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MATHEMATICS
V. A. GOLUBEVA
SINGULARITIES OF FUNDAMENTAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
(Presented by Academician I. G. Petrovskii on 9 X 1962)
Consider a solution of a linear partial differential equation with constant real coefficients in three independent variables:
\[ \mathcal L\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\mathcal E(x,y,z)=\delta(x,y,z),\qquad (x,y,z)\in R^3, \]
where \(\mathcal L(\alpha,\beta,\gamma)\) is a homogeneous polynomial of degree \(m>4\). We shall call the function \(\mathcal E(x,y,z)\) a fundamental solution of the operator \(\mathcal L\). It is clear that \(\mathcal E(x,y,z)\) is a homogeneous function.
In the case when the algebraic curve \(\mathcal L(\alpha,\beta,1)=0\) has no real singular points, the construction of the function \(\mathcal E(x,y,z)\) was given by Zeilon \((^1)\). In the case when the algebraic curve \(\mathcal L(\alpha,\beta,1)=0\) has real singular points, the construction of the function \(\mathcal E(x,y,z)\) was given by the author in \((^4)\).
The function \(\mathcal E(x,y,z)\) is locally analytic at all points of the space \(R^3\), except for the points of the characteristic cone \(K(x,y,z)=0\) of the operator \(\mathcal L\). This theorem, in the case when the algebraic curve \(\mathcal L(\alpha,\beta,1)=0\) has no real singular points, was proved by V. A. Borovikov \((^2)\). The proof in the case when the algebraic curve \(\mathcal L(\alpha,\beta,1)=0\) has real singular points is due to the author \((^3)\).
V. A. Borovikov investigated the behavior of the function \(\mathcal E(x,y,z)\) in a neighborhood of an ordinary point of the characteristic cone \(K(x,y,z)=0\) of the operator \(\mathcal L\) \((^2)\). In the present work we describe the behavior of the function \(\mathcal E(x,y,z)\) in a neighborhood of certain singular points of the characteristic cone. The method of investigation is the usual method of the theory of integrals on algebraic curves, the method of local expansions \((^{5,6})\).
By virtue of the homogeneity of the function \(\mathcal E(x,y,z)\), its study in a neighborhood of points of the cone \(K(x,y,z)=0\) can be reduced to the study of the function \(E(x,y)=\mathcal E(x,y,1)\) in a neighborhood of points of the algebraic curve \(\Gamma(x,y)=K(x,y,1)=0\). It can be shown that the algebraic curve \(\Gamma(x,y)=0\) is dual \((^4)\) to the algebraic curve \(Q(\alpha,\beta)=\mathcal L(\alpha,\beta,1)=0\). In the present work we use the terminology of the theory of algebraic curves adopted in the monograph of R. Walker \((^4)\).
We shall assume that the algebraic curve \(Q(\alpha,\beta)=0\) is irreducible and has the simplest real singular points (double points), and we shall describe the behavior of the function \(E(x,y)\) in a neighborhood of the singular points of the algebraic curve \(\Gamma(x,y)=0\). In this case the problem of investigating the fundamental solution of the operator \(\mathcal L\) at all points of the space \(R^3\) is completely solved. The same method of investigating the function \(E(x,y)\) can also be used in the case when the algebraic curve \(Q(\alpha,\beta)=0\) has more complicated singular points; however, here a painstaking study of the structure of the dual curve \(\Gamma(x,y)=0\) in a neighborhood of the corresponding points is required, and we have not undertaken this.
The function under consideration \(E(x,y)\) has the form
\[ E(x,y)=\operatorname{Re}\frac{i}{4\pi\Gamma(m-2)} \sum_\nu \int_{\beta_\mu(x,y)}^{\beta_\nu(x,y)} \frac{(\alpha_\nu x+\beta y+1)^{m-3}\alpha\beta}{Q_\alpha(\alpha_\nu,\beta)}; \]
the integration in each term of the sum is carried out along the algebraic curve \(Q(\alpha,\beta)=0\); the upper limits of the integrals in the sum are points of this curve, whose coordinates \(\beta_\nu(x,y)\) are complex roots with positive imaginary part for \(x>0\) (with negative imaginary part for \(x<0\)) of the algebraic equation
\[ Q\left(-\frac{1}{x}(1+\beta y),\beta\right)=0; \]
the summation in each region of the \(x,y\)-plane is performed over all such roots of this equation; \(\beta_\mu(x,y)\) is the root of the same equation complex-conjugate to the root \(\beta_\nu(x,y)\) in the region where these roots are complex; in the region of the \(x,y\)-plane in which the roots \(\beta_\nu(x,y)\) and \(\beta_\mu(x,y)\) are real, the value of the corresponding integral of the sum is determined by continuity.
