Gradient Structures Associated with a Polynomial Differential Equation

Abstract

In this paper, by using the characteristic system method, the kernel of a polynomial differential equation involving a derivation in ${\displaystyle R^n}$ is described by solving the Cauchy Problem for the corresponding first order system of PDEs. Moreover, the kernel representation has a special significance on the space of solutions to the corresponding system of PDEs. As very important applications, it has been established that the mathematical framework developed in this work can be used for the study of some second-order PDEs involving a finite set of derivations.

1. Introduction

Throughout time, gradient-type representations for some solutions, gradient systems in a Lie algebra and the algebraic representation of gradient systems, have been investigated, with remarkable results, by Vârsan [1]. Moreover, stochastic partial differential equations (SPDEs) of Hamilton–Jacobi type including non $F_T$-adapted solutions have been studied in Ijacu and Vârsan [2]. By using the commuting property of the drift and diffusion vector fields with respect to the usual Lie bracket, a representation for a classical solution of some nonlinear SPDEs driven by Fisk–Stratonovich stochastic integral was constructed by Iftimie et al. [3]. Furthermore, sufficient conditions for linear subspaces of smooth vector fields in order to be written as a kernel of some linear first order partial differential equations are have been formulated and proved in Parveen and Akram [4]. Further, Treanţă and Vârsan [5] proved that solutions associated with an extended affine control system can be obtained as a limit process using solutions for a parameterized affine control system and weak small controls. Recently, Treanţă [6] studied affine control systems with jumps for which the ideal generated by the drift vector field can be imbedded as a kernel of a linear first-order partial differential equation. Mainly, these references motivate the present study. For other different but connected viewpoints regarding this subject, the reader is directed to Friedman [7], Sussmann [8], Crandall and Souganidis [9], Sontag [10], Bressan and Shen [11], Nonlaopon [12], Saira et al. [13] and Treanţă [14,15,16].

In this paper, taking into account the results included in the quite recently work Treanţă [17] (the kernel of a polynomial of scalar derivations is described by solving Cauchy Problems for the corresponding system of ODEs; also, a gradient representation for the associated Cauchy Problem solution is derived), we investigate the kernel of a polynomial differential equation involving a derivation in ${\displaystyle R^n}$ by solving the Cauchy Problem for the corresponding first order system of PDEs. Furthermore, we extend a solution by considering Radon measures and their bounded variation functions, or Wiener and Levy processes. Moreover, it is established that the kernel representation has a special significance on the space of solutions to the corresponding system of PDEs.

This paper is organized as follows. In Section 2, in order to delineate certain steps in the solving algorithm proposed for the main result (Theorem 1), some preliminary results are formulated. More precisely, two crucial lemmas for the present paper are mentioned. Further, we establish two important remarks. A gradient structure for the associated Cauchy Problem solution is provided by Remark 1. The final part of this section, including Remark 2, extends a solution considering Radon measures and their bounded variation functions, or using Wiener (or Levy) processes. The aim of Section 3 is to provide a characterization for the kernel of a polynomial differential equation involving a derivation in ${\displaystyle R^n}$. Specifically, through the use of the characteristic system method and some results formulated in Section 2, the associated Cauchy Problem solution is derived (see Theorem 1). Moreover, this solution has a special significance on the space of solutions to the corresponding first order system of PDEs. Finally, Section 4 concludes the paper.

2. Preliminary Results

In this section, taking into account a very recent work (see Treanţă [17]), some auxiliary results are formulated.

Let ${\displaystyle 0\in I\subseteq R}$ be an open interval. Consider a polynomial of the scalar derivation ${\displaystyle \frac{d}{dt}}$,

$${\displaystyle P_m\left(t;\frac{d}{dt}\right)=a_1(t)+a_2(t)\left(\frac{d}{dt}\right)+\cdots +a_m(t){\left(\frac{d}{dt}\right)}^{m-1}-{\left(\frac{d}{dt}\right)}^m,}$$

(1)

where ${\displaystyle m\ge 1,a_j\in L_{\infty }\left(I\right),j\in \left\{1,2,\dots ,m\right\}}$. Define

$${\displaystyle H_{\infty }^m\left(I\right)=\left\{h\in C^{m-1}(I):{\left(\frac{d}{dt}\right)}^m(h)\in L_{\infty }\left(I\right)\right\}}$$

