A Harmonic-Polynomial Escape Sequence for an Ornstein–Uhlenbeck Transmission Problem Across Hilbert–Schmidt Ellipsoids

1. Introduction

The finite-dimensional Ornstein–Uhlenbeck operator

$$L_n=\Delta -x·\nabla$$

(1)

is the symmetric diffusion generator associated with the standard Gaussian measure. With respect to the normalized measure ${\left(2\pi \right)}^{-n/2}{\mathrm{e}}^{-{\left|x\right|}^2/2}\mathrm{d}x$, it is self-adjoint and non-positive on its natural $L^2$ domain. The operator is a central object in Gaussian Sobolev spaces, Gaussian Dirichlet forms, and finite- and infinite-dimensional Ornstein–Uhlenbeck semigroup theory; standard references include Bogachev’s survey on Ornstein–Uhlenbeck operators and semigroups [1], the Dirichlet-semigroup work of Da Prato and Lunardi [2], and the finite-dimensional survey of Lunardi, Metafune and Pallara [3].

This paper studies the positive spectral equation

$$L_{n}u=u.$$

(2)

Because $L_n$ is non-positive in the global Gaussian energy space, (2) has no non-zero global finite-energy solution. Exterior-domain solutions are different: the finite-energy exterior branch may be singular at a point hidden inside the obstacle. We exploit this observation for ellipsoidal obstacles whose axes are determined by a square-summable sequence.

Let $a_i>0$ and assume

$${\displaystyle \sum _{i=1}^{\infty }}{a}_i^2<\infty .$$

(3)

For each $n\ge 1$, define

$$Q_n\left(x\right)={\displaystyle \sum _{i=1}^n}{a}_i^2{x}_i^2,B_n=\{x\in {\mathbb{R}}^n:Q_n\left(x\right)<1\}.$$

(4)

The boundary $\partial B_n$ is a smooth ellipsoid. We consider the transmission class

$$\begin{array}{c}\hfill S_n=\{u\in C^{\infty }({\mathbb{R}}^n\setminus \partial B_n)\cap C\left({\mathbb{R}}^n\right):L_{n}u=u\mathrm{in}{\mathbb{R}}^n\setminus \partial B_n\}.\end{array}$$

(5)

No normal-flux matching is imposed at $\partial B_n$. The goal is to construct $u_n\in S_n$ such that

$${\int }_{{\mathbb{R}}^n}\left(|u_n{|}^2+{|\nabla u_n|}^2\right){\mathrm{e}}^{-{\left|x\right|}^2/2}\mathrm{d}x<\infty$$

(6)

and

$${\displaystyle \frac{{\displaystyle {\int }_{{\mathbb{R}}^n}\left(|u_n{|}^2+{|\nabla u_n|}^2\right){\mathrm{e}}^{-{\left|x\right|}^2/2}\mathrm{d}x}}{{\displaystyle {\int }_{B_n}\left(|u_n{|}^2+{|\nabla u_n|}^2\right){\mathrm{e}}^{-{\left|x\right|}^2/2}\mathrm{d}x}}}⟶+\infty .$$

(7)

The construction in this version uses only one angular mode. For $n\ge 2$ we take the real harmonic polynomial

$$H_m\left(x\right)=Re{(x_1+i{x}_2)}^m.$$

(8)

Multiplying $H_m$ by the Tricomi branch of the separated radial equation gives an explicit exterior solution. The boundary trace is then extended into $B_n$ as the minimizer of the weighted Dirichlet energy. The proof is organized so that every estimate used in the final ratio is stated separately: a separated-solution lemma, a radial Tricomi multiplier lemma, an exterior lower bound, and an interior trace-capacity upper bound.

2. Notation and Main Result

Throughout the paper

$$\mathrm{d}{\gamma }_n\left(x\right)={\mathrm{e}}^{-{\left|x\right|}^2/2}\mathrm{d}x$$

(9)

denotes the unnormalized Gaussian measure. The normalization factor ${\left(2\pi \right)}^{-n/2}$ is deliberately omitted because it cancels in all ratios. For a measurable set $D\subset {\mathbb{R}}^n$ and a weakly differentiable function v, define

$${\mathcal{E}}_n(v;D)={\int }_D\left({\left|v\right|}^2+{|\nabla v|}^2\right)\mathrm{d}{\gamma }_n.$$

(10)

The symbol C denotes a positive constant whose value may change from line to line. Constants may depend on the fixed sequence $\left(a_i\right)$ but never on n or on the degree $m=m_n$ unless this dependence is explicitly stated.