We shall give the expansion of the function \(E(x,y)\) in a neighborhood of a singular point of the curve \(\Gamma(x,y)=0\) corresponding to an ordinary point of the curve \(Q(\alpha,\beta)=0\) with zero curvature; in a neighborhood of an isolated singular point of the curve \(\Gamma(x,y)=0\); in a neighborhood of an inflection point of the curve \(\Gamma(x,y)=0\), corresponding to a cusp of the first kind of the curve \(Q(\alpha,\beta)=0\); in a neighborhood of the points of a real double tangent to the curve \(\Gamma(x,y)=0\), corresponding to a point of self-intersection of the curve \(Q(\alpha,\beta)=0\); in a neighborhood of a point of self-tangency of the curve \(\Gamma(x,y)=0\), corresponding to the same kind of point of the curve \(Q(\alpha,\beta)=0\).
The expansion of the function \(E(x,y)\) in a neighborhood of the point under consideration will in all cases have, in specially chosen local coordinates \(u,v\), analytically depending on the variables \(x,y\), analytically invertible and such that \(\partial(u,v)/\partial(x,y)=1\), the form
\[ E(x,y)=E'(u,v)=\Phi(u,v)+\Psi(u,v), \]
where \(\Psi(u,v)\) is an analytic function; the expansion of the function \(\Phi(u,v)\) will be specified in each case.
- Let the point \((1,0)\) of the algebraic curve \(Q(\alpha,\beta)=0\) be an ordinary point with zero curvature, and let the function \(Q(\alpha,\beta)\) in a neighborhood of this point have the expansion
\[ Q(\alpha,\beta)=(\alpha-1)+\frac{(n-1)^{n-1}}{n^n}\beta^n+\cdots . \]
Then the corresponding point \((-1,0)\) of the dual curve \(\Gamma(x,y)=0\) will be a singular point, in a neighborhood of which one can introduce such local coordinates \(u,v\) that the function \(\Gamma(x,y)\) in these coordinates takes the form:
\[ \Gamma(x,y)=\Gamma'(u,v)=[u^{\,n-1}+(-1)^{n-1}v^n]\Lambda(u,v), \]
where \(\Lambda(u,v)\) is an analytic function satisfying the condition \(\Lambda(0,0)=1\).
The function \(\Phi(u,v)\) in a neighborhood of the point \((0,0)\) has the expansion
\[ \Phi(u,v)=\sum_{i=1}^{i=[n/2]}\Phi_i(u,v), \]
where
\[ \Phi_1(u,v)= \begin{cases} |u|^{m-3+1/n} f_k(\varphi_1(k))+\chi(k,u), & \text{for } k<1,\\ 0, & \text{for } k>1; \end{cases} \]
\[ \Phi_i(u,v)=|u|^{m-3+1/n}f_k(\varphi_i(k))+\chi(k,u) \quad \text{for all } k,\quad i=2,\ldots,\left[\frac{n}{2}\right]; \]
\[ f_k(\varphi_i(k))= \frac{i}{4\pi\Gamma(m-2)} \int_{\varphi(k)}^{\varphi(k)} \left[ 1-zk^{1/n}+\frac{(n-1)^{n-1}}{n^n}z^n \right]^{m-3} dz; \]
\[
k=(-1)^n v^n/u^{n-1};
\]
the summation is carried out over the complex roots of the equation
\[ \frac{(n-1)^{n-1}}{n^n}\varphi^n-k^{1/n}\varphi+1=0, \]
satisfying the condition \(\operatorname{Im}\varphi(k)u^{1/n}>0\) (for \(k>1\) two roots of this equation become real); \(\chi(k,u)\) is a function containing, for every fixed \(k\), terms of higher order with respect to \(u\).