(2)

and consider ${\displaystyle Ker\left(P_m\right)\subseteq H_{\infty }^m\left(I\right)}$, where

$${\displaystyle Ker\left(P_m\right)=\left\{h\in H_{\infty }^m\left(I\right):P_m\left(t;\frac{d}{dt}\right)\left(h\right)(t)=0,a.e.t\in I\right\}.}$$

(3)

The procedure of characteristic systems (see Friedman [7], Vârsan [1]) allows us to describe ${\displaystyle Ker\left(P_m\right)}$ by solving Cauchy Problems for the corresponding system of ODEs using a vector variable

$${\displaystyle y=col\left(y_1,y_2,\dots ,y_m\right),\frac{dy}{dt}=Ay+\sum _{i=1}^m{a}_i(t)B_{i}y,y(0)=y_0\in R^m.}$$

(4)

Here, the ${\displaystyle (m\times m)}$ constant matrices ${\displaystyle A}$ and ${\displaystyle B_i,i=\overline{1,m}}$, are defined by

$${\displaystyle A=\left[0e_1\cdots e_{m-1}\right],B_1=\left[e_{1}0\cdots 0\right],\cdots ,B_m=\left[00\cdots e_m\right],}$$

(5)

where ${\displaystyle \left\{e_1,\dots ,e_m\right\}}$ is the canonical basis and ${\displaystyle 0\in R^m}$ is the origin. By definition

$${\displaystyle \left[B_i,B_j\right]:=B_j{B}_i-B_i{B}_j,i,j\in \left\{1,2,\dots ,m\right\},(Liebracket),}$$

(6)

and making a direct computation, we get

$${\displaystyle O=\left[B_i,B_j\right],i,j\in \left\{1,2,\dots ,m\right\},}$$

(7)

with ${\displaystyle O}$—null matrix, and

$${\displaystyle A^m=O,(Aisanilpotentmatrix).}$$

(8)

The Cauchy Problem solution for $(4)$ is represented by

$${\displaystyle y(t;y_0)=\left[expAt\right]\widehat{y}(t;y_0),t\in I,}$$

(9)

where ${\displaystyle \left\{\widehat{y}(t;y_0):t\in I\right\}}$ fulfils the following linear system (initial value problem)

$${\displaystyle \frac{dy}{dt}=\sum _{i=1}^m{a}_i(t)A_i(t)y,t\in I,y(0)=y_0\in R^m.}$$

(10)

Write the ${\displaystyle (m\times m)}$ matrices

$${\displaystyle A_i(t):=\left[exp(-tA)\right]B_i\left[exptA\right],i\in \left\{1,2,\dots ,m\right\},}$$

(11)

as follows

$${\displaystyle A_i(t)=\left[expt{ad}_A\right](B_i),i\in \left\{1,2,\dots ,m\right\},t\in I,}$$

(12)

where the linear mapping ${\displaystyle {ad}_A:M_{m\times m}\to M_{m\times m}}$ is given by ${\displaystyle {ad}_A(B):=BA-AB}$ (see ${\displaystyle \left[A,B\right]}$). In addition, using $(7)$, $(8)$ and $(12)$, we get

$${\displaystyle A_i(t)=B_i+\frac{t}{1!}{ad}_A(B_i)+\cdots +\frac{t^{m-1}}{(m-1)!}{({ad}_A)}^{m-1}(B_i),i\in \left\{1,\dots ,m\right\}.}$$

(13)

Denote ${\displaystyle N=m^2}$ and define ${\displaystyle N}$ matrices ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}\subseteq M_{m\times m}}$, as follows

$$\begin{array}{c}\left\{C_1,C_2,\dots ,C_N\right\}=\left\{a{d}_A^k(B_1):k\in \left\{0,1,2,\dots ,m-1\right\}\right\}\cup \cdots \cup \\ \left\{a{d}_A^k(B_m):k\in \left\{0,1,2,\dots ,m-1\right\}\right\}.\end{array}$$

(14)

