Abstract
In this article, we study Lie symmetries to fundamental solutions to the Leutwiler-Weinstein equation
in the upper half-space \(\mathbb {R}^{n}_{+}\). Starting from the infinitesimal generators of the equation Lu = 0, we deduce symmetries of the equation Lu = δ(x − x0), and using its invariant solutions, we construct a fundamental solution. As an application, we study a Green functions of the operator in the hyperbolic unit ball.
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References
Abramowitz, M., Stegun, I.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)
Akin, Ö., Leutwiler, H.: On the invariance of the solutions of the Weinstein equation under Möbius transformations. Classical and modern potential theory and applications (Chateau de Bonas, 1993), 19–29. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 430, Kluwer Acad. Publ., Dordrecht (1994)
Aksenov, A.V.: Fundamental solution of equations in displacements for a transversely isotropic elastic medium (Russian). Dokl. Akad. Nauk 470(5), 514–518 (2016). translation in Dokl. Math. 94 (2016), no. 2, 598–601
Aksenov, A.V.: Method of construction of the Riemann function for a second-order hyperbolic equation. J. Phys.: Conf. Ser. 937, 012001 (2017)
Aksenov, A.V.: Symmetries of linear partial differential equations, and fundamental solutions (Russian). Dokl. Akad. Nauk 342(2), 151–153 (1995)
Aksenov, A.V.: Symmetries and fundamental solutions of the multidimensional generalized axisymmetric Laplace equation (Russian). Differ. Uravn. 31 (10), 1697–1700, 1774 (1996) (1995)
Aksenov, A.V.: Symmetries of fundamental solutions and their application in continuum mechanics. Proc. Steklov Inst. Math. 300, 1 (2018)
Aksenov, A.V.: Symmetries of fundamental solutions of partial differential equations (Russian). In: Simmetrij Differentsialnyh Uravnenij, Sbornik nauchnyh trudov. Moscow Institute of Physics and Technology (Moscow State University), pp 6–35 (2009)
Berest, Y.U.: Weak invariants of local groups of transformations (Russian). Differentsialnye Uravneniya 29(10), 1796–1803 (1993)
Bluman, G.: Simplifying the form of Lie groups admitted by a given differential equation. J. Math. Anal. Appl. 145(1), 52–62 (1990)
Eriksson, S.-L., Orelma, H.: A New Cauchy type Integral Formula for Quaternionic k-hypermonogenic Functions. Modern Trends in Hypercomplex Analysis, Trends in Mathematics, pp. 175–189 (2016)
Eriksson, S.-L., Orelma, H.: General Integral Formulas for k-hypermogenic Functions. Adv. Appl. Clifford Algebras 27, 99–110 (2017)
Eriksson, S.-L., Orelma, H.: Hyperbolic Function Theory in the Skew-Field of Quaternions. Adv. Appl. Clifford Algebras 29, 97 (2019)
Eriksson, S.-L., Orelma, H.: Hyperbolic Laplace operator and the Weinstein equation in \(\mathbb {R}^{3}\). Complex Var. Elliptic Equ. 24(1), 109–124 (2014)
Eriksson, S.-L., Orelma, H.: Quaternionic hyperbolic function theory. In: Bernstein, S. (ed.) Topics in Clifford Analysis. Trends in Mathematics. Birkhäuser, Cham (2019)
Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin, New York-London, pp. xlv+ 1086 (1965)
Gurtin, M.E.: An Introduction to Continuum Mechanics, p 265. Academic Press, New York (1981)
Güngör, F.: The Schrödinger propagator on \((0,\infty )\) for a special potential by a Lie symmetry group method. Rend. Circ. Mat. Palermo II. Ser 70, 1609–1616 (2021)
Ibragimov, N.K.H.: Transformation groups applied to mathematical physics (Russian), p 280. Moscow, Nauka (1983)
