A Computer-Assisted Counterexample to a Sharp L^p-Dissipativity Conjecture for the Three-Dimensional Lamé Operator

1. Introduction

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a nonempty open set. With the negative-semidefinite convention

$$\mathrm{\Delta }={\displaystyle \sum _{j=1}^3}{𝜕}_j^2,$$

define, for every $\alpha >-1$, the constant-coefficient operator

$${\mathcal{L}}_{\alpha }u:=\mathrm{\Delta }u+\alpha \nabla divu.$$

(1)

The Lamé operator written in terms of the Poisson ratio $\nu$ is

$${\mathcal{E}}_{\nu }={\mathcal{L}}_{{\alpha }_{\nu }},{\alpha }_{\nu }={\displaystyle \frac{1}{1-2\nu }}>-1.$$

(2)

Changing the sign of both the Laplacian convention and the displayed operator gives the equivalent convention used in parts of the elasticity literature.

For $1<p<\infty$, the operator is called $L^p$-dissipative when

$${\int }_{\mathrm{\Omega }}{\mathcal{E}}_{\nu }u·{\left|u\right|}^{p-2}u\mathrm{d}x\le 0\mathrm{for}\mathrm{every}u\in C_c^{\infty }(\mathrm{\Omega };{\mathbb{R}}^3),$$

(3)

with the standard interpretation at the zeros of u when $1<p<2$. The theory of $L^p$-dissipativity for scalar operators and systems was developed systematically by Cialdea and Maz’ya; see [1,2,3,4,5].

For the planar Lamé operator, the exact criterion is

$${\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}\right)}^2\le {\displaystyle \frac{2(\nu -1)(2\nu -1)}{{(3-4\nu )}^2}}.$$

(4)

In dimension three, the same inequality is necessary, while its sufficiency was posed as Problem 43 in [6]. Later work developed functional dissipativity and p-ellipticity criteria in related settings [7,8], but does not establish (4) as a sufficient three-dimensional criterion.

The purpose of this paper is to disprove the proposed sufficiency. The proof combines four ingredients: an explicit periodic vector field, an elementary convex-duality lower bound, a finite Fourier certificate checked over $\mathbb{Q}$, and a volume-versus-boundary localization argument. The computational component is verification rather than numerical approximation: after the integer Fourier arrays are fixed, every decisive quantity is evaluated exactly.

Theorem 1. 

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be nonempty and open. Set

$$\nu =0,p_{+}=18+12\sqrt{2},p_{-}=18-12\sqrt{2}.$$

Then (4) holds with equality for $p=p_{\pm }$, but the Lamé operator

$${\mathcal{E}}_{0}u=\mathrm{\Delta }u+\nabla divu$$

is neither $L^{p_{+}}$-dissipative nor $L^{p_{-}}$-dissipative on Ω. Consequently, condition (4) is not sufficient in dimension three.

The exponents $p_{+}$ and $p_{-}$ are Hölder conjugates. The transformed coefficient used below depends on p only through ${(1-2/p)}^2$, so the same periodic obstruction applies to both.

2. A Self-Contained Transformed Identity

Put

$${\beta }_p=1-{\displaystyle \frac{2}{p}}.$$

The following identity contains the exact nonlinear transformation needed in the proof.

Lemma 1 

(Transformed identity). Let $1<p<\infty$, $\alpha >-1$, and let $v\in C_c^{\infty }(\mathrm{\Omega };{\mathbb{R}}^3)$. Put $s=\left|v\right|$ and assume the following:

(i)

the fields

$$u=s^{-{\beta }_p}v,\varphi =s^{{\beta }_p}v,$$

initially defined on $\{s>0\}$, extend to compactly supported $C^{\infty }$ fields on Ω;

(ii)

s is a $C^{\infty }$ scalar function on Ω;

(iii)

the scalar function $s^{-1}v·\nabla s$, initially defined on $\{s>0\}$ and extended by zero on $\{s=0\}$, is a continuous compactly supported function ψ on Ω.