- Let the algebraic curve \(\Gamma(x,y)=0\) have an isolated singular point with coordinates \((-1,0)\), and let the function \(\Gamma(x,y)\) in a neighborhood of this point have the expansion
\[ \Gamma(x,y)=(x+1)^2+y^2+a(x+1)^3+b(x+1)y^2+\cdots \]
It can be shown that the function \(E(x,y)\) is analytic in a neighborhood of the point \((-1,0)\).
- Let the algebraic curve \(Q(\alpha,\beta)=0\) have a cusp of the first kind with coordinates \((1,0)\), and let the function \(Q(\alpha,\beta)\) in a neighborhood of this point have the expansion
\[ Q(\alpha,\beta)=(\alpha-1)^2-\beta^3+\cdots \]
The corresponding part of the dual curve \(\Gamma(x,y)=0\) consists of two components: the branch of the curve having a point of inflection, whose equation in the corresponding local coordinates is
\[ u+\frac{4}{27}v^3=0, \]
and the tangent line to this branch with equation \(u=0\).
The function \(\Phi(u,v)\) in a neighborhood of the point \((0,0)\) has the form
\[ \Phi(u,v)= \begin{cases} u^{m-1/6}f_k(\varphi(k))+\chi(k,u), & \text{for } -\dfrac{4}{27}<k<\infty,\\ 0, & \text{for } -\infty<k<-\dfrac{4}{27}, \end{cases} \]
where
\[ f_k(\varphi(k))= \frac{i}{8\pi\Gamma(m-2)} \int_{\varphi(k)}^{\overline{\varphi(k)}} \frac{(1+zk^{1/3}-z^{1/2})^{m-3}-(1+zk^{1/3}+z^{1/2})^{m-3}}{z^{3/2}}\,dz; \]
\[
k=v^3/u;
\]
\(\varphi(k)\) is the complex root with positive imaginary part of the cubic equation
\[ \varphi^3-k^{2/3}\varphi^2-2k^{1/3}\varphi-1=0 \qquad \left(-\frac{4}{27}<k<\infty\right) \]
(for \(-\infty<k<-\frac{4}{27}\) the roots of this equation are real); \(\chi(k,u)\) is a function containing, for every fixed \(k\), terms of higher order with respect to \(u\).
- Let the algebraic curve \(Q(\alpha,\beta)=0\) have a self-intersection point with coordinates \((1,0)\), and let the function \(Q(\alpha,\beta)\) in a neighborhood of the point \((1,0)\) have the expansion
\[ Q(\alpha,\beta)=(\alpha-1)^2-\beta^2+a(\alpha-1)^3+b(\alpha-1)\beta^2+\cdots,\qquad a>0,\ b>0. \]
The corresponding part of the dual curve \(\Gamma(x,y)=0\) consists of three components: two branches of the curve, having the following local equations in the corresponding local coordinates:
\[ u=-\frac{1}{4l}v_i^2,\qquad l=\frac{a+b}{2},\qquad i=1,2 \]
(the first branch passes through the point \((-1,+1)\), the second through the point \((-1,-1)\)), and their common tangent with equation \(u=0\) (\(u=x+1\)).
The function \(\Phi(u,v_i)\) in a neighborhood of the point \((0,0)\) has the expansion
\[ \Phi(u,v_i)= \begin{cases} u^{m-1/2}f_k(\varphi(k))+\chi_i(k,u), & \text{for } k+4l<0,\\ 0, & \text{for } k+4l>0,\ v_1>0\ (v_2<0),\\ \sigma(u), & \text{for } k+4l>0,\ v_1<0\ (v_2>0). \end{cases} \]
where
\[ f_k(\varphi(k))=\frac{-i}{4\pi\Gamma(m-3)} \int_{\overline{\varphi}(k)}^{\varphi(k)} (1+zk^{1/2}-lz^2)^{m-4}\,dz; \]
\(k=v_i^2/u\); \(\varphi(k)\) is the complex root with positive imaginary part of the quadratic equation
\[ l\varphi^2-\varphi k^{1/2}-1=0 \qquad (k+4l<0) \]
(for \(k+4l>0\) the roots of this equation are real); \(\chi_i(k,u)\) is a function which, for every fixed \(k\), contains terms of higher order relative to \(u\); \(\sigma(u)\) is an analytic function.
