Abstract
In this paper we consider the problem of characterizing the sets of uniqueness for the solutions of the sandwich equation \(\partial _{\underline{x}}^3f\partial _{\underline{x}}\) = 0, where \(\partial _{\underline{x}}\) stands for the Dirac operator in \({{\mathbb {R}}}^m.\) These solutions are referred to as infrabimonogenic functions and can be viewed as a non-commutative version of biharmonic functions. Our main result states that a pair of distinct spheres is a set of uniqueness for infrabimonogenic functions in a convex domain of an odd-dimensional space.
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1 Introduction
Let f be a real-valued function defined in a domain \(\Omega \) of the Euclidean space \({{\mathbb {R}}}^m\). If f is 2k-times continuously differentiable and
then f is called polyharmonic of degree k. Here and in the sequel \(\Delta ^k\) denotes the k-th iteration of the Laplace operator \(\Delta \).
Polyharmonic functions play an important role in pure and applied mathematics. In particular, for \(k = 2\), biharmonic functions are specially important in elasticity theory.
The real Clifford algebra, \({{\mathbb {R}}}_{0,m}\) is generated by the orthonormal basis vectors \(e_1,e_2,\ldots e_m\) of the Euclidean space \({{\mathbb {R}}}^m\); with the relations
We will consider functions defined on subsets of \({{\mathbb {R}}}^m\) and taking values in \({{\mathbb {R}}}_{0,m}\). An \({{\mathbb {R}}}_{0,m}\)-valued function f is called left monogenic (right monogenic) in \(\Omega \) if \(\partial _{\underline{x}}f = 0\) (\(f\partial _{\underline{x}}= 0\)) in \(\Omega \), where \(\partial _{\underline{x}}\) stands for the so-called Dirac operator
It should be noticed that \(\partial _{\underline{x}}^2=\partial _{\underline{x}}\partial _{\underline{x}}=-\Delta .\)
Functions that are both left and right monogenic are called two-sided monogenic (see for example [2, 7, 8]). More generally, an \({{\mathbb {R}}}_{0,m}\)-valued function f in \(C^k(\Omega )\) is called left polymonogenic of order k, or simply k-monogenic (left) if \(\partial _{\underline{x}}^k\,f = 0\) in \(\Omega \). With regard to this generalization references may be made to [3, 4, 15, 18].
As a natural consequence of the non-commutativity of the Clifford product, a new class of functions arises, which can be seen as a non-commutative version of harmonic functions, namely the solutions of the sandwich equation \(\partial _{\underline{x}}f\partial _{\underline{x}}=0\). Such functions are termed inframonogenic (see [16, 17] ) and represent a refinement of the much more recognized biharmonic ones. Interesting connections have been found in [12,13,14], between them and the solutions of the Lamé–Navier system in linear elasticity theory. Infrapolymonogenic functions, introduced in [1], are the solutions of the generalized sandwich equation \(\partial _{\underline{x}}^{2k-1}f\partial _{\underline{x}}=0\), with \(k\in {{\mathbb {N}}}\).
Infrabimonogenic functions are, in particular, the \({{\mathbb {R}}}_{0,m}\)-valued solutions of the fourth-order partial differential equation \(\partial _{\underline{x}}^{3}f\partial _{\underline{x}}=0\) (a non-commutative version of the biharmonic equation). As easily seen, a function f is infrabimonogenic if and only if its Laplacian \(\Delta f\) is inframonogenic, or, equivalently, if and only if \(\partial _{\underline{x}}f\partial _{\underline{x}}\) is harmonic.
In [6], Edenhoffer proved that a polyharmonic function of order k is completely determined by its values on k concentric spheres. This result was extended by Hayman and Korenblum, who proved in [9] that the concentric assumption may be removed.
As proved in [10], one sphere represents a set of uniqueness for inframonogenic functions in odd-dimensional spaces. However, in even-dimensional spaces, it is possible to construct non-zero inframonogenic functions whose restrictions vanish on a sphere. More recently in [11] has been proved that k distinct concentric spheres is a set of uniqueness for infrapolymonogenic functions in odd dimensional spaces. The main result of this article ensures that, in the case of infrabimonogenic functions, the above concentric requirement may be removed in convex domains. Indeed, we prove that two distinct spheres in a convex domain uniquely determine an infrabimonogenic function. Incidentally, a characterization of sets of uniqueness for a generalized Lamé–Navier equation has been derived.