Then $\varphi ={\left|u\right|}^{p-2}u$ and

$$-{\int }_{\mathrm{\Omega }}{\mathcal{L}}_{\alpha }u·{\left|u\right|}^{p-2}u\mathrm{d}x={\mathcal{F}}_{p,\alpha }\left[v\right],$$

(5)

where the transformed form is unambiguously defined by

$${\mathcal{F}}_{p,\alpha }\left[v\right]:={\int }_{\mathrm{\Omega }}\left\{{|\nabla v|}^2+\alpha {|divv|}^2-{\beta }_p^2\left[{|\nabla s|}^2+\alpha {\psi }^2\right]\right\}\mathrm{d}x.$$

(6)

Equivalently, on $\{v\ne 0\}$ one has ${\psi }^2={\left|v\right|}^{-2}{(v·\nabla |v\left|\right)}^2$. In particular, if ${\mathcal{F}}_{p,\alpha }\left[v\right]<0$, then ${\mathcal{L}}_{\alpha }$ is not $L^p$-dissipative.

Proof. 

Write $\beta ={\beta }_p$. Since $\left|u\right|=s^{1-\beta }=s^{2/p}$, one has

$${\left|u\right|}^{p-2}u=s^{\beta }v=\varphi .$$

On the open set $\{s>0\}$,

$$\begin{array}{cc}\hfill {𝜕}_{j}u& =s^{-\beta }\left({𝜕}_{j}v-\beta s^{-1}\left({𝜕}_{j}s\right)v\right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {𝜕}_j\varphi & =s^{\beta }\left({𝜕}_{j}v+\beta s^{-1}\left({𝜕}_{j}s\right)v\right).\hfill \end{array}$$

(8)

Taking the Euclidean scalar product gives the pointwise cancellation

$${𝜕}_{j}u·{𝜕}_j\varphi ={\left|{𝜕}_{j}v\right|}^2-{\beta }^2{\left({𝜕}_{j}s\right)}^2.$$

(9)

Similarly,

$$\begin{array}{cc}\hfill divu& =s^{-\beta }\left(divv-\beta s^{-1}v·\nabla s\right),\hfill \\ \hfill div\varphi & =s^{\beta }\left(divv+\beta s^{-1}v·\nabla s\right),\hfill \end{array}$$

so

$$(divu)(div\varphi )={|divv|}^2-{\beta }^2{s}^{-2}{(v·\nabla s)}^2.$$

(10)

Integration by parts, using the compact support of u and $\varphi$, yields

$$-{\int }_{\mathrm{\Omega }}{\mathcal{L}}_{\alpha }u·\varphi \mathrm{d}x={\int }_{\mathrm{\Omega }}\left(\nabla u:\nabla \varphi +\alpha (divu)(div\varphi )\right)\mathrm{d}x.$$

Substitution of (9) and (10) proves (5) on $\{s>0\}$. By the hypotheses, both sides of the pointwise identities extend continuously across the zero set. On the interior of $\{s=0\}$ the fields u, $\varphi$, and v vanish identically, while $\nabla s=0$ and $\psi =0$, so the identities hold there as well. Thus the integral identity follows on all of $\mathrm{\Omega }$. The flat cutoffs used in Lemma 3 verify hypotheses (i)–(iii) explicitly.    □

Remark 1. 

The usual regularization $s\mapsto {(s^2+{\epsilon }^2)}^{1/2}$ extends the preceding calculation to the standard necessary-and-sufficient transformed criterion for general admissible fields; see [2,3,4]. The present proof only needs the explicit smooth fields produced in Lemma 3, for which Lemma 1 applies directly.

For $p=p_{+}$,

$${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{2}}-{\displaystyle \frac{\sqrt{2}}{3}},{\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}\right)}^2={\displaystyle \frac{2}{9}},{\beta }_p^2={\displaystyle \frac{8}{9}}.$$

(11)

At $\nu =0$, the right-hand side of (4) is also $2/9$. Hence the proposed condition is saturated exactly.