Since $\left(a_i\right)\in {\ell }^2$ and $a_i>0$, the two constants

$$M=\underset{i\ge 1}{sup}{a}_i<\infty ,K_a={\left(a_1^{-2}+a_2^{-2}\right)}^{1/2}$$

(11)

are finite. The number $K_a$ is used only for $n\ge 2$. The case $n=1$ is immaterial for the limiting statement and can be filled by any one-dimensional exterior branch with its variational interior extension.

Theorem 1 

(Constructive escape sequence). Let $a_i>0$ and ${\sum }_{i=1}^{\infty }{a}_i^2<\infty$. There exist functions $u_n\in S_n$ such that ${\mathcal{E}}_n(u_n;{\mathbb{R}}^n)<\infty$ for every n and

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{\mathcal{E}}_n(u_n;{\mathbb{R}}^n)}{{\mathcal{E}}_n(u_n;B_n)}}=+\infty .$$

(12)

For $n\ge 2$, one may take $m_n=\lfloor n^{1/8}\rfloor$ and construct the exterior part from the harmonic polynomial $Re{(x_1+i{x}_2)}^{m_n}$. More precisely, there are constants $C,c>0$ such that, for all sufficiently large n,

$${\mathcal{E}}_n(u_n;B_n^c)\ge c{\left(2\pi \right)}^{n/2}{n}^{-1/2}{\left({\displaystyle \frac{2m_n}{\mathrm{e}}}\right)}^{m_n},{\mathcal{E}}_n(u_n;B_n)\le C{\left(2\pi \right)}^{n/2}{n}^C{C}^{m_n}.$$

(13)

Consequently the quotient in (12) is bounded from below by

$$c{n}^{-C}{\left({\displaystyle \frac{2m_n}{\mathrm{e}C}}\right)}^{m_n},$$

(14)

which tends to $+\infty$.

3. The Weighted Dirichlet Principle

Let $D\subset {\mathbb{R}}^n$ be a bounded $C^2$ domain. For smooth functions $v,w$ on $\overline{D}$, Gaussian integration by parts gives

$${\int }_{D}v{L}_{n}w\mathrm{d}{\gamma }_n=-{\int }_D\nabla v·\nabla w\mathrm{d}{\gamma }_n+{\int }_{\partial D}v{\partial }_{\nu }w{\mathrm{e}}^{-{\left|x\right|}^2/2}\mathrm{d}S,$$

(15)

where $\nu$ is the outward Euclidean unit normal. Hence the bilinear form

$${\mathfrak{a}}_D(v,w)={\int }_D(vw+\nabla v·\nabla w)\mathrm{d}{\gamma }_n$$

(16)

is coercive on the weighted Sobolev space $H^1(D,{\gamma }_n)$. If g is the trace of at least one $H^1(D,{\gamma }_n)$ function, the direct method gives a unique minimizer of ${\mathcal{E}}_n(·;D)$ among all functions with trace g. Its Euler–Lagrange equation is

$$L_n{v=v\mathrm{in}D,v|}_{\partial D}=g$$

(17)

in the weak sense. Since $\partial B_n$ is smooth, standard elliptic regularity implies that the minimizer is $C^{\infty }$ in $B_n$ and continuous up to $\partial B_n$ whenever g is smooth. This is the only variational ingredient used for the interior part of the construction.

4. Separated Ornstein–Uhlenbeck Solutions

We recall the separated form of the finite-energy branch of (2). Let $H_m$ be a homogeneous harmonic polynomial of degree m in ${\mathbb{R}}^n$. In the construction below, $H_m$ is the real part of ${(x_1+i{x}_2)}^m$, but the separation calculation is general.

Lemma 1 

(Tricomi radial branch). Let $n\ge 2$ and $m\ge 0$. Put

$${\alpha }_m={\displaystyle \frac{m+1}{2}},{\beta }_{m,n}={\displaystyle \frac{n}{2}}+m,$$

(18)

and define

$${\mathcal{R}}_{m,n}\left(r\right)={\displaystyle \frac{U\left({\alpha }_m,{\beta }_{m,n},r^2/2\right)}{U\left({\alpha }_m,{\beta }_{m,n},n/2\right)}},r>0,$$

(19)

where $U(a,b,z)$ is Tricomi’s confluent hypergeometric function. Then

$$W_{m,n}\left(x\right)={\mathcal{R}}_{m,n}\left(\right|x\left|\right)H_m\left(x\right)$$

(20)

satisfies $L_n{W}_{m,n}=W_{m,n}$ on ${\mathbb{R}}^n\setminus \left\{0\right\}$, is normalized by ${\mathcal{R}}_{m,n}\left(\sqrt{n}\right)=1$, and has finite Gaussian $H^1$ energy on every exterior region $\left\{\right|x|>\rho \}$, $\rho >0$.