Furthermore, let $\left\{{\alpha }_1(t),{\alpha }_2(t),\dots ,{\alpha }_N(t):t\in I\right\}$ be given by

$$\begin{array}{c}\left\{{\alpha }_1(t),\dots ,{\alpha }_m(t)\right\}=a_1(t)\left[1,\frac{t}{1!},\dots ,\frac{t^{m-1}}{(m-1)!}\right],\\ ⋮\\ \left\{{\alpha }_{N-m+1}(t),\dots ,{\alpha }_N(t)\right\}=a_m(t)\left[1,\frac{t}{1!},\dots ,\frac{t^{m-1}}{(m-1)!}\right].\end{array}$$

(15)

With these notations, we write ODE $(10)$ as follows

$${\displaystyle \frac{dy}{dt}=\sum _{j=1}^N{\alpha }_j(t)Y_j(y),t\in I,y(0)=y_0,}$$

(16)

where ${\displaystyle Y_j(y):=C_{j}y,j\in \left\{1,2,\dots ,N\right\}}$.

Lemma 1

([17]). Consider ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}}$ defined in $(14)$, with ${\displaystyle N=m^2}$. Then ${\displaystyle \left\{C_1,C_2,\dots ,C_N\right\}}$ is a basis for ${\displaystyle M_{m\times m}}$ and

$${\displaystyle \left\{Y_1(y)=C_{1}y,Y_2(y)=C_{2}y,\dots ,Y_N(y)=C_{N}y\right\}}$$

is a system of generators for the Lie algebra ${\displaystyle L\left(Y_1,\dots ,Y_N\right)\subseteq C^{\infty }\left(R^m;R^m\right)}$ generated by ${\displaystyle \left\{Y_1,\dots ,Y_N\right\}}$.

Lemma 2

([17]). Assume ${\displaystyle a_j(t)\equiv 0,j\in \left\{1,2,\dots ,i\right\}}$, for some ${\displaystyle 1\le i\le m}$. Define a subspace ${\displaystyle S_i=span\left\{e_1,\dots ,e_i\right\}\subseteq R^m}$ and its orthogonal complement ${\displaystyle S_i^{\perp }\subseteq R^m}$, where ${\displaystyle \left\{e_1,\dots ,e_m\right\}\subseteq R^m}$ is the canonical basis. Then

$${\displaystyle \forall y_0\in S_{i}isastationarypointforODE(10).}$$

(17)

In addition, for each ${\displaystyle y_0\in S_i^{\perp },y_0\ne 0}$, the following statements are valid:

$${\displaystyle dimL\left(Y_{j(i)},\dots ,Y_N\right)(y_0)=dim\left[span\left\{Y_{j(i)}(y_0),\dots ,Y_N(y_0)\right\}\right]=m-i,}$$

(18)

$$\begin{array}{c}\left\{C_{j(i)},\dots ,C_N\right\}isabasisfor{M}_{m\times m}^i:=L\left(C_{j(i)},\dots ,C_N\right),j(i):=mi+1\\ \widehat{y}(t,y_0)\in S_i^{\perp },t\in I,foranysolutionofODE(10).\end{array}$$

(19)

In the following, we establish two important remarks for the main result associated with this paper.

Remark 1.

Any solution of ODE $(4)$ can be represented using

$${\displaystyle p(t):=\left(t_1(t),\dots ,t_m(t)\right),t_j(t)={\int }_0^t{a}_j(s)ds,j\in \left\{1,2,\dots ,m\right\},t\in I.}$$

(20)

In this regard, notice that the matrices ${\displaystyle \left\{B_1,\dots ,B_m\right\}\subseteq M_{m\times m}}$ commute (see $(6)$) and the linear mapping

$${\displaystyle G(p)\lambda :=\left[exp{t}_1{B}_1\right]\cdots \left[exp{t}_m{B}_m\right]\lambda ,p=\left(t_1,\dots ,t_m\right)\in R^m,\lambda \in R^m}$$

(21)

satisfies a gradient system

$${\displaystyle {\partial }_{t_1}G(p)\lambda =B_{1}G(p)\lambda ,\cdots ,{\partial }_{t_m}G(p)\lambda =B_{m}G(p)\lambda .}$$

(22)