For an integrable periodic function f on ${\mathbb{T}}^3={(\mathbb{R}/2\pi \mathbb{Z})}^3$, define

$$〈f〉:={\displaystyle \frac{1}{{\left(2\pi \right)}^3}}{\int }_{{[0,2\pi ]}^3}f\left(x\right)\mathrm{d}x.$$

(12)

For a nonvanishing periodic vector field v, set

$$\begin{array}{cc}\hfill E\left[v\right]& =〈{|\nabla v|}^2+{|divv|}^2〉,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill D\left[v\right]& =〈{|\nabla |v\left|\right|}^2+{\left|v\right|}^{-2}{(v·\nabla |v\left|\right)}^2〉.\hfill \end{array}$$

(14)

At the endpoint (11), the transformed mean is

$$\overline{\mathcal{F}}\left[v\right]=E\left[v\right]-{\displaystyle \frac{8}{9}}D\left[v\right].$$

(15)

3. An Explicit Periodic Obstruction

On ${\mathbb{T}}^3$, let

$$\begin{array}{cc}\hfill v_1(x,y,z)& =\left({\displaystyle \frac{4}{3}}-cosx\right)\left({\displaystyle \frac{11}{10}}+siny\right)\left({\displaystyle \frac{11}{10}}-sinz\right),\hfill \\ \hfill v_2(x,y,z)& ={\displaystyle \frac{11}{25}}sinxcosy\left({\displaystyle \frac{123}{100}}-sinz\right),\hfill \\ \hfill v_3(x,y,z)& =-{\displaystyle \frac{11}{25}}sinx\left({\displaystyle \frac{123}{100}}+siny\right)cosz.\hfill \end{array}$$

(16)

The first component satisfies

$$v_1\ge \left({\displaystyle \frac{4}{3}}-1\right){\left({\displaystyle \frac{11}{10}}-1\right)}^2={\displaystyle \frac{1}{300}},$$

(17)

so v never vanishes.

Proposition 1. 

For the field (16),

$$E\left[v\right]={\displaystyle \frac{83564077}{12500000}}=6.68512616.$$

(18)

Proof. 

Each component of v is a finite trigonometric polynomial. Exact Laurent-polynomial multiplication and extraction of the zero Fourier coefficient give

$$〈{|\nabla v|}^2〉={\displaystyle \frac{72439343}{12500000}},〈{|divv|}^2〉={\displaystyle \frac{5562367}{6250000}}.$$

Their sum is (18). Both supplied verifiers reconstruct these quantities directly from (16), before reading or evaluating the dual certificate.    □

The term $D\left[v\right]$ is rational in trigonometric polynomials and need not be evaluated symbolically. Instead, a strict lower bound is certified by an elementary duality identity.

4. The Exact Fourier Certificate

Set

$$r^2={\left|v\right|}^2,n_j=v·{𝜕}_{j}v,m={\displaystyle \sum _{j=1}^3}{v}_j{n}_j.$$

(19)

Since ${𝜕}_{j}r=n_j/r$ and $v·\nabla r=m/r$, one has

$$D\left[v\right]=〈{\displaystyle \sum _{j=1}^3}{\displaystyle \frac{n_j^2}{r^2}}+{\displaystyle \frac{m^2}{r^4}}〉.$$

(20)

Lemma 2 

(Dual lower bound). For arbitrary real periodic functions $g_1,g_2,g_3,h$,

$$D\left[v\right]\ge \mathcal{L}(g,h),$$

(21)

where

$$\mathcal{L}(g,h)=〈2{\displaystyle \sum _{j=1}^3}{n}_j{g}_j-r^2{\displaystyle \sum _{j=1}^3}{g}_j^2+2mh-r^4{h}^2〉.$$

(22)

Proof. 

Pointwise,

$${\displaystyle \frac{n_j^2}{r^2}}-\left(2n_j{g}_j-r^2{g}_j^2\right)={\left({\displaystyle \frac{n_j}{r}}-r{g}_j\right)}^2\ge 0$$

and

$${\displaystyle \frac{m^2}{r^4}}-\left(2mh-r^4{h}^2\right)={\left({\displaystyle \frac{m}{r^2}}-r^{2}h\right)}^2\ge 0.$$

Summing and averaging proves the claim.    □

The certificate consists of four real trigonometric polynomials

$$g_j\left(x\right)={\displaystyle \frac{1}{Q}}\sum _{k\in {[-d,d]}^3\cap {\mathbb{Z}}^3}{c}_k^{\left(j\right)}{e}^{ik·x},h\left(x\right)={\displaystyle \frac{1}{Q}}\sum _{k\in {[-d,d]}^3\cap {\mathbb{Z}}^3}{c}_k^{\left(4\right)}{e}^{ik·x},$$

(23)

where

$$d=22,Q=2^{16},c_k^{\left(\mathcal{l}\right)}\in \mathbb{Z}\left[i\right],c_{-k}^{\left(\mathcal{l}\right)}=\overline{c_k^{\left(\mathcal{l}\right)}}.$$

The conjugate symmetry makes the four functions real-valued. The fixed data are stored in supplement/certificate.npz, with a compressed readable CSV export.

