Homothetic Hodge$-$de Rham Theory and a Geometric Regularization of Elliptic Boundary Value Problems

Homothetic Ho dge–de Rham Theory and a Geometric Regularization of Elliptic Boundary V alue Problems F ereidoun Sab etghadam Me chanic al Engine ering F aculty, Scienc e and R ese ar ch Br anch, IA U, T ehr an, Ir an Email: fsab et@srbiau.ac.ir Abstract W e in tro duce a homothetic extension of classical W eyl–in tegrable geometry by generalizing the usual linear gauge transformations to affine homothetic transformations cen tered at a distinguished harmonic, scale–in v arian t form α d . After re–linearizing these affine gauge transformations via a suitable shift of v ariables, w e obtain a t wisted exterior calculus that is structurally equiv alen t to the Witten deformation of the de Rham complex. On this basis w e dev elop a corresponding homothetic Hodge theory: we define a t wisted adjoin t and homothetic Laplacian, and pro ve a homothetic Ho dge decomposition theorem on compact Riemannian manifolds. In the context of partial differen tial equations, we show that the scalar homothetic Laplacian pro vides a rigorous diffuse–interface (v olume–p enalization) representation of elliptic b oundary v alue problems. Mo deling the W eyl scale field as a fixed distribution lo calized near a h yp ersurface, the resulting low er–order geometric terms form a p enalt y lay er that enforces Diric hlet, Neumann, or Cauc hy data within a single geometric equation. This form ulation yields consisten t weak solutions even in the presence of classically incompatible Cauch y data. As an application, w e construct a non–singular model for p oin t sources in elliptic field equations, which preserv es the correct Coulombian far–field while removing the core singularit y and yielding finite field energy . K eywor ds: 2020 MSC: 53C07, 58A12, 58J05, 35J25 W eyl-in tegrable geometry, Witten deformation, weigh ted Ho dge theory, twisted de Rham complex, homothetic Laplacian, elliptic b oundary v alue problems 1. In tro duction Classical W eyl geometry w as originally prop osed as an elegan t attempt to unify gra vitation and electromagnetism b y introducing lo cal scale in v ariance [ 35 ]. While its original ph ysical in terpretation was largely sup erseded, W eyl-integrable geometry has since found profound applications in mo dern ph ysics, particularly in scalar-tensor theories of gra vity [ 31 ], conformal manifolds, and symmetry reduction in gauge theories [18, 32]. Bey ond its role in scalar–tensor gra vity and conformal geometry , W eyl–in tegrable struc- tures naturally accommo date generalized notions of scale symmetry that are w ell suited to geometric form ulations of differential op erators and field equations. Motiv ated by this p erspective, w e introduce in this work a homothetic extension of W eyl–in tegrable geom- etry . Instead of the usual linear scaling of a p -form, α 7→ e wσ α , w e p ostulate an affine transformation c haracterized by a distinguished, harmonic, scale-in v arian t p -form, α d , whic h acts as a “homothety center,” i.e., a fixed p oin t of the affine scaling. By re-linearizing this transformation via shifting the v ariables, we naturally induce a twist in the exterior calculus. In terestingly , this twisted differen tial conceptually and structurally coincides with the exact Witten deformation of the de Rham complex [ 36 , 30 ]. This geometric structure provides the mathematical foundation for our first main result: the construction of a homothetic Ho dge–de Rham theory , complete with a corresp onding twisted adjoint, a homothetic Laplacian, and a mo dified Ho dge decomp osition theorem on compact Riemannian manifolds. Bey ond its intrinsic geometric interest, this homothetic Ho dge theory offers a nov el and rigorous mechanism for tac kling classical elliptic b oundary v alue problems. T raditional approac hes to imp osing b oundary conditions typically rely on strict domain truncation and trace op erators. Here, we demonstrate that the scalar homothetic Laplace equation can act as a natural volume-penalization or diffuse-in terface metho d [ 25 ]. By considering the W eyl p oten tial λ as a lo calized, fixed distribution concen trated along a h yp ersurface, the lo wer-order terms of the homothetic op erator act as a thin p enalt y lay er. W e show that this single geometric equation can seamlessly enforce Cauc hy , Dirichlet, or Neumann b oundary data, yielding mathematically consistent w eak transmission solutions even when Cauch y data are classically incompatible. A compelling mathematical ph ysics application is the regularization of singular p oin t sources [ 4 , 17 ]. By framing the p oint source within our homothetic Laplace equation and enforcing b oundary data on a regularized hollo w-sphere interface, we obtain a non-singular mo del that decouples the interior and exterior fields. The resulting p otential maintains the correct Coulombic b ehavior in the far field while eliminating the singularity at the origin, yielding a finite total field energy . The pap er is organized as follo ws. Section 2 reviews classical W eyl-in tegrable geometry and in tro duces our homothetic extension. In Section 3, we define the t wisted differen tial op erators and establish their relation to Witten deformations. Section 4 develops the homothetic de Rham complex, follow ed by the form ulation of the homothetic Hodge theory and its decomp osition theorem in Section 5. In Section 6, we derive the explicit scalar homothetic Laplace equation. Section 7 explores the PDE applications, demonstrating how lo calized scale fields enforce interface conditions and resolving the p oin t-source singularity . Finally , Section 8 concludes the pap er, and App endix A pro vides the ODE analysis gov erning the lo calization of the scale field. 2. A homothetic extension of W eyl-Integrable Geometry The form ulation is provided consistent with our previous formulations [ 32 ], whic h in turn, is consisten t with the ma jor scalar-tensor theories of gra vity (see e.g., [31]). 