In particular, for ${\displaystyle p=p(t),t\in I}$ (see $(20)$), we get that any ${\displaystyle \left\{y(t,y_0):t\in I\right\}}$ satisfying ODE $(4)$ is represented by

$${\displaystyle y(t;y_0)=G\left(p(t)\right)\widehat{y}(t;y_0),t\in I,y_0\in R^m,}$$

(23)

where ${\displaystyle \left\{\widehat{y}(t;y_0):t\in I\right\}\subseteq C(I,R^m)}$ fulfils

$${\displaystyle \frac{dy}{dt}=G\left(-p(t)\right)AG\left(p(t)\right)y,y(0)=y_0,t\in I.}$$

(24)

In addition, ${\displaystyle A(t):=G\left(-p(t)\right)AG\left(p(t)\right),t\in I}$, defined in $(24)$, can be written as (see “${\displaystyle \circ }$” as the symbol for composing functions)

$${\displaystyle A(t)=\left[exp{t}_1(t){ad}_{B_1}\right]\circ \dots \circ \left[exp{t}_m(t){ad}_{B_m}\right](A),t\in I,}$$

(25)

where the adjoint mapping ${\displaystyle {ad}_B:M_{m\times m}\to M_{m\times m}}$ is defined by

$${\displaystyle {ad}_B(C)=\left[B,C\right]=CB-BC,(Liebracket).}$$

(26)

A direct computation leads us to

$${\displaystyle \left[expt{ad}_{B_m}\right](A)=A+\frac{t}{1!}{ad}_{B_m}(A)+\cdots +\frac{t^k}{k!}a{d}_{B_m}^k(A)+\cdots ,}$$

(27)

where ${\displaystyle a{d}_{B_m}^k(A)=C_m:=\left[0\dots 0e_{m-1}\right]}$, for any ${\displaystyle k\ge 1}$. Rewrite $(27)$ as

$${\displaystyle \left[expt{ad}_{B_m}\right](A)=A+\left[expt-1\right]C_m,C_m=\left[0\dots 0e_{m-1}\right].}$$

(28)

Notice that

$${\displaystyle {ad}_{B_{m-1}}(C_m)=C_{m-1}-B_m,{ad}_{B_{m-1}}(C_{m-1})=O,}$$

(29)

where ${\displaystyle C_{m-1}:=\left[0\dots 0e_{m-1}0\right]}$. Using $(28)$ and $(29)$, we obtain

$${\displaystyle \left[exp{t}_{m-1}{ad}_{B_{m-1}}\right](C_m)=C_m+t_{m-1}(C_{m-1}-B_m)}$$

and

$$\begin{array}{c}\left[exp{t}_{m-1}{ad}_{B_{m-1}}\right]\circ \left[exp{t}_m{ad}_{B_m}\right](A)\\ =\left[(exp{t}_m)-1\right]\left[exp{t}_{m-1}{ad}_{B_{m-1}}\right](C_m)+\left[exp{t}_{m-1}{ad}_{B_{m-1}}\right](A)\\ {\displaystyle =\left[(exp{t}_m)-1\right]\left[C_m+t_{m-1}\left(C_{m-1}-B_m\right)\right]+\left[exp{t}_{m-1}{ad}_{B_{m-1}}\right](A).}\end{array}$$

(30)

On the other hand, compute

$${\displaystyle \left[exp{t}_{m-1}{ad}_{B_{m-1}}\right](A)=A+t_{m-1}\left(C_{m-1}-B_m\right)-\frac{t_{m-1}^2}{2!}{B}_{m-1},}$$

(31)

where ${\displaystyle C_{m-1}=\left[0\dots 0e_{m-1}0\right]}$, and inserting $(31)$ into $(30)$, we get

$$\begin{array}{c}\left[exp{t}_{m-1}{ad}_{B_{m-1}}\right]\circ \left[exp{t}_m{ad}_{B_m}\right](A)\\ =A+(exp{t}_m)\left[C_m+t_{m-1}\left(C_{m-1}-B_m\right)\right]-C_m-\frac{t_{m-1}^2}{2!}{B}_{m-1}.\end{array}$$

(32)

Notice that (see $(31)$)

$${\displaystyle \left[exp{t}_{m-2}{ad}_{B_{m-2}}\right](A)=A+t_{m-2}\left(C_{m-2}-B_{m-1}\right),}$$