Kovalenko, S., Stogniy, V., Tertychnyi, M.: Lie symmetries of fundamental solutions of one (2 + 1)-dimensional ultra-parabolic Fokker–Planck–Kolmogorov equation. arXiv:1408.0166 (2014)
Leutwiler, H.: Best constants in the Harnack inequality for the Weinstein equation. Aequationes Math. 34(2-3), 304–315 (1987)
Olver, P.: Applications of Lie groups to differential equations, Second edition, Graduate Texts in Mathematics, 107. Springer-Verlag, New York, pp. xxviii+ 513 (1993)
Ortner, N., Wagner, P.: On Green’s functions in generalized axially symmetric potential theory. Appl. Anal. 99(7), 1171–1180 (2020)
Ovsiannikov, L.V.: Group Analysis of Differential Equations, p xvi+ 416. Academic Press, Inc., New York, London (1982)
Weinstein, A.: Generalized axially symmetric potential theory. Bull. Amer. Math. Soc. 59, 20–38 (1953)
Acknowledgements
This paper was started during a visit by the second author to Moscow State University (MGU). The second author wishes to thank all the nice people at MGU for the inspirational atmosphere and hospitality, and YTK for financial support to complete this job. The authors wish to thank Prof. Sirkka-Liisa Eriksson for her important comments and discussions related to the Dirichlet problem. The authors wishes to thank unknown referees for valuable suggestions and commends.
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Appendix A: Proof of Proposition 6
Appendix A: Proof of Proposition 6
Using Eqs. 13 and 14, we have (for each i≠j)
and we may write Eqs. 11 and 12 of the form
From these, applying Eq. 13, we obtain the formulas
Substituting the preceding formulas into Eq. 10 (multiplied by 2), we obtain the equation
We assume that n ≥ 3 and k(2 − k) + 4ℓ≠ 0. Using Eqs. 13 and 14, we have
and
It is a third-order linear ordinary differential equation with respect to xn, and it has three linearly independent solutions. We look for solutions of the form \(\xi ^{n}(x)=h(\widetilde {x})(x^{n})^{\alpha }\), where \(\widetilde {x}=(x^{1},...,x^{n-1})\). Making the substitution, we obtain the equation (putting β = 2k − k2 + 4ℓ)
Assume that the root α≠ 1. Then
and by Eq. 14
and by Eq. 13
that is, h = 0, and we see that these solutions do not give us a nontrivial symmetry. Let us now look for symmetries for α = 1, i.e.,
Then we have by (13) and (14) that
and
Since \(\xi ^{n}_{x^{j}x^{j}}(x)=0\) for all j = 1,...,n, it has to be of the form
and hence
From (A.2), we infer
that is, \(\eta =\eta (\widetilde {x})\). Using (A.1), (A.3) and (A.4), we compute
for i = 1,...,n − 1. We infer that
Let us now compute the coefficients ξi. By Eq. 14, we obtain
for i = 1,...,n − 1 and we have
where ci(x) = ci(x1,...,xi− 1,xi+ 1,...,xn). If i≠p and
we have by Eq. 13 that
then we obtain that
for any p = 1,...,n. We have that
Since
we have that
Then we compute
Using Eq. 13, we obtain
where we assume i≠r≠q≠i. Since \(\xi ^{i}_{x^{r}x^{q}}=d^{i}_{rq}+d^{i}_{qr}\), Eq. A.5 gives
Changing the role of r and q in the last equation, we obtain that
Then the term
and
Then, by Eq. 13, we have
that is, \({e_{j}^{i}}=-{e_{i}^{j}}\) for i≠j and \({e^{j}_{j}}=0\). We obtain
where \({e_{j}^{i}}=-{e_{i}^{j}}\). Assume again, that j≠i. Then we compute
and Eq. 13 gives
and
Scaling the coefficients ai by 2, we have the solution
where \({e_{j}^{i}}=-{e_{i}^{j}}\).
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Aksenov, A.V., Orelma, H. Lie Symmetries of Fundamental Solutions to the Leutwiler-Weinstein Equation. Potential Anal 59, 789–821 (2023). https://doi.org/10.1007/s11118-022-10002-3
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DOI: https://doi.org/10.1007/s11118-022-10002-3