2.1. Classic al W eyl-inte gr able structur e Let M b e a smo oth d -dimensional manifold. A W eyl structur e [ 35 ] on M consists of a conformal class of (pseudo)-Riemannian metrics [ g ] together with a one-form ϕ ∈ Ω 1 ( M ) , 2 and a torsion-free connection ˆ ∇ satisfying the non-metricity condition ˆ ∇ g = − 2 ϕ ⊗ g . (1) Giv en ( M , [ g ] , ϕ ) , a W eyl gauge (scale) transformation is sp ecified by g 7− → e − 2 σ g , ϕ 7− → ϕ + dσ, (2) in whic h σ ∈ C ∞ ( M ) . Under (2) , the connection ˆ ∇ remains unc hanged, and (1) is preserv ed. The W eyl structure is called inte gr able if dϕ = 0 [ 33 , 31 ]. Therefore, it is exact globally if ϕ = dλ for some smo oth function λ ∈ C ∞ ( M ) . Along with the gauge freedom (2), one usually consider the transformation α 7→ e − wσ α, (3) for a differential p -form α ∈ Ω p ( M ) [ 18 ], in which w is called the W eyl weight and in physics it usually determines suc h that a physical theory b e scale in v arian t. In the globally exact case ϕ = dλ , choosing σ = − λ in (2) pro duces the Einstein (Riemann) gauge , in whic h ϕ E = 0 , g E = e 2 λ g . (4) In this gauge, the W eyl connection ˆ ∇ reduces to the Levi–Civita connection of the metric g E . 2.2. Homothetic extension: affine sc aling with a fixe d p oint Here w e propose our extension to the ab o v e classical W eyl integrable geometry b y generalization of the linear transformation (3) to a homothetic one. Fix p ∈ { 0 , 1 , . . . , d } . Assume there exists a distinguished form α d ∈ Ω p ( M ) that is sc ale invariant under (2) . W e in terpret α d as a “homothety center” (fixed p oin t). Let α E ∈ Ω p ( M ) denote a p -form field in the Einstein gauge (i.e., for ϕ = ϕ E = 0 ). Giv en a W eyl w eigh t w ∈ R , first we define the homothetic al ly sc ale d form α := e − wλ α E + (1 − e − wλ ) α d = e − wλ ( α E − α d ) + α d . (5) Then, w e p ostulate the gauge transformation g E 7→ g := e − 2 λ g E , ϕ = 0 7→ 0 + dλ, α E 7→ α. (6) Therefore, for the W eyl-integrable manifold ( M , [ g ] , ϕ ) , the W eyl gauge (scale) freedom (2) – (3) is c hanged to g 7→ g := e − 2 σ g , ϕ 7→ ϕ + dσ, α E 7→ e − w ( λ + σ ) α E + (1 − e − w ( λ + σ ) ) α d . (7) W e call this setting the homothetic extension of the W eyl in tegrable geometry , and w e shall sho w that a consistent Ho dge-de Rham theory can b e constructed on it. Remark 2.1. 3 1. F or the special case α d = 0 , the con ven tional W eyl gauge freedom (2) - (3) is retriev ed for ( M , [ g ] , ϕ ) g 7→ g := e − 2 σ g , ϕ 7→ ϕ + dσ, α 7→ e − wσ α. (8) Ho wev er, for general α d , (6) in tro duces a homothety on the affine space α d + Ω p ( M ) , with the homothet y center α d and scale factor e − wλ . 2. On the ab o ve setting, the metric tensor is a 2-form with zero homothet y cen ter g d = 0 , and the W eyl w eight w = 2 . 3. W e should emphasize that ˆ ∇ is still in v arian t under the scale freedom (7). Ho wev er, since transformation (5) is not linear, but affine, the transformation of the differen tial forms in (7) is not linear as well. So, we first re-linearize it. 2.3. Line arization by shifting T o reco ver a linear nature of the transformations (under the gauge freedom), we define the shifted v ariables β E := α E − α d , β := α − α d . (9) Then (5) b ecomes linear again β = e − wλ β E . (10) Remark 2.2 (Relation to comp ensator/dressing) . After the shift β := α − α d (and similarly β E := α E − α d ), the affine homothety ab out the fixed p oin t α d b ecomes a purely multiplicativ e W eyl scaling, β = e − wλ β E . This is structurally the same mec hanism as the standard comp ensator (dilaton) / Stuec kelberg realization of lo cal scale inv ariance, where one forms W eyl-co v ariant “dressed” v ariables by multiplying by an exp onen tial of the scale field; see e.g. [ 12 ]. It also fits the general viewpoint of the dressing field metho d (DFM) for gauge- symmetry reduction, which has b een applied in conformal/W eyl Cartan geometry (including the extraction of a dilaton from tractor data) [3, 11]. The linearization may b e ac hieved via adding the dimension, as w e hav e done b efore [ 32 ]. The resulting form ulations are equiv alen t, but the set up of [ 32 ] is more clear for the ph ysics applications (b ecause it separates clearly the group of symmetry and the space of solutions). The present linearization b y shifting stresses more the mathematical foundations, and therefore is more suitable for the PDE application. No w, we can construct differential op erators acting on suc h spaces. 3. T wisted Op erators W e shall construct a Ho dge-de Rham theory based on the ab o ve geometry . T o this end, w e define some twisted op erators. Similar op erators w ere used widely in the literature for the other purp oses [36, 27, 14]. W e fix a background (pseudo)-Riemannian metric g E , in the Einstein frame, and t wist the exterior calculus b y the scalar field λ . Define the twisted exterior deriv ativ e ˜ d := e − wλ d e wλ = d + w d λ ∧ . (11) 4 It satisfies the iden tity ˜ d  e − wλ β E  = e − wλ d β E , (12) and, since d ϕ = 0 with ϕ = d λ , it is nilp oten t: ˜ d 2 = 0 . (13) Let δ and ∆ = d δ + δ d denote the usual Ho dge co differen tial and Ho dge Laplacian for the fixed metric g E . W e use the w eighted pairing ⟨ η , ξ ⟩ λ,w := Z M e 2 wλ η ∧ ∗ ξ , (14) and define the corresp onding twisted adjoint and Laplacian by conjugation: ˜ δ := e − wλ δ e wλ , ˜ ∆ := ˜ d ˜ δ + ˜ δ ˜ d = e − wλ ∆ e wλ . (15) Conjugation immediately implies ˜ δ 2 = 0 (16) and yields a self-adjoin t elliptic op erator ˜ ∆ on p -forms under standard h yp otheses (made precise later). These three t wisted op erators are enough for our final goals. Ho wev er, b efore dev eloping the theory further, lets sp ecify their relations with the other theories. 3.1. R elation to the other works Set f := w λ, S ( η ) := e f η . Then our t wisted differential ˜ d = e − f d e f = S − 1 dS is precisely the Witten deformation (exact t wist) of the de Rham differential [ 36 ]. More generally , one ma y twist by a closed 1 -form θ via d θ := d + θ ∧ , which yields the Morse– No viko v/Lic hnerowicz cohomology in the non-exact case [ 27 , 10 ]. (W e restrict throughout to the globally exact W eyl-integrable case ϕ = dλ .) Distinct, genuinely c onformal ly invariant “detour”/Branson–Go ver-t yp e de Rham complexes also exist in conformal geometry , but they are differen t ob jects from the presen t first-order twist [5, 14]. Con ven tion (adjoin t vs. inner pro duct). There are tw o common wa ys to incorp orate the Witten t wist at the Ho dge setting. In this pap er we enco de the twist in the Hilb ert-space structure by using the w eighted pairing (14) , so that the formal adjoint and Laplacian satisfy the clean conjugation iden tities ˜ δ = e − f δ e f = S − 1 δ S, ˜ ∆ = ˜ d ˜ δ + ˜ δ ˜ d = e − f ∆ e f = S − 1 ∆ S. Man y references instead keep the unw eigh ted L 2 pairing and work with the corresp onding adjoin t d ∗ f = e f δ e − f . 