Proof. 

Write $x=r\omega$ with $\omega \in {\mathbb{S}}^{n-1}$ and $H_m\left(x\right)=r^m{Y}_m\left(\omega \right)$. The polar form of $L_n$ gives

$$L_n\left\{R\left(r\right)Y_m\left(\omega \right)\right\}=\left(R^{\prime \prime }+{\displaystyle \frac{n-1}{r}}{R}^{\prime }-r{R}^{\prime }-{\displaystyle \frac{m(m+n-2)}{r^2}}R\right)Y_m\left(\omega \right).$$

(21)

Taking $R\left(r\right)=r^{m}h(r^2/2)$, the equation $L_n\left(R{Y}_m\right)=R{Y}_m$ becomes

$$s{h}^{\prime \prime }+(n/2+m-s)h^{\prime }-{\displaystyle \frac{m+1}{2}}h=0,s=r^2/2.$$

(22)

This is Kummer’s equation. Its Tricomi solution is $h\left(s\right)=U\left(\right(m+1)/2,n/2+m,s)$, using the standard notation for $U(a,b,z)$[4]. Since $r^m{Y}_m\left(\omega \right)=H_m\left(x\right)$, the expression in (20) follows. The normalization is immediate from (19). Finally, the large-z asymptotic $U(a,b,z)=O\left(z^{-a}\right)$ gives ${\mathcal{R}}_{m,n}\left(r\right)H_m\left(x\right)=O\left(r^{-1}\right)$ and its gradient is $O\left(r^{-2}\right)$ as $r\to \infty$ for fixed $m,n$; Gaussian integrability on $\left\{\right|x|>\rho \}$ follows. □

The following estimate is the technical point where the high-dimensional normalization at $r=\sqrt{n}$ is used. It is a standard consequence of the integral representation and Laplace estimates for U; details are included to fix constants and remove ambiguity about the inward growth of the irregular branch.

Lemma 2 

(Radial multiplier bounds). Fix $r_0>0$. There are constants $C,N_0>0$ depending on $r_0$ but independent of n and m such that, whenever $n\ge N_0$ and $1\le m\le n^{1/4}$, the following hold.

(a)If $I_{-}\left(\rho \right)=(-log\rho -1+\rho )11_{(0,1]}\left(\rho \right)$, then, for all $r\ge r_0$,

$$|{\mathcal{R}}_{m,n}\left(r\right)|+{\displaystyle \frac{1}{\sqrt{n}}}\left|{\mathcal{R}}_{m,n}^{\prime }\left(r\right)\right|\le C{n}^C{\mathrm{e}}^{Cm}exp\left\{{\displaystyle \frac{n}{4}}{I}_{-}\left({\displaystyle \frac{r^2}{n}}\right)\right\}.$$

(23)

(b)Uniformly for $n\le r^2\le n+1$,

$${\mathcal{R}}_{m,n}\left(r\right)=1+O\left({\displaystyle \frac{m+1}{n}}\right),\left|{\mathcal{R}}_{m,n}^{\prime }\left(r\right)\right|\le C{\displaystyle \frac{m+1}{\sqrt{n}}}.$$

(24)

(c)With $Q_n$ as in (4),

$$\begin{array}{c}\hfill {\int }_{\{1/2<Q_n<1\}}\left(|{\mathcal{R}}_{m,n}{\left(\right|x\left|\right)|}^2+|{\mathcal{R}}_{m,n}^{\prime }{\left(\right|x\left|\right)|}^2\right)\mathrm{d}{\gamma }_n\left(x\right)\le C{\left(2\pi \right)}^{n/2}{n}^C{\mathrm{e}}^{Cm}.\end{array}$$

(25)

Proof. 