(33)

and applying ${\displaystyle \left[exp{t}_{m-2}{ad}_{B_{m-2}}\right]}$ to $(32)$, we obtain

$$\begin{array}{c}\left[exp{t}_{m-2}{ad}_{B_{m-2}}\right]\circ \left[exp{t}_{m-1}{ad}_{B_{m-1}}\right]\circ \left[exp{t}_m{ad}_{B_m}\right](A)\\ =A+t_{m-2}\left(C_{m-2}-B_{m-1}\right)+(exp{t}_m)\left[C_m+t_{m-1}\left(C_{m-1}-B_m\right)\right]\\ {\displaystyle -C_m-\frac{t_{m-1}^2}{2!}{B}_{m-1}+t_{m-2}\left[(exp{t}_m)-1\right]C_{m-2}.}\end{array}$$

(34)

This computation shows that

$${\displaystyle \hat{A}(t_1,\dots ,t_m):=\left[exp{t}_1{ad}_{B_1}\right]\circ \dots \circ \left[exp{t}_m{ad}_{B_m}\right](A)}$$

and, in particular, ${\displaystyle A(t)=\hat{A}(t_1(t),\dots ,t_m(t))}$ (see $(25)$), can be written as

$${\displaystyle \hat{A}(t_1,\dots ,t_m)=A+\sum _{k=1}^m{\alpha }_k(t_1,\dots ,t_m)B_k+\sum _{k=1}^m{\beta }_k(t_1,\dots ,t_m)C_k,}$$

(35)

for ${\displaystyle {\alpha }_k,{\beta }_k\in C^{\omega }(R^m)}$.

Remark 2.

Any solution of ODE $(4)$ can be represented as in $(23)$ (see Remark 1) and it allows us to extend a solution considering Radon measures and their bounded variation functions

$${\displaystyle p(t):=\left(t_1(t),\dots ,t_m(t)\right),t_j(t)={\mu }_j(t)={\int }_0^{t}d{\mu }_j(s),j\in \left\{1,2,\dots ,m\right\},}$$

(36)

with ${\displaystyle t\in [0,T]\subseteq I}$, replacing ${\displaystyle p\in C([0,T];R^m)}$ defined in $(20)$.

In addition, any solution of ODE $(4)$ can be represented as in $(23)$ using Wiener, ${\displaystyle W(t,w):[0,T]\times \Omega \to R^m}$, or Levy processes, ${\displaystyle p(t,w):[0,T]\times \Omega \to R^m}$, replacing ${\displaystyle p\in C([0,T];R^m)}$ in $(20)$. If it is the case, then any solution of ODE $(4)$ and, as a consequence, ${\displaystyle Ker\left(P_m\right)}$ defined in $(3)$, changes accordingly. We get (see $(23)$)

$$\begin{array}{c}y(t,w;y_0)=G(W(t,w))\hat{y}(t,w;y_0),t\in [0,T],\\ y(t,w;y_0)=G(p(t,w))\hat{y}(t,w;y_0),t\in [0,T],\end{array}$$

(37)

where ${\displaystyle \left\{\hat{y}(t,w;y_0):t\in [0,T],w\in \Omega \right\}}$ is a continuous process satisfying a system of ODEs

$${\displaystyle \frac{dy}{dt}=G\left(-\hat{p}(t,w)\right)AG\left(\hat{p}(t,w)\right)y,y(0)=y_0,t\in [0,T],}$$

(38)

with ${\displaystyle \hat{p}(t,w)=W(t,w),t\in [0,T],w\in \Omega }$, or ${\displaystyle \hat{p}(t,w)=p(t,w),t\in [0,T],w\in \Omega }$.

3. Main Results

This section contains the main result associated with the present paper. In order to formulate and prove it, we start with the following mathematical tools and hypotheses.