5 4. Homothetic de Rham Complex 4.1. The homothetic (twiste d) c omplex Fix w ∈ R and a smo oth function λ ∈ C ∞ ( M ) suc h that ϕ = d λ . Let ˜ d = e − wλ d e wλ b e the t wisted differential defined in (11) ; equiv alen tly , ˜ d is the exact (Witten-type) twist of d [36]. Since ˜ d 2 = 0 (cf. (13)), the sequence 0 − → Ω 0 ( M ) ˜ d − − → Ω 1 ( M ) ˜ d − − → · · · ˜ d − − → Ω d ( M ) − → 0 (17) is a co c hain complex, whic h we call the homothetic de Rham c omplex . Definition 4.1 (Homothetic de Rham cohomology) . F or p = 0 , 1 , . . . , d , define H p hom ( M ; λ, w ) := k er  ˜ d : Ω p ( M ) → Ω p +1 ( M )  im  ˜ d : Ω p − 1 ( M ) → Ω p ( M )  . (18) 4.2. Chain isomorphism and r elation to de Rham c ohomolo gy Define the m ultiplication op erator S : Ω p ( M ) → Ω p ( M ) , S ( η ) := e wλ η . (19) Then ˜ d = S − 1 d S and hence S is a c hain map b et ween ˜ d and d : S ◦ ˜ d = d ◦ S. (20) In particular, S induces an isomorphism on cohomology whenev er λ is globally defined. Prop osition 4.2 (Cohomological equiv alence) . A ssume λ ∈ C ∞ ( M ) is glob al ly define d. Then multiplic ation by e wλ induc es a c anonic al isomorphism H p hom ( M ; λ, w ) ∼ = H p dR ( M ) , p = 0 , 1 , . . . , d. (21) Pr o of. Let η ∈ Ω p ( M ) . Since ˜ d = S − 1 d S , w e ha ve ˜ dη = 0 if and only if d( S η ) = 0 , and η = ˜ dξ if and only if S η = d( S ξ ) . Thus S maps k er ˜ d on to k er d and im ˜ d on to im d , yielding (21). 4.3. Homothetic close d and exact forms Recall the affine homothet y α = e − wλ ( α E − α d ) + α d and the shifted field β = α − α d = e − wλ β E . The t wisted differen tial ˜ d is designed so that ˜ dβ transforms with the same w eight as d β E in the Einstein gauge: ˜ dβ = ˜ d  e − wλ β E  = e − wλ d β E . (22) A ccordingly , in the homothetic gauge picture one naturally regards β as homothetic al ly close d if ˜ dβ = 0 and homothetic al ly exact if β = ˜ dγ for some γ . 6 5. Homothetic Ho dge Theory 5.1. W eighte d p airing and twiste d adjoint Fix a background (pseudo)-Riemannian metric g E on M . Let ∗ denote the corresp onding Ho dge star, and let δ b e the usual Ho dge co differen tial (the L 2 -adjoin t of d with resp ect to R η ∧ ∗ ξ ).Throughout this section, whenever we app eal to adjointness/in tegration b y parts, w e either assume M is compact without b oundary , or we w ork with compactly supp orted forms, so that the relev an t boundary terms v anish and the pairings b elo w are finite. F or λ ∈ C ∞ ( M ) and w eight w ∈ R , define the w eighted inner pro duct ⟨ η , ξ ⟩ λ,w := Z M e 2 wλ η ∧ ∗ ξ , η , ξ ∈ Ω p ( M ) . (14 revisited) On noncompact M this is understo od on Ω p c ( M ) , or on a w eighted L 2 -domain where the in tegral conv erges. Prop osition 5.1 (T wisted adjoin t by conjugation) . With r esp e ct to (14) , the formal adjoint of ˜ d = e − wλ d e wλ is ˜ δ := e − wλ δ e wλ . (23) A nd we have ˜ δ 2 = 0 . Pr o of. Let η ∈ Ω p ( M ) and ξ ∈ Ω p +1 ( M ) with compact supp ort (or assume h yp otheses ensuring v anishing of b oundary terms, e.g. M compact without b oundary). Set ˆ η := e wλ η and ˆ ξ := e wλ ξ . Then ⟨ ˜ dη , ξ ⟩ λ,w = Z M e 2 wλ ( e − wλ d ˆ η ) ∧ ∗ ξ = Z M (d ˆ η ) ∧ ∗ ˆ ξ . By definition of δ as the adjoint of d (for the unw eigh ted pairing), this equals Z M ˆ η ∧ ∗ ( δ ˆ ξ ) = Z M e 2 wλ η ∧ ∗  e − wλ δ ( e wλ ξ )  = ⟨ η , ˜ δ ξ ⟩ λ,w . Finally , ˜ δ 2 = 0 follo ws from δ 2 = 0 and the conjugation (23). 5.2. Homothetic L aplacian and c onjugation Define the homothetic (Witten form) Laplacian b y ˜ ∆ := ˜ d ˜ δ + ˜ δ ˜ d. (24) Let ∆ := d δ + δ d b e the usual Ho dge Laplacian for the fixed metric g E . Prop osition 5.2 (Conjugation identit y) . L et S ( η ) := e wλ η . Then ˜ d = S − 1 d S, ˜ δ = S − 1 δ S, ˜ ∆ = S − 1 ∆ S. (25) Pr o of. The first t wo identities are the definitions of ˜ d and ˜ δ . F or the Laplacian, ˜ ∆ = ( S − 1 d S )( S − 1 δ S ) + ( S − 1 δ S )( S − 1 d S ) = S − 1 (d δ + δ d) S = S − 1 ∆ S. Remark 5.3 (Ellipticit y and low er-order terms) . Since ˜ ∆ is conjugate to ∆ by the p oint wise m ultiplication S = e wλ , it has the same principal sym b ol as ∆ and hence is elliptic whenever ∆ is elliptic (e.g. for Riemannian g E ). The dep endence on λ en ters only through lo wer-order terms. 7 5.3. Homothetic harmonic forms and Ho dge de c omp osition Define the space of homothetic harmonic p -forms b y H p hom ( M ; λ, w ) := ker  ˜ ∆ : Ω p ( M ) → Ω p ( M )  . (26) Prop osition 5.4 (T ransp ort of harmonic forms) . A ssume g E is R iemannian. If M is c omp act without b oundary, or if one works in a class of forms for which ˜ ∆ and ∆ ar e define d and the c onjugation (25) is valid (e.g. c omp act supp ort or suitable de c ay/b oundary c onditions), then H p hom ( M ; λ, w ) = e − wλ H p ( M ; g E ) , (27) wher e H p ( M ; g E ) = ker(∆ : Ω p → Ω p ) denotes the usual sp ac e of harmonic p -forms for g E . Pr o of. By (25) , ˜ ∆ η = 0 if and only if ∆( S η ) = 0 . Thus η ∈ k er ˜ ∆ iff S η = e wλ η ∈ k er ∆ , i.e. iff η = e − wλ ω with ω ∈ H p ( M ; g E ) . Theorem 5.5 (Homothetic Ho dge decomp osition) . A ssume M is c omp act, oriente d, and R iemannian (and without b oundary). Then for e ach p ther e is an ortho gonal de c omp osition with r esp e ct to ⟨· , ·⟩ λ,w : Ω p ( M ) = im ˜ d ⊕ im ˜ δ ⊕ H p hom ( M ; λ, w ) , (28) and every c ohomolo gy class in H p hom ( M ; λ, w ) has a unique r epr esentative in H p hom ( M ; λ, w ) . On nonc omp act manifolds such as R d , an analo gous glob al ortho gonal de c omp osition r e quir es additional functional-analytic hyp otheses (choic e of Hilb ert sp ac es, closur es of r anges, and b oundary/de c ay c onditions), and is not use d in the PDE applic ations b elow. Pr o of sketch. Since ˜ ∆ = S − 1 ∆ S is conjugate to the usual Ho dge Laplacian, (28) follo ws b y transp orting the classical Ho dge decomposition for ∆ via the isomorphism S and using that ˜ d = S − 1 d S and ˜ δ = S − 1 δ S . A full pro of is standard in w eighted Ho dge theory and is therefore omitted. Remark 5.6 (Manifolds with b oundary) . On manifolds with b oundary , one may imp ose absolute/relativ e (or other) b oundary conditions to obtain corresp onding Ho dge decomp osi- tions. F or the PDE applications b elo w, w e will mainly w ork on domains in R d and discuss b oundary/in terface conditions directly at the lev el of the scalar homothetic Laplace equation. In particular, on R d w e only use the scalar conjugation iden tity ˜ ∆ u = e − wλ ∆( e wλ u ) and do not app eal to Theorem 5.5. 6. A Homothetic Laplace equation for the Scalar Fields F rom here on, w e restrict ourselv es on the Riemannian metrics, resulting in, the elliptic ∆ and ˜ ∆ . 