retained in the closure defining $H_0^1$. It also gives zero Sobolev trace in the usual sense. In a flattened strip $0<y_n<a$, a smooth $v$ supported away from the lateral edges satisfies $$\|v(\cdot,0)\|_2^2 \le a^{-1}\|v\|_2^2+2\|v\|_2\|\partial_nv\|_2 \le (a^{-1}+1)\|v\|_2^2+\|\partial_nv\|_2^2.$$ For each fixed $y'$, integrate the derivative of $|v(y',t)|^2$ between zero and $t$, average $t$ over $(0,a)$, and then integrate in $y'$; this proves the first inequality. Smooth functions on a neighborhood of the closed smaller half-strip are dense in its $H^1$ space, locally away from the artificial edges. To see this directly, first translate the function and each of its weak derivatives by $\delta e_n$ toward the interior and restrict to the smaller half-strip. Extend the original function and each derivative by zero only for comparing their $L^2$ translations; translation continuity gives convergence as $\delta\downarrow0$. For fixed $\delta$, mollify with radius less than $\delta/2$. Every convolution used on the smaller half-strip then lies strictly in the original open half-strip, so derivatives commute with convolution there. Choose that radius small enough to make all the finitely many $L^2$ errors tend to zero. A cutoff away from the artificial edges supplies the stated local support. Thus the inequality extends boundary restriction continuously to $H^1$, independently of the chosen approximation. Coordinate integration and the smooth chart bounds transfer it to the boundary patch. The approximating $u_j\in C_c^\infty(\Omega)$ have trace zero, so their $H^1$ limit does too. Only this implication is needed here.

4. The exact operator realization

Theorem 4.1. On $L^2(\Omega)$, the Dirichlet Laplacian is the positive self-adjoint operator $$A=-\Delta,\qquad D(A)=H^2(\Omega)\cap H_0^1(\Omega). \tag{15}$$ It satisfies $A\geq C_P^{-2}I$. Its inverse $A^{-1}=K$ is bounded and compact, and every nonreal resolvent $(A-z)^{-1}$ is compact. Its graph norm is equivalent to the $H^2$ norm on (15).

Proof. Define initially $A=K^{-1}$ on $\operatorname{Ran}K$, which is dense by Section 2. If $u=Kf$, (3) identifies $-\Delta u=f$ in distributions, and Theorem 3.1 places $u$ in (15). Conversely, for $u$ in (15), $f=-\Delta u\in L^2$. Testing first against $C_c^\infty$ and then using its $H^1$ density in $H_0^1$ proves (3); uniqueness gives $u=Kf$. This proves both domain inclusions and the action in (15), without changing the realization.

The energy identity makes $A$ symmetric and $(Au,u)=\|\nabla u\|_2^2\geq C_P^{-2}\|u\|_2^2$. Its self-adjointness follows directly on its original domain. If $v\in D(A^*)$ and $A^*v=g$, substitute $u=Kf$ in $(Au,v)=(u,g)$ to obtain $(f,v)=(Kf,g)=(f,Kg)$ for every $f\in L^2$. Thus $v=Kg\in D(A)$ and $Av=g$. Symmetry gives the opposite adjoint inclusion. Also $A$ is closed: $u_j\to u$ and $Au_j\to f$ imply $u=Kf$ by boundedness of $K$.

For nonreal $z$, symmetry gives $\|(A-z)u\|_2\geq|\operatorname{Im}z|\|u\|_2$. Closedness makes its range closed: a convergent sequence of images gives convergence of the arguments and of their $A$-images. The orthogonal complement of that range is $\ker(A-\overline z)=0$, so the range is the whole space. Hence $R(z)=(A-z)^{-1}$ is bounded and maps into (15). The inverse identity gives $$R(z)=K(I+zR(z)),$$ which is compact since $K$ is. Finally (6), with $f=Au$, bounds the $H^2$ norm by the graph norm, while $\|\Delta u\|_2\leq C_n\|u\|_{H^2}$ gives the reverse bound. $\square$

The graph-norm inclusion $D(A)\to L^2$ is compact as well: $(Au,u)=\|\nabla u\|_2^2$ makes its graph-norm unit ball bounded in $H_0^1$, and Lemma 1.1 applies. The generic compact positive inverse and diagonal-domain provider can then supply an eigenbasis and spectral multiplier domains. That Hilbert-space result alone does not prove any of the boundary or compact Sobolev steps established here.