2 Auxiliary Results
In this section, some definitions and basic properties of a Clifford algebra will be recalled. Besides, we will provide some auxiliary results before stating the main theorems. The elements of \({{\mathbb {R}}}_{0,m}\) will be described in the form \(a =\sum _{A}a_Ae_A\), where as indices the elements A of the set containing the ordered subsets of \(\{1,2,\ldots ,m\}\) will be used, with the empty subset corresponding to the index 0. An arbitrary element \(a\in {{\mathbb {R}}}_{0,m}\) may be written in a unique way as
where \([\,]_{p}\) denotes the projection of \({{\mathbb {R}}}_{0,m}\) onto the subspace \({{\mathbb {R}}}_{0,m}^{(p)}\) of p-vectors defined by
We will make repeated use of the operator \(\Psi :{{\mathbb {R}}}_{0,m}\mapsto {{\mathbb {R}}}_{0,m}\) given by
for \(a\in {{\mathbb {R}}}_{0,m}\).
The operator \(\Psi \) keeps the subspace \({{\mathbb {R}}}_{0,m}^{(p)}\) invariant and, moreover, we have (see [16])
for a p-vector \(Y^p\).
When restricting to odd dimension m, the operator \(\Psi \) becomes a bijection and its inverse is given by
We will also deal with \({{\mathbb {R}}}_{0,m}\)-valued homogeneous polynomials of degree k given by
where \(\textbf{k}=(k_1,k_2,\dots ,k_m)\) denotes a multiindex, \(|\textbf{k}|=k_1+k_2+\cdots +k_m\) and \(\underline{x}^{\textbf{k}}=x_1^{k_1}x_2^{k_2}\cdots x_m^{k_m}\).
Lemma 1
Let f be a two-sided 3-monogenic function in a convex set \(\Omega \), and \(\underline{y}\in \Omega \). Then, there exists a unique harmonic function h, and a unique two-sided monogenic function \(\phi \), such that
Proof
The Almansi-type decomposition obtained in [15, Theorem 2.1] enables us to infer the following representation (uniquely determined)
with \(f'_1(\underline{x})\), \(f'_2(\underline{x})\) and \(f'_3(\underline{x})\) monogenic in \(\Omega \). Moreover from [14, Proposition 2] it follows that \(\underline{x}f'_2(\underline{x})\) is harmonic, then also \(f_1(\underline{x})=f'_1(\underline{x}) +\underline{x}f'_2(\underline{x})\). Hence (3) can be rewritten as
where \(f_1\), and \(f_2\) are harmonic, and left monogenic, respectively.
It will be proved that \(f_2\) is also right monogenic. If we apply (on the left) the Dirac operator \(\partial _{\underline{x}}\) to both sides of (4), we get
where \(\partial _{\underline{x}}f_1\) and \(f_2\) are both left monogenic functions.
Since, by assumption \(\partial _{\underline{x}}f\) is bimonogenic (harmonic), the uniqueness of the above representation is guaranteed by [15, Theorem 2.1].
On the other hand, by assumption we have that \(\partial _{\underline{x}}f\) is also inframonogenic and so, we can apply [5, Corollary 2.5] to yield the alternative (but unique!) representation
where \(f_1^*\) and \(f_2^*\) are left and two-sided monogenic, respectively.
The uniqueness of (5) and (6) yields \(2f_2=f_2^*\) and hence \(f_2\) is two-sided monogenic.
Then, due to the translation invariance of monogenic functions, \(f(\underline{x}+\underline{y})\) is a two-sided 3-monogenic function as well, and
with \(\partial _{\underline{x}}^2f_1^*=0\), and \(\partial _{\underline{x}}f_2^*=f_2^*\partial _{\underline{x}}=0\).
Hence
The proof is completed by defining \(h(\underline{x})=f_1^*(\underline{x}-\underline{y})\), and \(\phi (\underline{x})=f_2^*(\underline{x}-\underline{y})\).
\(\square \)
Lemma 2
Suppose that \(\Omega \) is a star-like domain with center 0, if f is two-sided 5-monogenic then f admits in \(\Omega \) the unique representation
where h is 4-monogenic and \(\psi \) is two-sided monogenic in \(\Omega \).