8 6.1. 0-forms: explicit form of the homothetic L aplacian In this section w e sp ecialize to p = 0 , i.e. scalar fields (functions). Let ( M , g ) b e a Riemannian manifold with fixed metric g , and let ∆ denote the (p ositiv e) Laplace–Beltrami op erator on functions. 1 F or u ∈ C ∞ ( M ) , the homothetic Laplacian is, b y (25), ˜ ∆ u = e − wλ ∆  e wλ u  . (29) Expanding (29) using the pro duct rule yields the drift–p oten tial form ˜ ∆ u = ∆ u + 2 w ⟨∇ λ, ∇ u ⟩ +  w (∆ λ ) + w 2 |∇ λ | 2  u, (30) where ⟨· , ·⟩ and | · | are tak en with resp ect to g . Remark 6.1. W e should remark that ˜ ∆ u = 0 ⇐ ⇒ ∆  e wλ u  = 0 . (31) Th us the homothetic harmonicity of u is equiv alen t to the classical harmonicity of the “dressed” field e wλ u . 6.2. Homothetic L aplac e e quation for the homothetic al ly tr ansforme d fields with the homothety c enter α d Let α E b e a scalar field in the Einstein gauge and let α d b e a fixed, scale-inv ariant 0 -form. The homothetic transformation (5) reads α = e − wλ α E + (1 − e − wλ ) α d , (32) and the shifted v ariable u := α − α d (33) transforms linearly: u = e − wλ ( α E − α d ) . Definition 6.2 (Homothetic scalar Laplace equation) . W e sa y that α satisfies the homothetic L aplac e e quation (with cen ter α d ) if ˜ ∆( α − α d ) = 0 , i.e. ˜ ∆ α = ˜ ∆ α d . (34) Expanding the ab o v e equation, and using (30) giv es ∆( α − α d ) + 2 w ⟨∇ λ, ∇ ( α − α d ) ⟩ +  w (∆ λ ) + w 2 |∇ λ | 2  ( α − α d ) = 0 . (35) And finally , if we consider the harmonic α d , in the classical sense, ∆ α d = 0 , then (35) reduces to ∆ α + 2 w ⟨∇ λ, ∇ ( α − α d ) ⟩ +  w (∆ λ ) + w 2 |∇ λ | 2  ( α − α d ) = 0 . (36) W e call this equation and its solutions the homothetic L aplac e e quation , and the homothetic harmonic functions resp ectiv ely . This equation will b e employ ed for the imp osition of the b oundary conditions on the classical Laplace equations in the next sections. 1 Throughout, one may equiv alen tly use the geometric sign con ven tion ∆ = δ d = − div ∇ . 9 6.3. Brief r emarks on wel l-p ose dness fr om the el liptic the ory viewp oint On a b ounded domain Ω ⊂ M with smo oth boundary and Riemannian metric g , the op erator L λ,w u := ˜ ∆ u = ∆ u + 2 w ⟨∇ λ, ∇ u ⟩ + w (∆ λ ) u + w 2 |∇ λ | 2 u (37) is a second-order elliptic op erator with the same principal symbol as ∆ . In particular, stan- dard existence and uniqueness results apply under standard b oundary conditions (Dirichlet, Neumann, or mixed) pro vided the co efficients determined by λ are sufficien tly regular. In later sections w e fo cus on a complemen tary viewp oin t: choosing λ to lo calize near an interface so that the single equation L λ,w u = 0 effectively enforces b oundary or matc hing data for u = α − α d . 7. PDE applications The scalar homothetic Laplace equation (36) can b e used to imp ose b oundary/in terface conditions for classical elliptic problems [ 9 , 13 ]. The idea is to c ho ose λ so that the low er-order co efficien ts in the homothetic op erator concentrate near a hypersurface S , to p enalize the mismatc h of prescrib ed data encapsulated in ϕ d . Thus λ is treated as a giv en fixed parameter, not a dynamical one. This viewp oin t is particularly appropriate for PDE applications, and geometrically justifies the volume-penalization/diffuse-interface approaches are widely in use in the computational ph ysics [ 2 , 25 , 28 ]. W e use the normalizing c hoice w = 1 , whic h yields particularly transparen t form ulas. W e restrict to the Euclidean domains R d , on which the inner pro ducts simplify to the dot pro ducts. Moreo ver, to emphasize that w e are treating 0-forms, w e will use ϕ instead of α . Therefore, Eq. (36) reads ∆ ϕ + 2 ∇ λ · ∇ ( ϕ − ϕ d ) +  (∆ λ ) + |∇ λ | 2  ( ϕ − ϕ d ) = 0 . (38) W e alw ays assume ϕ d is harmonic in the classical sense ∆ ϕ d = 0 . F or instance, ϕ d ma y b e constructed in a tubular neighborho o d of S as the harmonic extension of the prescrib ed b oundary data. Then, by its smo oth extension outside this neigh b orho o d, the p enalization terms b elo w are lo calized near S (cf. standard elliptic regularit y and extension results [ 13 , 9 ]). 7.1. Cauchy, Dirichlet and Neumann pr oblems Let S ⊂ R d b e a closed, connected, orien table, embedded C ∞ h yp ersurface where d ∈ { 1 , 2 , 3 } . By the Jordan–Brou wer separation theorem, R d \ S has precisely tw o connected comp onen ts, one b ounded and one unbounded [ 26 ]. W e denote them by Ω i (in terior) and Ω o (exterior), and w e ha ve Ω i ∩ Ω o = S . Let ν b e the unit normal on S p oin ting from Ω i in to Ω o , and for a sufficien tly regular function u define ∂ ν u := ∇ u · ν on S . Moreo ver, let T ε n ( S ) denote the tubular neigh b orho od of S of radius ε n , where lim n →∞ ε n = 0 . Let ξ ( x ) := dist ( x, S ) denote the (unsigned) distance function, whic h is smo oth in eac h of the t wo comp onen ts Ω i and Ω o a wa y from S . Using these definitions, we set λ n := ln f n ( ξ ( x )) , (39) in whic h f n b elongs to a family ( f n ) n ≥ 1 of smo oth cutoff functions (see App endix A for the details). Note that suc h a b eha vior results in ∇ λ n = ∇ f n f n and ∆ λ n + |∇ λ n | 2 = ∆ f n f n . (40) 10 In particular, since f n ≡ 1 on R d \ T ε n ( S ) , one has ∇ λ n ≡ 0 and ∆ λ n + |∇ λ n | 2 ≡ 0 there, so all lo wer–order homothetic terms v anish iden tically aw a y from T ϵ n ( S ) . Cauc hy data on S . Fix b oundary data g := ϕ S and h := ∂ ν ϕ S on S , the problem ϕ ( x ) :=        ∆ ϕ = 0 in R d ϕ | S = g , ∂ ν ϕ | S = h, (41) under a suitable condition at infinity in the exterior domain (e.g. deca y , b oundedness, or a prescrib ed far–field b ehavior), defines a Cauchy pr oblem for the Laplace equation on R d . It is well kno wn that (41) is o v erdetermined. Generically , a pair ( g , h ) does not arise as the trace and normal trace of a global harmonic function on R d , and the corresp onding con tinuation problem is ill-p osed in the Hadamard sense (see, e.g., [15, 23]). A p enalized homothetic formulation. Giv en a fixed function ϕ d whic h is harmonic in T ϵ n ( S ) and matches the desired b oundary data on S ϕ d | S = g , ∂ ν ϕ d | S = h, By c ho osing λ n from (39) resulting in (40) co efficien ts, Eq. (38) b ecomes the p enalize d Cauchy pr oblem ∆ ϕ n + 2 ∇ f n f n ·  ∇ ϕ n − ∇ ϕ d  + ∆ f n f n ( ϕ n − ϕ d ) = 0 in R d \ S. (42) F or each fixed n , (42) is a linear second–order elliptic equation with smo oth co efficien ts on Ω i and on Ω o . As n → ∞ , the lo wer–order terms concentrate near S and act as a thin p enalt y la yer which attempts to enforce simultane ously the Diric hlet and Neumann data enco ded in ϕ d . Strong vs. weak solutions. Since, in the limit, the solutions ma y fail to b e C 1 across S , it is con venien t to w ork with a piecewise Sob olev space [1, 23]. Define the space H 1 (Ω i ⊔ Ω o ) := n u : u | Ω i ∈ H 1 (Ω i ) , u | Ω o ∈ H 1 (Ω o ) o , ∥ u ∥ 2 := ∥ u ∥ 2 H 1 (Ω i ) + ∥ u ∥ 2 H 1 (Ω o ) . F or u ∈ H 1 (Ω i ⊔ Ω o ) , denote by γ i u and γ o u the (Sob olev) traces on S from the interior and exterior side, resp ectiv ely . When the traces exist, w e define the jump by [ u ] S := γ o u − γ i u. Similarly , for sufficien tly regular u (e.g. piecewise H 2 ), one ma y define the normal deriv ativ e traces ∂ ν u | Ω i and ∂ ν u | Ω o and their jump [ ∂ ν u ] S := ( ∂ ν u | Ω o ) − ( ∂ ν u | Ω i ) . With these definitions in place, w e analyze Eq. (42). Consisten t and inconsistent Cauc hy data. F or Eq. (42) t wo different situations are distinguishable. 11 (A) Consistent Cauchy data (a glob al harmonic function exists). Assume that there exists a harmonic function ϕ ∈ C ∞ ( R d ) satisfying the Cauch y data, that is, (41) holds (together with the chosen b eha vior at infinit y in the exterior domain). In this situation, the p enalized Cauc hy form ulation (42) is consistent with a global classical harmonic function. It is then natural (and can b e pro ved under standard uniform elliptic estimates) that the solutions ϕ n con verge, as n → ∞ , to that global harmonic function in the str ong sense (e.g., in H 1 loc ( R d ) , and in C ∞ loc ( R d \ S ) b y elliptic regularit y). In particular, the restrictions ϕ i := ϕ | Ω i , ϕ o := ϕ | Ω o are harmonic and satisfy the full matc hing conditions [ ϕ ] S = 0 , [ ∂ ν ϕ ] S = 0 , so that ϕ i and ϕ o are harmonic con tinuations of one another across S . Remark 7.1 (Internal surface) . In this case, the h yp ersurface S do es not act as a b oundary for the limiting harmonic field: the in terior and exterior solutions glue to a single ϕ ∈ C ∞ ( R d ) with [ ϕ ] S = [ ∂ ν ϕ ] S = 0 . In this case, S is an internal surfac e (or r emovable interfac e ). (B) Inc onsistent Cauchy data (no glob al harmonic function exists). Assume no w that the Cauch y data are inconsistent, that is, there is no global harmonic function on R d satisfying b oth ϕ | S = g and ∂ ν ϕ | S = h . In other w ords, one cannot enforce simultane ously the tw o constraints enco ded in ϕ d and ∇ ϕ d on S . Definition 7.2 (Homothetic b oundary) . In this case, w e call S a homothetic b oundary . Equiv alen tly , there is no global harmonic ϕ on R d satisfying b oth ϕ | S = g and ∂ ν ϕ | S = h , so the limiting homothetic form ulation necessarily yields a problem in which at least one of the jumps [ ϕ ] S or [ ∂ ν ϕ ] S is nonzero. When there is not a consistent Cauch y data in hand, it is usually natural to think of imp osing only one of the t wo b oundary conditions. It leads to the follo wing tw o reduced p enalized problems. • P enalized Diric hlet problem. Solv e ∆ ϕ n + ∆ f n f n ( ϕ n − ϕ d ) = 0 in R d \ S. (43) F ormally , the p enalt y term forces γ i ϕ n ≈ γ o ϕ n ≈ g on S as n → ∞ , but it do es not enforce matc hing of normal deriv ativ es. Hence, in the limit one generically obtains a piecewise harmonic function ϕ ∈ H 1 (Ω i ⊔ Ω o ) satisfying γ i ϕ = γ o ϕ = g on S, while the normal deriv ativ e ma y jump: [ ∂ ν ϕ ] S  = 0 in general . Consequen tly there is, in general, no global strong (classical) harmonic solution on R d ; rather, the limit ob ject is a weak solution in the space H 1 (Ω i ⊔ Ω o ) . 12 • P enalized Neumann problem. Here w e assume h satisfies the con ven tional consistency condition for a homogeneous Neumann problem R S hdS = 0 . Solve ∆ ϕ n + 2 ∇ f n f n ·  ∇ ϕ n − ∇ ϕ d  = 0 in R d \ S. (44) Here the p enalization enforces ∂ ν ϕ | S = h in the limit (in the natural w eak trace sense), but it do es not enforce matching of v alues across S . Th us the limiting ob ject is again piecewise harmonic, satisfies the Neumann condition on S (on eac h side, with the appropriate sign con ven tion), and ma y exhibit a jump in the trace: [ ϕ ] S  = 0 in general . As ab o v e, this precludes a global strong solution on R d in general, but admits a w eak solution; cf. the standard v ariational theory for Neumann problems [23, 9]. The ab o v e issues can b e summarized as follo ws. Theorem 7.3. Given the ab ove definitions, assume ϕ n is a family of solutions to either (43) or (44) , subje ct to a fixe d b ehavior at infinity in the exterior domain. A ssume further that ϕ n is uniformly b ounde d in H 1 loc (Ω i ) and H 1 loc (Ω o ) . Then, after extr action of a subse quenc e, ϕ n c onver ges we akly in H 1 loc (Ω i ) × H 1 loc (Ω o ) to a p air ( ϕ i , ϕ o ) such that: 1. ∆ ϕ i = 0 in Ω i and ∆ ϕ o = 0 in Ω o in the we ak sense (and henc e ϕ i , ϕ o ar e smo oth by el liptic r e gularity). 2. In the Dirichlet c ase (43) , the tr ac es satisfy γ i ϕ i = γ o ϕ o = g on S , i.e. ( ϕ i , ϕ o ) solve the classic al L aplac e–Dirichlet pr oblems on Ω i and Ω o with b oundary value g . 3. In the Neumann c ase (44) , the (we ak) normal tr ac es satisfy the pr escrib e d Neumann c ondition determine d by h on S (with the standar d sign c onventions on Ω i and Ω o ), i.e. ( ϕ i , ϕ o ) solve the classic al L aplac e–Neumann pr oblems on Ω i and Ω o with b oundary flux h d (with the usual additive-c onstant ambiguity, fixe d by the b oundary c ondition at infinity if ne e de d). In gener al, the glob al glue d function ϕ ( x ) :=    ϕ i ( x ) , x ∈ Ω i , ϕ o ( x ) , x ∈ Ω o , is a we ak solution on R d and may exhibit a jump in ϕ or in ∂ ν ϕ acr oss S , dep ending on whether Dirichlet or Neumann data wer e imp ose d. The ab o ve issues can b e more developed if w e put them inside the framework of classical p oten tial theory . 