For $a>0$ and $z>0$, Tricomi’s function admits the positive integral representation

$$U(a,b,z)={\displaystyle \frac{1}{\Gamma \left(a\right)}}{\int }_0^{\infty }{\mathrm{e}}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}\mathrm{d}t,$$

(26)

which is recorded in the DLMF integral-representation section for confluent hypergeometric functions [5]. Apply (26) with $a={\alpha }_m$, $b={\beta }_{m,n}$, and $z=r^2/2$. The exponent in the integrand is

$${\Phi }_z\left(t\right)=-zt+({\beta }_{m,n}-{\alpha }_m-1)log(1+t)+({\alpha }_m-1)logt.$$

(27)

Laplace’s method, uniformly for $m\le n^{1/4}$, gives the comparison between $z=r^2/2$ and the normalization point $z_0=n/2$:

$${\displaystyle \frac{U({\alpha }_m,{\beta }_{m,n},z)}{U({\alpha }_m,{\beta }_{m,n},z_0)}}\le C{n}^C{\mathrm{e}}^{Cm}exp\left\{{\displaystyle \frac{n}{4}}{I}_{-}\left({\displaystyle \frac{2z}{n}}\right)\right\},z\ge r_0^2/2.$$

(28)

For $z\ge z_0$ the maximum of (27) does not exceed the maximum at $z_0$ up to the factor $C{n}^C{\mathrm{e}}^{Cm}$; for $z<z_0$ the difference of the leading maxima is ${\displaystyle \frac{n}{4}}[-log(2z/n)-1+2z/n]$, again with an $O\left(m\right)$ and polynomial error. This proves the bound for $|{\mathcal{R}}_{m,n}|$ in (23). Differentiating U under the integral, or equivalently using ${\partial }_{z}U(a,b,z)=-aU(a+1,b+1,z)$, gives the derivative estimate in the same form.

For (24), set $z=n/2+\sigma$ with $0\le \sigma \le 1/2$. The logarithmic derivative obtained from (26) is $O\left(\right(m+1)/n)$ uniformly in this interval, and the mean-value theorem gives (24).

It remains to justify (25). On $\{Q_n>1/2\}$ one has $\left|x\right|\ge r_0={\left(2M^2\right)}^{-1/2}$. Hence (23) applies. In polar coordinates with $\rho =r^2/n$, the Gaussian radial density is

$$c_n{n}^{n/2}{\rho }^{n/2-1}{\mathrm{e}}^{-n\rho /2}\mathrm{d}\rho =C_{n}exp\left\{-{\displaystyle \frac{n}{2}}(-log\rho -1+\rho )\right\}{\rho }^{-1}\mathrm{d}\rho ,$$

(29)

where $C_n$ is of polynomial size after the total mass ${\left(2\pi \right)}^{n/2}$ is factored out, by Stirling’s formula. Squaring (23) cancels at most the large-deviation exponent in (29) for $0<\rho \le 1$ and does not create growth for $\rho \ge 1$. Since $r\ge r_0$ means $\rho \ge r_0^2/n$, the remaining radial integral is bounded by $C{n}^C{\int }_{r_0^2/n}^1{\rho }^{-1}\mathrm{d}\rho +C{n}^C\le C{n}^C$. The angular proportion of the set $\{1/2<Q_n<1\}$ is at most one, so (25) follows. The derivative term is covered by the derivative part of (23) and the same calculation. □

5. Exterior Lower Bound

For $n\ge 2$ let

$$m_n=\lfloor n^{1/8}\rfloor ,H_{m_n}\left(x\right)=Re{(x_1+i{x}_2)}^{m_n},$$

(30)

and define the exterior field

$$W_n\left(x\right)={\mathcal{R}}_{m_n,n}\left(\right|x\left|\right)H_{m_n}\left(x\right),x\in B_n^c.$$

(31)

Let

$${\alpha }_a=min(a_1,a_2)>0.$$

(32)

For large n, the set

$$\begin{array}{cc}\hfill E_n=\{x=(z,y)\in {\mathbb{R}}^2\times {\mathbb{R}}^{n-2}:& m_n\le {\left|z\right|}^2\le 2m_n,{cos}^2(m_{n}argz)\ge 1/2,\hfill \\ \hfill & n\le {\left|z\right|}^2+{\left|y\right|}^2\le n+1\}\hfill \end{array}$$

(33)

is contained in $B_n^c$, because $Q_n\left(x\right)\ge {\alpha }_a^2{\left|z\right|}^2\ge {\alpha }_a^2{m}_n>1$.