Proof
By similar arguments to those given in the proof of Lemma 1. It follows from the Almansi-type decomposition [15, Theorem 2.1] and [14, Proposition 2] that
with \(\partial _{\underline{x}}\psi =\psi ^*\partial _{\underline{x}}=0\) and \(\partial _{\underline{x}}^4h=\partial _{\underline{x}}^4h^*=0\). By (7) and (8) we have
then
where \(r_1\) and \(r_2\) have been chosen such that \(\overline{B(0,r_1)}\cup \overline{B(0,r_2)}\subset \Omega \). Hence, \(\psi ^*-\psi \equiv 0\) in \(\Omega \), since two distinct spheres is a set of uniqueness for biharmonic functions. \(\square \)
Lemma 3
Suppose that g is two-sided monogenic in a convex \(\Omega \), then
where \(\phi \) is two-sided monogenic, and \(\underline{y}\in \Omega \).
Proof
We have that \(|\underline{x}{|}^4g=|\underline{x}{|}^2\big (|\underline{x}{|}^2g\big )\), and \(\partial _{\underline{x}}^3\big (|\underline{x}{|}^2g\big )=\big (|\underline{x}{|}^2g\big )\partial _{\underline{x}}^3=0\). Indeed, from Lemma 1, there exist unique functions h and \(\phi \), such that
and by direct calculations we obtain the desired result. \(\square \)
Lemma 4
Let \(\mathcal {P}_k\) be a two-sided monogenic homogeneous polynomial of degree k in the odd-dimensional space \({{\mathbb {R}}}^m\), and let
Then,
Proof
First, we consider
and for simplicity, in what follows we use the notation \(\alpha _k=-(m+2k)\). Therefore \(\partial _{\underline{x}}(\underline{x}\mathcal {P}_k)=\alpha _k\mathcal {P}_k\), and we have the following chain of identities:
Then,
where use has been made of [14, Lemma 2.1]. \(\square \)
3 Main results
We state and prove our main results in this section. In the sequel, it will be assumed that one is working in an odd-dimensional Euclidean space \({{\mathbb {R}}}^m\). The standard notations B(0, r) and \(\overline{B(0,r)}\) for the open and closed ball with radius r centered at the origin will be used, respectively. In a similar manner, the sphere with radius r centered at the origin is denoted by \(\partial B(0,r)\).
Theorem 1
Let \(\overline{B(\underline{y},r)}\subset \Omega \), \(A\in {{\mathbb {R}}}\), \(A\ne \pm 1\), and \([u]_k\in C^2(\Omega )\). If
then the problem
has only the trivial solution \([u]_k\equiv 0\).
Proof
Suppose that \([u]_k\) is a solution of the above problem. Let \(R\in {{\mathbb {R}}}\), such that \(\overline{B(\underline{y},r)}\subset \overline{B(\underline{y},R)}\subset \Omega \). It follows from (10) that
and so, we have
Hence \([u]_k\partial _{\underline{x}}^3=\partial _{\underline{x}}^3[u]_k=0\), since \(A\ne \pm 1\).
Next, by Lemma 1, we have
where h and \(\phi \) are harmonic and two-sided monogenic, respectively.
Now, we introduce the auxiliary function \(G=r^2\phi \) which is obviously harmonic. Then, (11) can be rewritten as
Since
thus
The harmonicity of \(h+G\) yields \(h\equiv -G\) in \(B(\underline{y},r)\), and so in \(B(\underline{y},R)\).
Therefore,
Since \(\phi =[\phi ]_k\) is two-sided monogenic it can be expanded into the converging Taylor series in \(B(\underline{y},r)\)
with \([P_j]_k\) being two-sided monogenic as well.
Consequently,
and
On the other hand,
using [14, Lemma 2.1]. Summarizing, we have
Under the assumption (9), the last identity yields
and hence \([u]_k\equiv 0\). \(\square \)
A simple corollary is the following.