Let ${\displaystyle D\subseteq R^n}$ be an open and convex set. Consider a derivation mapping ${\displaystyle \overrightarrow{X}:C^{\infty }(D)\to C^m(D)}$ given by ${\displaystyle \overrightarrow{X}(\phi )(x):=\sum _{i=1}^n{X}_i(x){\partial }_i\phi (x)}$, where ${\displaystyle X_i\in C^m(D),1\le i\le n,\phi \in C^{\infty }(D)}$. Associate a polynomial differential equation

$${\displaystyle P_m\left(x;\overrightarrow{X}\right)(\phi )(x):=f(x)+\sum _{j=1}^m{a}_j(x)y_j(x)-y_{m+1}(x)=0,}$$

(39)

where ${\displaystyle f,a_j\in C^1(D),x\in \hat{B}:=B(x_0,\rho )\subseteq D,y_1=\phi ,\overrightarrow{X}(y_j)=y_{j+1}}$, and ${\displaystyle \left(X_1(x_0),\dots ,X_n(x_0)\right)\ne 0_{R^n}}$. Assume that

$${\displaystyle X_n(x)\ne 0,\forall x\in B(x_0,\rho )\subseteq D,}$$

(40)

where ${\displaystyle X=\left(X_1,\dots ,X_n\right)}$. Define a restricted kernel of ${\displaystyle P_m}$ to ${\displaystyle \hat{B}}$ by

$${\displaystyle Ker{P}_m{|}_{\hat{B}}:=\left\{\phi \in C^m(\hat{B}\subseteq D):P_m\left(x;\overrightarrow{X}\right)(\phi )(x)=0,x\in \hat{B}\right\}.}$$

(41)

A procedure for describing ${\displaystyle \left\{Ker{P}_m{|}_{\hat{B}}\right\}}$ is to associate a first order system of PDEs involving a vector state variable ${\displaystyle y=\left(y_1,y_2,\dots ,y_m\right)}$, ${\displaystyle y_i(x):={(\overrightarrow{X})}^{(i-1)}(\phi )(x),1\le i\le m}$. More precisely, consider the following first order system of PDEs

$$\begin{array}{c}{\displaystyle \overrightarrow{X}(y_1)(x)=y_2(x),\dots ,\overrightarrow{X}(y_{m-1})(x)=y_m(x)}\\ {\displaystyle \overrightarrow{X}(y_m)(x)=f(x)+\sum _{i=1}^m{a}_i(x)y_i(x).}\end{array}$$

(42)

A Cauchy Problem (CP) for $(42)$ is defined considering a hyperplane ${\displaystyle H:=\left\{(\hat{x},x_{0n})\in R^n:\hat{x}\in R^{n-1}\right\}}$ (see ${\displaystyle x_0:=({\hat{x}}_0,x_{0n})}$) and some fixed Cauchy conditions on ${\displaystyle H\cap D}$

$${\displaystyle y_1(\hat{x},x_{0n})=y_1^0(\hat{x}),\dots ,y_m(\hat{x},x_{0n})=y_m^0(\hat{x}),\hat{x}\in B({\hat{x}}_0,2\rho )\subseteq R^{n-1},}$$

(43)

where ${\displaystyle y_{i+1}^0\in C^{m-i}\left(B({\hat{x}}_0,2\rho )\right)}$ satisfies ${\displaystyle y_{i+1}^0={(\overrightarrow{X})}^{(i)}(y_1^0),1\le i\le m-1}$, and ${\displaystyle \left\{(\hat{x},x_{0n}):\hat{x}\in B({\hat{x}}_0,2\rho )\subseteq R^{n-1}\right\}\subseteq H\cap D}$.

A solution for (CP) (see $(42)$ and $(43)$) is found by solving the corresponding characteristic system

$$\begin{array}{c}{\displaystyle \frac{d{\hat{y}}_1}{d\sigma }(\sigma ,\lambda )={\hat{y}}_2(\sigma ,\lambda ),\dots ,\frac{d{\hat{y}}_{m-1}}{d\sigma }(\sigma ,\lambda )={\hat{y}}_m(\sigma ,\lambda ),}\\ {\displaystyle \frac{d{\hat{y}}_m}{d\sigma }(\sigma ,\lambda )=\hat{f}(\sigma ,\lambda )+\sum _{i=1}^m{\hat{a}}_i(\sigma ,\lambda ){\hat{y}}_i(\sigma ,\lambda ),\sigma \in [-\alpha ,\alpha ],}\end{array}$$