Proposition 1 

(Exterior energy lower bound). There is $c>0$ such that, for all sufficiently large n,

$${\mathcal{E}}_n(W_n;B_n^c)\ge c{\left(2\pi \right)}^{n/2}{n}^{-1/2}{\left({\displaystyle \frac{2m_n}{\mathrm{e}}}\right)}^{m_n}.$$

(34)

Proof. 

The $L^2$ part of the energy is enough. By Lemma 2(b), $|{\mathcal{R}}_{m_n,n}\left(\right|x\left|\right)|\ge 1/2$ on the one-unit shell $n\le {\left|x\right|}^2\le n+1$ for large n. Hence

$${\mathcal{E}}_n(W_n;B_n^c)\ge {\int }_{E_n}|W_n{|}^2\mathrm{d}{\gamma }_n\ge {\displaystyle \frac{1}{4}}{\int }_{E_n}{\left|H_{m_n}\left(x\right)\right|}^2\mathrm{d}{\gamma }_n.$$

(35)

For $z=(x_1,x_2)=t(cos\theta ,sin\theta )$, one has $H_m\left(x\right)=t^{m}cos\left(m\theta \right)$. The angular subset $\{{cos}^2\left(m\theta \right)\ge 1/2\}$ has positive measure independent of m. Therefore

$$\begin{array}{cc}\hfill {\int }_{\begin{array}{c}m\le {\left|z\right|}^2\le 2m\\ {cos}^2(margz)\ge 1/2\end{array}}{\left|H_m\left(z\right)\right|}^2{\mathrm{e}}^{-{\left|z\right|}^2/2}\mathrm{d}z& \ge c{\int }_{\sqrt{m}}^{\sqrt{2m}}{t}^{2m+1}{\mathrm{e}}^{-t^2/2}\mathrm{d}t\hfill \\ \hfill & =c{\int }_{m/2}^m{\left(2s\right)}^m{\mathrm{e}}^{-s}\mathrm{d}s\ge c{\left({\displaystyle \frac{2m}{\mathrm{e}}}\right)}^m.\hfill \end{array}$$

(36)

The last inequality follows by integrating over $s\in [m-1,m]$.

For every z with $m\le {\left|z\right|}^2\le 2m$ and $m=o\left(\sqrt{n}\right)$, Stirling’s formula and the local form of the ${\chi }_{n-2}^2$ density give

$${\int }_{n-{\left|z\right|}^2\le {\left|y\right|}^2\le n+1-{\left|z\right|}^2}{\mathrm{e}}^{-{\left|y\right|}^2/2}\mathrm{d}y\ge c{\left(2\pi \right)}^{(n-2)/2}{n}^{-1/2}.$$

(37)

Combining (35), (36), and (37) proves (34), after absorbing the fixed factor ${\left(2\pi \right)}^{-1}$ into c. □

6. Interior Extension and Trace-Capacity Bound

The boundary trace used for the transmission construction is

$$g_n=W_n{|}_{\partial B_n}.$$

(38)

Let $v_n$ be the unique minimizer of ${\mathcal{E}}_n(·;B_n)$ among all $H^1(B_n,{\gamma }_n)$ functions with trace $g_n$. Equivalently,

$$L_n{v}_n=v_n\mathrm{in}{B}_n,v_n=g_n\mathrm{on}\partial B_n.$$

(39)

The next estimate controls the minimizing energy from above by testing against a cutoff of the exterior field.

Proposition 2 

(Interior energy upper bound). There is a constant $C>0$ such that, for all sufficiently large n,

$${\mathcal{E}}_n(v_n;B_n)\le C{\left(2\pi \right)}^{n/2}{n}^C{C}^{m_n}.$$

(40)

Proof. 

Choose $\chi \in C^{\infty }(\mathbb{R};[0,1])$ satisfying

$$\chi \left(t\right)=0(t\le 1/2),\chi \left(t\right)=1(t\ge 3/4).$$

(41)

Define the admissible interior competitor

$${\tilde{v}}_n\left(x\right)=\chi \left(Q_n\left(x\right)\right)W_n\left(x\right),x\in B_n.$$

(42)

Since $\chi \left(1\right)=1$, the trace of ${\tilde{v}}_n$ on $\partial B_n$ is $g_n$. By the Dirichlet principle,

$${\mathcal{E}}_n(v_n;B_n)\le {\mathcal{E}}_n({\tilde{v}}_n;B_n).$$

(43)