13 7.2. R elation to classic al p otential the ory The ab o ve discussions are closely related to classical p oten tial theory , particularly the b eha vior of single-la yer and double-lay er p oten tials [ 20 , 21 ]; see also [ 24 ]. Sp ecifically , a single- layer p otential yields a con tinuous p otential across the in terface but typically introduces a discontin uit y in its normal deriv ativ e. In con trast, a double-layer p otential maintains con tinuit y of the normal deriv ativ e but allo ws for a jump in the potential itself [ 7 ]. The b eha vior of equation (42) in the limit n → ∞ is thus analogous to a c ombine d layer p otential , where b oth Dirichlet and Neumann data are enforced. Therefore, the ab ov e discussions can b e rephrased in the following theorems. Prop osition 7.4 (Distributional Laplacian across a h yp ersurface) . L et S ⊂ R d b e a C ∞ emb e dde d hyp ersurfac e and write R d = Ω i ⊔ S ⊔ Ω o . Fix a unit normal field ν on S p ointing fr om Ω i to Ω o and write ∂ ν := ∇ ( · ) · ν . L et ϕ i ∈ H 2 loc (Ω i ) and ϕ o ∈ H 2 loc (Ω o ) satisfy ∆ ϕ i = 0 in Ω i , ∆ ϕ o = 0 in Ω o , and define the glue d function ϕ on R d by ϕ ( x ) :=    ϕ i ( x ) , x ∈ Ω i , ϕ o ( x ) , x ∈ Ω o . A ssume the tr ac es γ i ϕ, γ o ϕ and normal tr ac es exist so that the jumps [ ϕ ] S := γ o ϕ − γ i ϕ, [ ∂ ν ϕ ] S :=  ∂ ν ϕ o | S  −  ∂ ν ϕ i | S  ar e wel l-define d. Then, for every ψ ∈ C ∞ c ( R d ) one has ⟨ ∆ ϕ, ψ ⟩ = Z S [ ∂ ν ϕ ] S ψ dσ − Z S [ ϕ ] S ∂ ν ψ dσ. (45) Equivalently, in D ′ ( R d ) , ∆ ϕ = [ ∂ ν ϕ ] S δ S + [ ϕ ] S ∂ ν δ S , wher e ⟨ ∂ ν δ S , ψ ⟩ := − Z S ∂ ν ψ dσ. In p articular, ϕ is a glob al harmonic function on R d if and only if [ ϕ ] S = 0 and [ ∂ ν ϕ ] S = 0 (i.e. S is a r emovable interfac e for ϕ ). Corollary 7.5 (Poten tial theory setting) . In the setting of The or em 7.3, assume additional ly that the limiting p air ( ϕ i , ϕ o ) admits tr ac es and normal tr ac es on S so that the jumps [ ϕ ] S and [ ∂ ν ϕ ] S ar e wel l-define d, and let ϕ b e the glue d function. Then ∆ ϕ is a distribution supp orte d on S and is given by ∆ ϕ = [ ∂ ν ϕ ] S δ S + [ ϕ ] S ∂ ν δ S in D ′ ( R d ) . Mor e over: 1. In the Dirichlet p enalization (43) , one has γ i ϕ i = γ o ϕ o = g on S , henc e [ ϕ ] S = 0 and ∆ ϕ = [ ∂ ν ϕ ] S δ S , i.e. S c arries a single-lay er sour c e unless [ ∂ ν ϕ ] S = 0 . 14 2. In the Neumann p enalization (44) , the enfor c e d flux c ondition yields the c orr esp onding normal-tr ac e c onstr aint on S (with the sign c onvention of The or em 7.3); in the flux- matching c ase [ ∂ ν ϕ ] S = 0 and ∆ ϕ = [ ϕ ] S ∂ ν δ S , i.e. S c arries a double-lay er sour c e unless [ ϕ ] S = 0 . Conse quently, whenever at le ast one of the jumps is nonzer o, S is dete cte d by the classic al L aplacian as a genuine ge ometric b oundary/interfac e for the limit n → ∞ . In the next section we employ the dev elop ed theory to address the longstanding problem of non-singular p oin t source in the p oten tial theory . 7.3. A non-singular mo del for a p oint sour c e Homothetic harmonic functions are more flexible than the classical harmonic functions due to the presence of λ and ϕ d , which allo w lo cal deformations. F or example, from Liouville’s theorem, one can infer that an y harmonic function on R d is either constan t or not b ounded. But this is not true for the homothetic harmonic functions. Here, we utilise this flexibilit y to construct a non-singular mo del for a p oin t source. In fact, there is more than one type of suc h regularization. W e hav e already pro vided a non-singular mo del for a p oin t source as a solution of Eq. (41) , suitable for an elementary particle (an electron in particular) [ 32 ]. Here, another non-singular mo del is presented based on Eq. (43) , which mo dels a hollow-sphere p oin t source (c.f. classical electrostatics and p oten tial theory [17, 19].) The p oten tial of a p oin t charge lo cated at the origin of R 3 tak es the form ϕ ( r ) = C r , where C is a constan t and r = ∥ x ∥ . While this solution satisfies the Laplace equation aw a y from the origin, it b ecomes singular at r = 0 , and the corresp onding energy , E = 1 2 Z R 3 |∇ ϕ | 2 dV , div erges due to the singularit y at the origin (indeed |∇ ϕ ( r ) | ∼ r − 2 and R 1 0 r − 2 dr = + ∞ ). This infinite self-energy has b een a foundational issue in classical field theory , prompting n umerous attempts at regularization [ 17 , 4 ]. No w, in the framew ork of homothetic harmonic functions one can av oid such singularities by enforcing the boundary data on a spherical in terface rather than at a p oin t. W e consider a spherical in terface ∂ B ( R ) ⊂ R 3 , and solv e the Diric hlet homothetic Laplace equation (43) adapted for this problem ∇ 2 ϕ n + ∆ f R,n f R,n ( ϕ n − ϕ d ) = 0 , (46) where f R,n is the cutoff function asso ciated with the surface ∂ B ( R ) constructed from the distance-to- ∂ B ( R ) as explained in App endix A, that is, f R,n ( x ) = dist ( x, ∂ B ( R )) in a la yer 15 of thic kness η ε n in the neigh b orho o d of S , and f R,n ≡ 1 outside a la yer of thickness ε n ; and ϕ d is the prescrib ed p oten tial ϕ d = C R , (47) enforcing a Coulom b-like condition at r = R , where r = ∥ x ∥ . Remark 7.6. It should b e emphasized that in con trast to the classical p oin t source mo dels, the p enalization la yer here is supp orted near the surface ∂ B ( R ) , not at the origin. F or ev ery fixed R > 0 the interface is a smo oth h yp ersurface ∂ B ( R ) ∼ = S 2 , not a single p oin t. F rom the viewp oin t of p oten tial theory , this distinction is reflected in capacit y: the Newtonian capacity of a sphere of radius R is prop ortional to R (in fact, cap ( ∂ B ( R )) = 4 π R in R 3 ), whereas a p oin t has zero Newtonian capacit y; see, e.g., [19, 22]. Due to the highly lo calized nature of the p enalt y term, the solution ϕ n adjusts sharply near the in terface to matc h the b oundary v alue, and decays as ϕ n ( r ) ∼ C /r for large r , consisten t with the b eha vior of the classical p oten tial in the far field [17, 19]. The structure of the equation ensures a natural decoupling of the interior and exterior domains. In the limit n → ∞ , the p enalization enforces the b oundary condition only on ∂ B ( R ) , and the in terior and exterior solutions b ecome indep enden t (see the discussion in §7.1). Thus, the b eha vior of the solution in R 3 \ B ( R ) is determined en tirely b y the enforced Diric hlet data and is unaffected by the field inside the ball. In particular, the exterior solution is the unique harmonic function on R 3 \ B ( R ) that deca ys at infinit y and satisfies ϕ | r = R = C /R [9, 13], hence ϕ o ( r ) = C r , r ≥ R . (48) On the other hand, the in terior solution on B ( R ) is determined solely b y the same b oundary condition on ∂ B ( R ) ; the unique regular harmonic solution in the ball satisfying ϕ | r = R = C /R is the constan t p oten tial (by the maximum principle) [9, 13] ϕ i ( r ) = C R , r ≤ R . (49) In particular, ϕ i and ϕ o mo del a hollo w spherical source of radius R with the surface p oten tial C /R placed at the origin of R 3 , and they arise as the limiting interior/exterior fields asso ciated with the single p enalized equation (46) . Ho wev er, this solution allo ws a jump in the normal deriv ativ e across ∂ B ( R ) , in accordance with the discussion in §7.1. One may now observe the b eha vior of the mo del for R ↓ 0 . F or each fixed R > 0 , the field energy is finite, since ∇ ϕ i ≡ 0 in B ( R ) and |∇ ϕ o ( r ) | = | C | r − 2 in the exterior, hence E = 1 2 Z R 3 |∇ ϕ | 2 dV = 1 2 Z R 3 \ B ( R ) C 2 r 4 dV = 2 π C 2 R < ∞ ( R > 0) . (50) In particular, the mo del remov es the singularity at r = 0 for every R > 0 . Therefore, b y subtraction of the single p oin t r = 0 from R 3 , the energy remains non-infinite on R 3 \ { 0 } for all R . 