Corollary 1
Let \(\Omega \subset {{\mathbb {R}}}^3\), and suppose that \(A\in {{\mathbb {N}}}\) is even. Then a sphere \(\partial B(\underline{y},r)\) is a set of uniqueness for the vector solutions of the system
Proof
It follows directly from Theorem 1. Indeed, since in this case \(m=3\) and \(k=1\), we have
for any even number A. \(\square \)
Remark 1
When u is a bivector, i.e. \(u=[u]_2\), an analogous statement holds, since
It is important to mention that Corollary 1 and Remark 1 can be generalized in case that m is odd and A is even. We are now in a position to state and prove our main result.
Theorem 2
Let \(\Omega \) be a convex domain containing two distinct balls \(B_1\) and \(B_2\), such that \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}B_1\cup B_2\hspace{-0.83328pt}}\hspace{0.83328pt}\subset \Omega \). If f is infrabimonogenic in \(\Omega \) and \(f|_{\partial B_1}=f|_{\partial B_2}=0\), then \(f=0\) identically in \(\Omega \). In other words, two distinct spheres is a set of uniqueness for infrabimonogenic functions.
Proof
Without loss of generality, assume that one of the balls is centered at 0. The another center will be denoted by \(\underline{y}\). Since \(\partial _{\underline{x}}^3f\partial _{\underline{x}}=0\), it follows that \(\partial _{\underline{x}}^5f=f\partial _{\underline{x}}^5=0\). From Lemma 2, there exist \(h^*\) and \(\psi \), such that
where \(h^*\) is 4-monogenic and \(\psi \) is two-sided monogenic.
Then
and by Lemma 3, there exists a two-sided monogenic function \(\phi \), such that
Therefore,
and
Now let r and R be the radius of the corresponding balls centered at 0 and \(\underline{y}\).
Notice that
and hence
Since two distinct spheres are a set of uniqueness for biharmonic functions, one has then
On the other hand, direct calculations give
for \(\underline{x}\in B(0,\delta )\subset B(0,r)\).
Next, we expand the two-sided monogenic function \(\phi \) in \(B(0,\delta )\subset \overline{B(0,r)}\subset \Omega \), into the converging Taylor series
Accordingly, we have
where use has been made of Lemma 4, the bijectivity of \(\Psi \), and the fact that \(\partial _{\underline{x}}(\underline{x}P_j)=\alpha _jP_j\).
Moreover, define \(G=8\partial _{\underline{x}}(\underline{x}\Psi (\phi ))-16\Psi (\phi )-4\partial _{\underline{x}}(\underline{y}\Psi (\phi ))\). Of course, since G represents a real analytic function, which vanishes in the open set \(B(0,\delta )\subset \Omega \), it follows that \(G\equiv 0\) in the whole domain \(\Omega \).
Then, we indeed have in \(\Omega \)
from this identity and [14, Lemma 2.1] we have
where \(\varepsilon \) was chosen such that \(B(\underline{y}/2,\varepsilon )\subset \Omega \). Similarly,
and so
after taking the k-vector part.
Applying \(\Psi ^{-1}\) to (12), yields
thus
Let us denote \([\omega ]_k=(|\underline{x}-\underline{y}/2{|}^2-\varepsilon ^2)[\phi ]_k\). It follows from (13) and (14) that
or equivalently
On the other hand,
which is obvious from the definition of \([\omega ]_k\).
Thus by Theorem 1, \([\omega ]_k\equiv 0\) in \(\Omega \) for every k. Then \(\phi \equiv 0\), and finally \(f\equiv 0\) in \(\Omega \), as desired. \(\square \)
4 Concluding Remark
In the context of the search for possible generalizations we can ask whether Theorem 2 remains valid for infrapolymonogenic functions of arbitrary order k if instead of two distinct spheres we consider k of them. This conclusion cannot be reached using the inductive method similar to that used in the proof of [9, Theorem 4], because it is based on a direct application of the maximum principle for harmonic functions. The above question will inspire further analysis and researches.
Data Availability Statement
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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Luis Miguel Martín Álvarez gratefully acknowledges the Financial support of the Postgraduate Study Fellowship of the Consejo Nacional de Ciencia y Tecnología (CONACYT) (Grant Number 962684)
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Alvarez, L.M.M., García, A.M., Alejandre, M.P.Á. et al. Two Spheres Uniquely Determine Infrabimonogenic Functions. Mediterr. J. Math. 20, 318 (2023). https://doi.org/10.1007/s00009-023-02523-x
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DOI: https://doi.org/10.1007/s00009-023-02523-x