The support of ${\tilde{v}}_n$ is contained in $\{1/2<Q_n<1\}$. On this set,

$$|x_1+i{x}_2|\le K_a,|H_{m_n}\left(x\right)|\le K_a^{m_n},|\nabla H_{m_n}\left(x\right)|\le m_n{K}_a^{m_n-1}.$$

(44)

Moreover

$$|\nabla Q_n{\left(x\right)|}^2=4{\displaystyle \sum _{i=1}^n}{a}_i^4{x}_i^2\le 4M^2{Q}_n\left(x\right)\le 4M^2.$$

(45)

Using (44), (45), and the product rule in (42), we obtain

$$|{\tilde{v}}_n{|}^2+{|\nabla {\tilde{v}}_n|}^2\le C{m}_n^2{K}_a^{2m_n}\left(|{\mathcal{R}}_{m_n,n}{\left(\right|x\left|\right)|}^2+|{\mathcal{R}}_{m_n,n}^{\prime }{\left(\right|x\left|\right)|}^2\right)11_{\{1/2<Q_n<1\}}.$$

(46)

Lemma 2(c) gives

$${\mathcal{E}}_n({\tilde{v}}_n;B_n)\le C{\left(2\pi \right)}^{n/2}{n}^C{\left(C{K}_a^2\right)}^{m_n}.$$

(47)

Renaming the base of the exponential constant proves (40). □

7. Construction and Proof of the Main Theorem

For $n\ge 2$, define

$$u_n\left(x\right)=\left\{\begin{array}{cc}{v}_n\left(x\right),\hfill & Q_n\left(x\right)<1,\hfill \\ W_n\left(x\right),\hfill & Q_n\left(x\right)>1,\hfill \end{array}\right.$$

(48)

where $W_n$ is given by (31) and $v_n$ by (39). For $n=1$, choose any finite-energy one-dimensional construction of the same type; for instance, use the degree-zero Tricomi branch on $B_1^c$ and the corresponding variational interior extension. This finite initial choice has no effect on the limit.

Proposition 3 

(Membership and finite energy). For every n, the function $u_n$ defined by (48) belongs to $S_n$ and satisfies ${\mathcal{E}}_n(u_n;{\mathbb{R}}^n)<\infty$.

Proof. 

The two pieces agree continuously on $\partial B_n$ because both have trace $g_n$. The interior piece is smooth in $B_n$ by elliptic regularity. The exterior piece is smooth on $B_n^c$ because $B_n$ contains the Euclidean ball $B(0,M^{-1})$, while the only possible singularity of the Tricomi branch in Lemma 1 is at the origin. Both pieces solve $L_{n}u=u$ in their respective domains. The interior energy is finite by Proposition 2. The exterior energy is finite by Lemma 1, applied with $\rho =M^{-1}$. □

Proof 

(Proof of Theorem 1). Proposition 3 gives $u_n\in S_n$ and finite energy. Proposition 1 and Proposition 2 give, for all sufficiently large n,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\mathcal{E}}_n(u_n;{\mathbb{R}}^n)}{{\mathcal{E}}_n(u_n;B_n)}}& \ge {\displaystyle \frac{{\mathcal{E}}_n(u_n;B_n^c)}{{\mathcal{E}}_n(u_n;B_n)}}\hfill \\ \hfill & \ge c{n}^{-C}{\left({\displaystyle \frac{2m_n}{\mathrm{e}C}}\right)}^{m_n}.\hfill \end{array}$$

(49)

Since $m_n=\lfloor n^{1/8}\rfloor$, the logarithm of the last factor is

$$m_{n}log{m}_n-O\left(m_n\right)-O(logn)\to +\infty .$$

(50)

Therefore (12) follows. □

8. Examples and Checks

The theorem applies to every positive ${\ell }^2$ sequence. The constants in the proof depend only on $a_1,a_2$ and on the finite number $M={sup}_i{a}_i$; no lower bound on the remaining axes is needed.

Example 1 

(The benchmark sequence $a_i=1/i$). Let

$$a_i={\displaystyle \frac{1}{i}},{\displaystyle \sum _{i=1}^{\infty }}{a}_i^2={\displaystyle \frac{{\pi }^2}{6}}.$$

(51)

Then $M=1$, ${\alpha }_a=1/2$, and

$$K_a=\sqrt{a_1^{-2}+a_2^{-2}}=\sqrt{5}.$$

(52)

For all n with $m_n>4$, the exterior test region $E_n$ in (33) is contained in $B_n^c$, since $Q_n\left(x\right)\ge {\left|z\right|}^2/4>1$ on $E_n$. The exterior growth is at least ${(2m_n/\mathrm{e})}^{m_n}$, while the interior trace cost is at most $n^C{C}^{m_n}$. Thus the ratio still diverges even though ${\sum }_i{a}_i^2$ is larger than one.