Table 1. Certificate metadata. Here “maximum coefficient” is the maximum absolute value of either the real or imaginary integer part.

Polynomial	Nonzero Fourier coefficients	Maximum coefficient
$g_1$	$54164$	$25617$
$g_2$	$51992$	$38292$
$g_3$	$51992$	$38292$
h	$88988$	$248025$

Table 1. Certificate metadata. Here “maximum coefficient” is the maximum absolute value of either the real or imaginary integer part.

Polynomial	Nonzero Fourier coefficients	Maximum coefficient
$g_1$	$54164$	$25617$
$g_2$	$51992$	$38292$
$g_3$	$51992$	$38292$
h	$88988$	$248025$

Each coefficient array has shape ${45}^3$, corresponding to the Fourier cube ${[-22,22]}^3$.

Finite verification algorithm.

For clarity, the proof-critical computation can be specified independently of any programming language. A finite trigonometric polynomial $A\left(x\right)={\sum }_k{a}_k{e}^{ik·x}$ is represented by the finitely supported map $k\mapsto a_k\in \mathbb{Q}\left(i\right)$. The verifier performs the following exact operations:

(1)

encode the three components in (16); differentiation multiplies the kth coefficient by $i{k}_j$, and multiplication is finite discrete convolution;

(2)

construct ${𝜕}_j{v}_i$, $r^2$, $r^4$, $n_j$, m, and ${|\nabla v|}^2+{|divv|}^2$ over $\mathbb{Q}\left(i\right)$;

(3)

read the four Gaussian-integer arrays, verify the fixed SHA-256 digest, required keys, degree, denominator, dimensions, and conjugate symmetry, and divide all coefficients by Q;

(4)

use

$$〈AB〉=\sum _k{a}_k{b}_{-k},〈A{C}^2〉=\sum _k{a}_k\sum _{\mathcal{l}}{c}_{\mathcal{l}}{c}_{-k-\mathcal{l}},$$

where every sum is finite, to evaluate the linear and weighted-square terms in (22);

(5)

combine those rational values according to (22), verify that the imaginary part is zero, and compare the result and the gap with the exact fractions in (24)–(25).

No analytic truncation or floating-point inequality is involved in these steps.

Proposition 2 

(Exact certificate evaluation). For the trigonometric polynomials encoded by the supplementary certificate,

$$\mathcal{L}(g,h)={\displaystyle \frac{364124663688583479446684197}{48318382080000000000000000}}.$$

(24)

Moreover,

$$\mathcal{L}(g,h)-{\displaystyle \frac{9}{8}}E\left[v\right]={\displaystyle \frac{733373630214865046684197}{48318382080000000000000000}}>0.$$

(25)

Proof. 

The accelerated verifier implements the preceding finite algorithm with sparse Laurent polynomials over $\mathbb{Q}\left(i\right)$. Final accumulation uses arbitrary-precision rational arithmetic. Its convolution stage uses signed 64-bit integer arrays only after an explicit frequency-by-frequency bound proves that every product and accumulated sum lies strictly inside the signed 64-bit range. The reference verifier is a standalone implementation: it duplicates the exact construction of the field and all polynomial operations, reads the certificate independently, and performs its convolution stage with Python arbitrary-precision integers stored in object arrays. It imports no code, constants, or expected values from the accelerated verifier.

Both verifiers contain their own hard-coded expected digest and exact target fractions. They terminate with a nonzero exit status if any checksum, key, dimension, symmetry, overflow, imaginary-part, or value check fails. The certificate file has SHA-256 digest

ee4faf41ff53d3d87ac00da6e7cc814051c0f7aaa38ce90d9134a8ee26b84d16

The resulting exact output is (24)–(25). □

Corollary 1. 

For the field (16),

$$E\left[v\right]-{\displaystyle \frac{8}{9}}D\left[v\right]\le -{\displaystyle \frac{733373630214865046684197}{54358179840000000000000000}}<0.$$

(26)

Proof. 