16 8. Conclusions In this pap er, we hav e prop osed an extension to classical W eyl-integrable geometry b y generalizing standard linear gauge transformations in to affine homothetic transformations cen tered around a distinguished, harmonic, scale-in v ariant form α d . By defining shifted v ariables to recov er the linearity , w e constructed a twisted Hodge–de Rham theory . W e demonstrated that our twisted differential op erators inheren tly corresp ond to Witten-t yp e exact deformations, enabling us to establish a homothetic Ho dge decomposition theorem for compact, orien ted Riemannian manifolds. T ransitioning from the abstract geometric framew ork to practical PDE applications, we sp ecialized this theory for 0 -forms to deriv e the scalar homothetic Laplace equation. W e sho wed that by treating the t wist parameter λ as a fixed, non-dynamical field determined b y a regularized distribution tuned to concentrate near a h yp ersurface, this equation acts as a p ow erful diffuse-in terface to ol for p enalizing mismatc hes in prescrib ed b oundary data. Sp ecifically , setting λ via a regularized surface delta distribution allows the single homoth- etic equation to enforce Cauch y , Dirichlet, or Neumann b oundary conditions. In cases of incompatible Cauc hy data, w e prov ed that the p enalized equations naturally relax to yield w eak solutions, p ermitting jumps in either the p oten tial or its normal deriv ativ e in a manner reminiscen t of classical single and double lay er p oten tials. F urthermore, w e highlighted the physical relev ance of this geometric flexibilit y by con- structing a non-singular mo del for a classical p oin t source. By mo deling a hollo w-sphere regularization that enforces a surface p oten tial on an in terface ∂ B ( R ) , our approach entirely a voids the infinite self-energy at the origin. The p enalization cleanly decouples the interior and exterior domains in the limit, ensuring a constan t finite in terior p oten tial while preserving the ph ysically observ able deca y in the far field. Finally , our analytical approach to determining the highly lo calized profile of λ —utilizing the well-established theory of regular singular p oints for second-order ODEs—pro vides a systematic (if not quite general) approach for b oundary data imp osition. This metho dology not only resolves foundational singularities in classical field theory but also demonstrates significan t p oten tial for future algorithmic implementations in computational physics. App endix A. T uning λ near the b oundary In § 7, λ ( x ) is a giv en fixed function, that is, it do es not ha ve a separate dynamics. In this app endix, we find this fixed distribution in the neigh b orho od of the b oundary , which allo ws us to apply our desired b oundary conditions. The result has the p oten tial for applications in computational ph ysics. Our goal here is to pro vide a practical approac h for obtaining sufficien tly regular b oundaries/in terfaces, rather than a complete classification of all admissible near-b oundary b eha vior. W e employ the well-established theory of regular singular p oin ts of second-order ordinary differen tial equations (see, e.g., [6, 34]). Consider the one-dimensional version of the homothetic Laplace equation (38) , and set u := ϕ − ϕ d to get: u ′′ + 2 λ ′ u ′ + ( λ ′′ + ( λ ′ ) 2 ) u = 0 . (A.1) where ( · ′ ) denotes differen tiation with resp ect to x ∈ (0 , ∞ ) , and x = 0 is the p oin t that w e w ant to imp ose our b oundary conditions. Moreov er, we assumed ϕ d is harmonic ϕ ′′ d = 0 in 17 the neigh b orho o d x = 0 + . The co efficien ts λ ′ and λ ′′ + ( λ ′ ) 2 should b ecome large as x → 0 + , and v anish a wa y from it. A standard sufficient condition for x = 0 to b e a r e gular singular p oint of (A.1) is the F uc hsian scaling λ ′ ( x ) ∼ a x , λ ′′ ( x ) + ( λ ′ ( x )) 2 ∼ a 2 − a x 2 ( x → 0 + ) , (A.2) for some real parameter a ∈ R . W e in terpret a as a calibration parameter controlling the sharpness at the b oundary . Moreo ver, we can v anish the lo wer-order terms, aw a y from x = 0 , as follo ws. Let ( ε n ) n ≥ 1 b e a sequence with ε n ↓ 0 defining the collar neighborho o d of the origin x = 0 , and fix η ∈ (0 , 1) . Cho ose a family of functions f n : (0 , ∞ ) → (0 , 1] such that f n ( x ) =    x, for 0 < x ≤ η ε n , 1 , for x ≥ ε n , f n ∈ C ∞ ( ]0 , ∞ [ ) . (A.3) Suc h f n are obtained by comp osing x with a standard C ∞ cutoff profile that in terp olates smo othly b etw een x and 1 on [ η ε n , ε n ] (see e.g.[16]). W e then define λ n ( x ) := a ln  f n ( x )  . (A.4) In the inner part 0 < x ≤ η ε n where f n ( x ) = x , w e obtain the regular singular b eha vior, ob eying the Euler–Cauch y equation u ′′ + 2 a x u ′ + a 2 − a x 2 u = 0 (0 < x ≤ η ε n ) , (A.5) while in the outer part x ≥ ε n , the co efficients v anish identically and the Laplace equation u ′′ = 0 retriev es. T rying a F rob enius/Euler ansatz u ( x ) = C x m giv es the indicial equation m ( m − 1) + 2 a m + ( a 2 − a ) = 0 , whose ro ots are m 1 = − a, m 2 = 1 − a. Hence the lo cal b eha vior near x = 0 + is u ( x ) ∼ C 1 x − a + C 2 x 1 − a . (A.6) The parameter a con trols the v anishing/blo w-up rates of these tw o branches. The branch that is selected in a given PDE application is determined by the criterion one imp oses (e.g. b oundedness, finite energy , or a w eak trace requirement). If one selects a branc h u ∼ x m with exp onen t m > 0 , then u (0) = 0 (suitable for the Dirichlet BCs). More generally , u ( k ) (0) = 0 holds pro vided m > k . F or example, for the branch u ∼ x − a one has u (0) = 0 if a < 0 , u ′ (0) = 0 if a < − 1 , u ′′ (0) = 0 if a < − 2 , 18 while for the branc h u ∼ x 1 − a one has u (0) = 0 if a < 1 , u ′ (0) = 0 if a < 0 , u ′′ (0) = 0 if a < − 1 . Th us a should b e view ed as a tunable p enalt y-strength parameter, by