Example 2 

(An ${\ell }^1$-small sequence). Let

$$a_i={\displaystyle \frac{3}{4}}{2}^{-i},i\ge 1.$$

(53)

Then

$${\displaystyle \sum _{i=1}^{\infty }}{a}_i={\displaystyle \frac{3}{4}}<1,{\displaystyle \sum _{i=1}^{\infty }}{a}_i^2={\displaystyle \frac{3}{16}}<1.$$

(54)

Here $a_1=3/8$, $a_2=3/16$, and therefore

$${\alpha }_a={\displaystyle \frac{3}{16}},K_a={\left({(8/3)}^2+{(16/3)}^2\right)}^{1/2}={\displaystyle \frac{8\sqrt{5}}{3}}.$$

(55)

The proof is unchanged. Once $m_n>{(16/3)}^2$, the same exterior region $E_n$ lies outside the ellipsoid because $Q_n\left(x\right)\ge {(3/16)}^2{\left|z\right|}^2>1$ on $E_n$. This example shows explicitly that the construction is not a consequence of a large ${\ell }^1$ or mean-volume size of the ellipsoid: the sequence is small in both ${\ell }^1$ and ${\ell }^2$, but the two-coordinate harmonic spike still creates exterior energy that is super-exponential in $m_n$ relative to the interior cost.

The preceding examples also provide a simple way to reproduce the proof numerically. The two-dimensional part of the exterior lower bound is the explicit integral

$$J_m={\int }_0^{2\pi }{\int }_{\sqrt{m}}^{\sqrt{2m}}{t}^{2m}{cos}^2\left(m\theta \right){\mathrm{e}}^{-t^2/2}t\mathrm{d}t\mathrm{d}\theta =\pi {\int }_{m/2}^m{\left(2s\right)}^m{\mathrm{e}}^{-s}\mathrm{d}s,$$

(56)

which satisfies $J_m\ge c{(2m/\mathrm{e})}^m$. The companion script included with the source package evaluates the constants in Table 1 and the integral (56) for sample values of m.

9. Discussion

The construction separates three points that are important for interpreting (7). First, the equation $L_{n}u=u$ is a positive spectral equation. The finite-energy exterior solution is therefore necessarily irregular somewhere; in this construction the singularity is at the origin, and the ellipsoid contains a fixed Euclidean ball around the origin. Second, the class $S_n$ imposes continuity across $\partial B_n$ but not equality of conormal derivatives. The distributional normal jump at the interface is part of the transmission mechanism. Third, the square-summability assumption (3) is used only to guarantee a uniform inner ball and bounded coefficients; the exterior amplification itself is generated by the first two coordinates through the harmonic polynomial $Re{(x_1+i{x}_2)}^{m_n}$.

The use of a single harmonic mode also avoids possible cancellation issues in high-degree spherical-polynomial expansions. On the Gaussian shell ${\left|x\right|}^2\approx n$, the radial Tricomi multiplier is close to one, so the exterior lower bound reduces to an elementary two-dimensional integral. On the ellipsoid, the first two coordinates are uniformly bounded, so the boundary trace grows only exponentially in $m_n$. Since ${(2m_n/\mathrm{e})}^{m_n}$ grows like $exp(m_{n}log{m}_n)$, the exterior energy eventually dominates every exponential trace cost of the form $C^{m_n}$ and every fixed polynomial in n.

10. Conclusions

For every positive square-summable sequence $\left(a_i\right)$, we constructed a sequence $u_n\in S_n$ with finite Gaussian $H^1$ energy such that the total energy divided by the energy inside the ellipsoid $B_n=\{Q_n<1\}$ diverges. The exterior field is explicit: a real harmonic polynomial in the first two coordinates multiplied by the normalized Tricomi branch of the separated Ornstein–Uhlenbeck equation. The interior field is canonical: the weighted-energy minimizer with the same boundary trace. The proof gives quantitative lower and upper bounds, and the two examples show that the mechanism works both for a standard sequence $a_i=1/i$ and for an ${\ell }^1$-small geometric sequence.