By Lemma 2 and Proposition 2, $D\left[v\right]\ge \mathcal{L}(g,h)>9E\left[v\right]/8$. Multiplication by $-8/9$ gives (26). □

Table 2. Certified quantities. Decimal values are included only for readability; the proof uses the exact fractions.

Quantity	Exact source	Decimal value
$E\left[v\right]$	(18)	$6.6851261600$
$9E\left[v\right]/8$	(18)	$7.5207669300$
$\mathcal{L}(g,h)$	(24)	$7.5359448726$
Certificate gap	(25)	$0.0151779426$
Upper bound for $E\left[v\right]-8D\left[v\right]/9$	(26)	$-0.0134915045$

Table 2. Certified quantities. Decimal values are included only for readability; the proof uses the exact fractions.

Quantity	Exact source	Decimal value
$E\left[v\right]$	(18)	$6.6851261600$
$9E\left[v\right]/8$	(18)	$7.5207669300$
$\mathcal{L}(g,h)$	(24)	$7.5359448726$
Certificate gap	(25)	$0.0151779426$
Upper bound for $E\left[v\right]-8D\left[v\right]/9$	(26)	$-0.0134915045$

5. Quantitative Localization to Compact Support

For a sufficiently regular vector field z, define the transformed density

$${\mathcal{F}}_c\left[z\right]:={|\nabla z|}^2+{|divz|}^2-c\left({|\nabla |z\left|\right|}^2+\mathrm{\Psi }{\left[z\right]}^2\right),\mathrm{\Psi }\left[z\right]:={\left|z\right|}^{-1}z·\nabla \left|z\right|,$$

(27)

whenever $\mathrm{\Psi }\left[z\right]$, initially defined on $\{z\ne 0\}$, has a continuous extension; that extension is used in (27).

Lemma 3 

(Periodic localization). Let $v\in C^{\infty }({\mathbb{R}}^3;{\mathbb{R}}^3)$ be $2\pi$-periodic and satisfy ${inf}_{{\mathbb{R}}^3}\left|v\right|>0$. Suppose, for some $c\ge 0$, that

$$\mu :=〈{\mathcal{F}}_c\left[v\right]〉<0.$$

Then there exists $w\in C_c^{\infty }({\mathbb{R}}^3;{\mathbb{R}}^3)$ such that

$${\int }_{{\mathbb{R}}^3}{\mathcal{F}}_c\left[w\right]\mathrm{d}x<0.$$

Moreover, for every $1<p<\infty$, the fields ${\left|w\right|}^{-(1-2/p)}w$ and ${\left|w\right|}^{1-2/p}w$ are smooth whenever w is constructed as below.

Proof. 

Choose a standard nondecreasing flat step $\rho \in C^{\infty }\left(\mathbb{R}\right)$ such that $0\le \rho \le 1$, $\rho =0$ on $(-\infty ,-1]$, $\rho =1$ on $[0,\infty )$, and every positive power ${\rho }^{\gamma }$ is smooth. Such a function is obtained from $\eta \left(t\right)=0$ for $t\le 0$ and $\eta \left(t\right)=e^{-1/t}$ for $t>0$ by the usual quotient construction. For $N\in \mathbb{N}$, set

$${\theta }_N\left(t\right)=\rho \left(t\right)\rho (2\pi N-t),{\chi }_N\left(x\right)={\displaystyle \prod _{j=1}^3}{\theta }_N\left(x_j\right),w_N={\chi }_{N}v.$$

Then ${\chi }_N=1$ on ${[0,2\pi N]}^3$, $supp{\chi }_N\subset {[-1,2\pi N+1]}^3$, and

$$K:=\underset{N\ge 1}{sup}{\parallel \nabla {\chi }_N\parallel }_{L^{\infty }}<\infty .$$

(28)

Because v is nonvanishing, ${\chi }_N\ge 0$, and ${\chi }_N$ is flat at its zero set, $w_N\in C_c^{\infty }$, $|w_N|={\chi }_N\left|v\right|\in C_c^{\infty }$, and every positive power of ${\chi }_N$ is smooth.