whic h one can determine ho w many deriv ativ es of the selected lo cal branc h v anish at the singular p oin t. Multi-dimensional in terfaces. F or a smo oth embedded hypersurface S ⊂ R d , d > 1 , one w orks in a tubular neigh b orho o d T ε n ( S ) with distance co ordinate ξ ( x ) = dist ( x, S ) , and sets f n ( x ) = F n ( ξ ( x )) , λ n ( x ) = a ln  f n ( x )  , where F n is the one-dimensional profile from (A.3) . In lo cal normal co ordinates, the leading singular b eha vior of the homothetic co efficients is gov erned b y the same ξ − 1 and ξ − 2 scalings as in the one-dimensional Euler–Cauch y mo del, while curv ature and tangential contributions en ter as lo wer-order terms. Consequently , the ab o ve one-dimensional analysis pro vides (at least in principle) a lo cal procedure for tuning a . How ev er, we do not assert a univ ersal dimension-indep enden t threshold for a in full generalit y . Moreo v er, note that ξ is smo oth on eac h side in Ω i and Ω o (in § 7 setting) within the tubular neigh b orho od of S . References [1] R. A. A dams and J. J. F ournier, Sob olev Sp ac es , 2nd ed., Academic Press, 2003. [2] A. Angot, C.-H. Bruneau, and P . F abrie, “A p enalization metho d to take into account obstacles in incompressible viscous flows,” Numerische Mathematik 81 (1999), 497–520. https://doi.org/10.1007/s002110050401 [3] J. Attard, J. F rançois, S. Lazzarini, and T. Masson, “The dressing field metho d of gauge symmetry reduction, a review with examples,” arXiv:1702.02753 [math-ph] (2017). [4] M. Born and L. Infeld, “F oundations of the new field theory ,” Pr o c e e dings of the R oyal So ciety A 144 (1934), 425–451. doi:10.1098/rspa.1934.0059 [5] T. Branson and A. R. Go v er, Conformally in v ariant op erators, differential forms, cohomology and a generalisation of Q -curv ature, arXiv:math/0309085 (2003). [6] E. A. Co ddington and N. Levinson, The ory of Or dinary Differ ential Equations , McGraw– Hill, 1955. [7] D. Colton and R. Kress, Inverse A c oustic and Ele ctr omagnetic Sc attering The ory , Springer, 1998. https://doi.org/10.1007/978- 1- 4612- 0611- 6 [8] Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou, “Analysis and approximation of nonlo cal diffusion problems with v olume constraints,” SIAM R eview 54 (2012), no. 4, 667–696. https://doi.org/10.1137/110833907 19 [9] L. C. Ev ans, Partial Differ ential Equations , 2nd ed., Graduate Studies in Mathematics, V ol. 19, Amer. Math. So c., 2010. [10] M. F arb er, T op olo gy of Close d One-F orms , Mathematical Surveys and Monographs, V ol. 108, Amer. Math. So c., Providence, RI, 2004. [11] J. F rançois, “Dilaton from tractor and matter field from twistor,” JHEP 06 (2019) 018, doi:10.1007/JHEP06(2019)018, arXiv:1810.07976 [math-ph]. [12] D. M. Ghilencea, “Stuec kelberg breaking of W eyl conformal geometry and applications to gravit y ,” Ph ys. Rev. D 101 (2020) 045010, doi:10.1103/Ph ysRevD.101.045010, arXiv:1904.06596 [hep-th]. [13] D. Gilbarg and N. S. T rudinger, El liptic Partial Differ ential Equations of Se c ond Or der , Springer, 2001 (reprin t of the 2nd ed.). [14] A. R. Go ver, Conformal de Rham Ho dge theory and op erators generalising the Q - curv ature, in Pr o c e e dings of the 24th Winter Scho ol “Ge ometry and Physics” (P alermo, 2004), Circolo Matematico di P alermo, 2005, pp. 109–137; arXiv:math/0404004. [15] J. Hadamard, L e ctur es on Cauchy’s Pr oblem in Line ar Partial Differ ential Equations , Y ale Univ. Press, 1923 (Do ver reprint av ailable). [16] L. Hörmander, The A nalysis of Line ar Partial Differ ential Op er ators I: Distribution The ory and F ourier A nalysis , Springer, 2nd ed., 1983. [17] J. D. Jac kson, Classic al Ele ctr o dynamics , 3rd ed., John Wiley & Sons, 1998. [18] G. K. Karananas and A. Monin, “W eyl vs. conformal,” Physics L etters B 757 (2016), 257–260. [19] O. D. Kellogg, F oundations of Potential The ory , Springer, Berlin–Heidelb erg, 1929. [20] R. Kress, Line ar Inte gr al Equations (3rd ed.), Springer, 2014. https://doi.org/10. 1007/978- 3- 642- 37338- 7 [21] O. A. Ladyzhenska ya, The Boundary V alue Pr oblems of Mathematic al Physics , Springer, 1985. https://doi.org/10.1007/978- 3- 0348- 9271- 2 [22] N. S. Landkof, F oundations of Mo dern Potential The ory , Springer, Berlin–Heidelb erg, 1972. [23] J.-L. Lions and E. Magenes, Non-Homo gene ous Boundary V alue Pr oblems and A pplic a- tions , V ol. I, Springer, 1972. [24] W. McLean, Str ongly El liptic Systems and Boundary Inte gr al Equations , Cam bridge Univ. Press, 2000. [25] R. Mittal and G. Iaccarino, “Immersed b oundary metho ds,” A nnual R eview of Fluid Me- chanics 37 (2005), 239–261. https://doi.org/10.1146/annurev.fluid.37.061903. 175743 20 [26] J. R. Munkres, T op olo gy , 2nd ed., Prentice Hall, 2000. [27] S. P . No viko v, Multiv alued functions and functionals. An analogue of the Morse theory , Dokl. A kad. Nauk SSSR 260 (1981), no. 1, 31–35. [28] R. Nguy en V an Y en, D. K olomenskiy , and K. Schneider, “Approximation of the Laplace and Stok es op erators with Dirichlet b oundary conditions through v olume p enalization: A sp ectral viewp oin t,” Numerische Mathematik 128 (2014), 301–338. https://doi. org/10.1007/s00211- 014- 0610- 8 [29] A. Piatnitski, V. Sloushc h, T. Suslina, and E. Zhizhina, “On op erator estimates in homogenization of nonlo cal op erators of conv olution t yp e,” Journal of Differ ential Equations 352 (2023), 153–188. https://doi.org/10.1016/j.jde.2022.12.036 [30] J. Ro e, El liptic Op er ators, T op olo gy and A symptotic Metho ds , 2nd ed., Chapman and Hall/CR C, 1998. [31] C. Romero, J. B. F onseca-Neto and M. L. Puc heu, “General relativity and W eyl geometry ,” Classic al and Quantum Gr avity 29 (2012) 155015, doi:10.1088/0264-9381/29/15/155015, arXiv:1201.1469 [gr-qc]. [32] F. Sab etghadam, “A homothetic Gauge Theory and the Regularization of the P oin t Charge,” submitted to Elsevier, https://doi.org/10.48550/arXiv.2507.06153 [33] J. M. Salim and S. L. Sautu, “Gra vitational theory in W eyl integrable space-time,” Classic al and Quantum Gr avity 13 (1996) 353–360, doi:10.1088/0264-9381/13/3/004. [34] G. F. Simmons, Differ ential Equations with A pplic ations and Historic al Notes (3rd ed.), CR C Press, 2016, ISBN: 978-1498702591 [35] H. W eyl, “Gravitation und Elektrizität,” Sitzungsb erichte der Königlich Pr eußischen A kademie der Wissenschaften zu Berlin (1918) 465–480. [36] E. Witten, Sup ersymmetry and Morse theory , J. Differ ential Ge om. 17 (1982), no. 4, 661–692. doi:10.4310/jdg/1214437492. 21