The expansion of the function \(E(x,y)\) along the normal to the double tangent has the form
\[ E(x,y)= \begin{cases} -\dfrac{(x+1)^{m-3}}{2\pi\Gamma(m-2)} +o\!\left[(x+1)^{m-2}\right], & \text{for } |y|<1,\ u>0,\\[6pt] 0, & \text{for } |y|>1,\ u>0,\\ 0, & \text{for any } y,\ u<0. \end{cases} \]
-
If the algebraic curve \(\Gamma(x,y)=0\) has an isolated double tangent, then one can show that the function \(E(x,y)\) is analytic in a neighborhood of this line.
-
Let the algebraic curve \(Q(\alpha,\beta)=0\) have a point of self-tangency with coordinates \((1,0)\), and let the function \(Q(\alpha,\beta)\) in a neighborhood of this point have the expansion
\[ Q(\alpha,\beta)=[(\alpha-1)-b_1\beta^2]\,[(\alpha-1)-b_2\beta^2]+\ldots,\qquad b_2<b_1<0. \]
The corresponding point \((1,0)\) of the dual curve \(\Gamma(x,y)=0\) is also a point of self-tangency, in a neighborhood of which one can introduce local coordinates \(u,v\), in which the function \(\Gamma(x,y)\) takes the form
\[ \Gamma(x,y)=\Gamma'(u,v)= \left(u+\frac{1}{4b_1}v^2\right) \left(u+\frac{1}{4b_2}\chi(v)\right)\Lambda(u,v), \]
where \(\chi(v)\) is an analytic function satisfying the conditions \(\chi(0)=\chi'(0)=0,\ \chi''(0)=2\); \(\Lambda(u,v)\) is an analytic function satisfying the condition \(\Lambda(0,0)=1\).
In this case, in a neighborhood of the point \((0,0)\), the function \(\Phi(u,v)\) has the expansion
\[ \Phi(u,v)= \begin{cases} u^{m-7/2}\,[f_{1k}(\varphi_1(k))+f_{2k}(\varphi_2(k))]+\chi_1(k,u), & \text{for } k+4b_1<0,\\ u^{m-7/2}f_{2k}(\varphi_2(k))+\chi_2(k,u), & \text{for } k+4b_2<0,\\ 0, & \text{for } k+4b_2>0, \end{cases} \]
where
\[ f_{jk}(\varphi_j(k))= \frac{i}{4\pi\Gamma(m-2)} \int_{\overline{\varphi}_j(k)}^{\varphi_j(k)} \frac{(1+zk^{1/2}-b_1z^2)^{m-3}-(1+zk^{1/2}-b_2z^2)^{m-3}} {(b_1-b_2)z^2}\,dz; \]
\(k=v^2/u\); \(\varphi_j(k)\) is the complex root with positive imaginary part of the quadratic equation
\[ b_j\varphi^2-\varphi k^{1/2}-1=0 \qquad (k+4b_j<0;\ j=1,2) \]
(for \(k+4b_j>0\) the roots of this equation are real); \(\chi_1(k,u)\), \(\chi_2(k,u)\) are functions which, for every fixed \(k\), contain terms of higher order relative to \(u\).
Received
5 X 1962
References
- N. Zeilon, Ark. f. Math., Astr. och Fys., 9 (1913).
- V. A. Borovikov, Tr. Mosk. matem. obshch., 8, 199 (1959).
- V. A. Golubeva, Dokl. AN AzSSR, 18, No. 2 (1962).
- R. Walker, Algebraic Curves, IL, 1952.
- P. Appell, E. Goursat, Théorie des fonctions algébriques et de leurs intégrales, Paris, 1929.
- F. Klein, Gesammelte mathematische Abhandlungen, 2, Berlin, 1922.