Write $r=\left|v\right|$, $q=\nabla {\chi }_N$, and

$$a={\displaystyle \frac{v·\nabla r}{r}}.$$

On $\{{\chi }_N>0\}$,

$$\begin{array}{cc}\hfill \nabla w_N& ={\chi }_N\nabla v+v\otimes q,\hfill \\ \hfill div{w}_N& ={\chi }_{N}divv+v·q,\hfill \\ \hfill \nabla |w_N|& ={\chi }_N\nabla r+rq,\hfill \\ \hfill |w_N{|}^{-2}(w_N·\nabla |w_N{\left|\right)}^2& ={({\chi }_{N}a+v·q)}^2.\hfill \end{array}$$

The quantity $\mathrm{\Psi }\left[w_N\right]={\chi }_{N}a+v·q$ extends smoothly by zero across $\{{\chi }_N=0\}$, because all derivatives of ${\chi }_N$ vanish there. Consequently the last displayed square is precisely the extended value of $\mathrm{\Psi }{\left[w_N\right]}^2$. Expanding the four squares gives

$${\mathcal{F}}_c\left[w_N\right]={\chi }_N^2{\mathcal{F}}_c\left[v\right]+R_N,$$

(29)

where, with

$$M_0={\parallel v\parallel }_{L^{\infty }\left({[0,2\pi ]}^3\right)},M_1={\parallel \nabla v\parallel }_{L^{\infty }\left({[0,2\pi ]}^3\right)},$$

one has the explicit pointwise estimate

$$|R_N|\le 2M_0{M}_1(1+\sqrt{3}+2c)\left|q\right|+2(1+c)M_0^2{\left|q\right|}^2.$$

(30)

Indeed, $|\nabla r|\le M_1$, $\left|a\right|\le M_1$, and $|divv|\le \sqrt{3}{M}_1$, after which (30) follows term by term from the expanded squares.

Let

$${\mathrm{\Lambda }}_N={[-1,2\pi N+1]}^3\setminus {[0,2\pi N]}^3.$$

The remainder and the difference between ${\chi }_N^2{\mathcal{F}}_c\left[v\right]$ and the core density are supported in ${\mathrm{\Lambda }}_N$. Since

$$|{\mathrm{\Lambda }}_N{|\le (2\pi N+2)}^3-{\left(2\pi N\right)}^3=24{\pi }^2{N}^2+24\pi N+8,$$

(31)

and since (28) and (30) are uniform in N, one may take

$$C_{*}:={\parallel {\mathcal{F}}_c\left[v\right]\parallel }_{L^{\infty }\left({[0,2\pi ]}^3\right)}+2M_0{M}_1(1+\sqrt{3}+2c)K+2(1+c)M_0^2{K}^2.$$

(32)

Then $C_{*}$ is finite, independent of N, and

$$\begin{array}{cc}\hfill \left|{\int }_{{\mathrm{\Lambda }}_N}\left({\chi }_N^2{\mathcal{F}}_c\left[v\right]+R_N\right)\mathrm{d}x\right|& \le C_{*}\left(24{\pi }^2{N}^2+24\pi N+8\right).\hfill \end{array}$$

(33)

On the core ${[0,2\pi N]}^3$, periodicity gives exactly

$${\int }_{{[0,2\pi N]}^3}{\mathcal{F}}_c\left[v\right]\mathrm{d}x=N^3{\left(2\pi \right)}^3\mu .$$

Combining this identity with (29) and (33) yields

$${\int }_{{\mathbb{R}}^3}{\mathcal{F}}_c\left[w_N\right]\mathrm{d}x=N^3{\left(2\pi \right)}^3\mu +O\left(N^2\right).$$

(34)

In particular, the integral is negative for every integer N satisfying the explicit inequality

$$N^3{\left(2\pi \right)}^3\left|\mu \right|>C_{*}\left(24{\pi }^2{N}^2+24\pi N+8\right).$$

(35)

Such integers exist because the left-hand side is cubic and the right-hand side is quadratic.

Finally, if $\beta =1-2/p$, then on $\{{\chi }_N>0\}$

$$|w_N{|}^{-\beta }{w}_N={\chi }_N^{1-\beta }{r}^{-\beta }v={\chi }_N^{2/p}{r}^{-\beta }v,$$

while

$$|w_N{|}^{\beta }{w}_N={\chi }_N^{1+\beta }{r}^{\beta }v={\chi }_N^{2-2/p}{r}^{\beta }v.$$

Both exponents of ${\chi }_N$ are positive, so flatness gives smooth compactly supported extensions. This proves the final assertion. □

Proof of Theorem 1. 

For $p=p_{+}$, identities (11) show that the proposed condition (4) holds with equality and that ${\beta }_p^2=8/9$. By Corollary 1 and Lemma 3, there exists $w\in C_c^{\infty }({\mathbb{R}}^3;{\mathbb{R}}^3)$ such that

$${\mathcal{F}}_{p,1}\left[w\right]={\int }_{{\mathbb{R}}^3}{\mathcal{F}}_{8/9}\left[w\right]\mathrm{d}x<0.$$

The last assertion of Lemma 3 implies that

$$u={\left|w\right|}^{-{\beta }_p}w\in C_c^{\infty }({\mathbb{R}}^3;{\mathbb{R}}^3).$$

Therefore Lemma 1 gives

$${\int }_{{\mathbb{R}}^3}(\mathrm{\Delta }u+\nabla divu)·{\left|u\right|}^{p-2}u\mathrm{d}x=-{\mathcal{F}}_{p,1}\left[w\right]>0,$$

which contradicts $L^{p_{+}}$-dissipativity.

To place the test field in an arbitrary nonempty open set $\mathrm{\Omega }$, choose a ball $B(x_0,R)⋐\mathrm{\Omega }$. For sufficiently small $s>0$, the rescaled field

$$w_s\left(x\right)=w\left({\displaystyle \frac{x-x_0}{s}}\right)$$

has support in $B(x_0,R)$. Every term in the transformed form scales by the same factor $s^{3-2}=s$, so the strict sign is unchanged.

Finally, ${(1-2/p_{-})}^2=8/9$ as well. Repeating the same transformation with $p=p_{-}$ and the same field $w_s$ proves failure of $L^{p_{-}}$-dissipativity. □

6. Reproducibility and Scope of the Computer-Assisted Step

The analytic proof consists of Lemmas 1–3. The computer is used only to evaluate the finite Fourier expression in Proposition 2; background on rigorous computer-assisted arguments in partial differential equations can be found in [9]. The supplementary package contains:

(i)

certificate.npz, the fixed Gaussian-integer coefficient arrays;

(ii)

verify_certificate.py, the accelerated exact verifier with explicit failure checks;

(iii)

a standalone arbitrary-precision integer reference verifier,

verify_certificate_reference.py;

(iv)

a convenience driver that checks file digests and runs both verifiers,

verify_all.py;

(v)

certificate_coefficients.csv.gz, a readable coefficient export;

(vi)

export_certificate_csv.py, the export script;

(vii)

clean output logs for both verification scripts;

(viii)

a minimal Python dependency file, explicit licenses, repository metadata, and SHA-256 manifests for the supplementary files and full source package.

Neither verifier uses floating-point quadrature, interval enclosures, or an unproved Fourier truncation estimate. The calibrating polynomials are finite, and the lower bound follows from the completed-square identity in Lemma 2. Floating-point computations were used only to discover a useful certificate; they play no role after the integer coefficients are fixed. The complete versioned manuscript and exact reproducibility package are permanently archived on Zenodo at 10.5281/zenodo.20722485.

Remark 2. 

A direct high-resolution torus quadrature gives approximately $D\left[v\right]\approx 7.561440506$ and $E\left[v\right]-8D\left[v\right]/9\approx -0.03615429$. These values are not used in the proof; the weaker exact bound (26) is sufficient.

7. Concluding Remarks

The counterexample shows that the sharp planar Lamé dissipativity condition does not persist as a sufficient condition in dimension three. The failure occurs at the simple elasticity parameter $\nu =0$ and at the endpoint of the proposed range. The construction also indicates a mechanism behind the dimensional distinction: in three dimensions, angular variations of a vector field can produce a negative integral interaction not detected by the planar algebraic condition.

Two questions remain natural. First, one may seek a sharp replacement for (4) in dimension three. Second, because the certified inequality is strict, non-dissipativity is expected to persist under small perturbations of $\nu$ and p; obtaining an explicit optimized neighborhood requires a separate quantitative analysis.