(44)

with Cauchy conditions ${\displaystyle {\hat{y}}_i(0,\lambda )=y_i^0(\lambda ),\lambda \in B({\hat{x}}_0,2\rho )\subseteq R^{n-1},1\le i\le m}$. Here, the smooth functions ${\displaystyle y_i^0}$ are fixed in $(43)$ and ${\displaystyle \hat{f},{\hat{a}}_i}$ are given by

$${\displaystyle \hat{f}(\sigma ,\lambda )=f(x(\sigma ,\lambda )),{\hat{a}}_i(\sigma ,\lambda )=a_i(x(\sigma ,\lambda )),1\le i\le m,}$$

(45)

for ${\displaystyle \sigma \in [-\alpha ,\alpha ],\lambda \in B({\hat{x}}_0,2\rho )\subseteq R^{n-1}}$, where the local flow ${\displaystyle \{x(\sigma ,\lambda )\}}$ satisfies the following system of ODEs

$${\displaystyle \frac{dx}{d\sigma }(\sigma ,\lambda )=X\left(x(\sigma ,\lambda )\right),x(0,\lambda )=(\lambda ,x_{0n}),\lambda \in B({\hat{x}}_0,2\rho )\subseteq R^{n-1}.}$$

(46)

The analysis of ODEs $(44)$ relies on the results derived in Section 2 (see Lemmas 1 and 2 and Remarks 1 and 2), for each ${\displaystyle \lambda \in B({\hat{x}}_0,2\rho )}$. This will lead us to the corresponding results for $(44)$ which can be transfered to $(42)$ and $(43)$ provided the local flow ${\displaystyle \{x(\sigma ,\lambda )\}}$ is written as follows

$${\displaystyle x(\sigma ,\lambda )=G(\sigma )[\lambda ,x_{0n}],\sigma \in [-\alpha ,\alpha ],\lambda \in B({\hat{x}}_0,2\rho )\subseteq R^{n-1},}$$

(47)

where ${\displaystyle \{G(\sigma )\left[z\right]\in B(x_0,3\rho )\subseteq D:\sigma \in [-2\alpha ,2\alpha ],z\in B(x_0,2\rho )\}}$ is the local flow generated by ${\displaystyle X\in C^m(D\subseteq R^n;R^n)}$, with initial condition ${\displaystyle G(0)\left[z\right]=z}$. In addition, the following equation must be solved

$${\displaystyle x(\sigma ,\lambda )=x\in B(x_0,\rho )\subseteq D,x_0:=({\hat{x}}_0,x_{0n}),}$$

(48)

for ${\displaystyle \lambda =({\lambda }_1,\dots ,{\lambda }_{m-1})\in B({\hat{x}}_0,2\rho )\subseteq R^{n-1}}$ and ${\displaystyle \sigma \in [-\alpha ,\alpha ]}$. The unique solution ${\displaystyle \lambda =\hat{\psi }(x)}$ and ${\displaystyle \sigma =\hat{\sigma }(x)}$ of $(48)$ satisfies

$${\displaystyle \hat{\psi }(\hat{x},x_{0n})=\hat{x},\hat{\sigma }(\hat{x},x_{0n})=0,}$$

(49)

for ${\displaystyle \hat{x}\in B({\hat{x}}_0,2\rho )\subseteq R^{n-1}}$.

For each ${\displaystyle x\in IntB(x_0,\rho )}$, let ${\displaystyle \{G(\sigma )\left[x\right]\in B(x_0,\rho ):\sigma \in [-\epsilon ,\epsilon ]\}}$ be the corresponding solution of $(46)$ and notice that (see $(47)$ and $(48)$)

$${\displaystyle G(\sigma )\left[x\right]=G(\sigma )\circ G(\hat{\sigma }(x))[\hat{\psi }(x),x_{0n}]=G\left(\sigma +\hat{\sigma }(x)\right)[\hat{\psi }(x),x_{0n}],}$$

(50)

for any ${\displaystyle \sigma \in [-\epsilon ,\epsilon ]}$. Using $(50)$ and the unique solution of $(48)$, we get

$${\displaystyle \hat{\sigma }\left(G(\sigma )\left[x\right]\right)=\sigma +\hat{\sigma }(x)\in [-\alpha ,\alpha ],\hat{\psi }\left(G(\sigma )\left[x\right]\right)=\hat{\psi }(x)\in B({\hat{x}}_0,2\rho ),}$$

(51)

for any ${\displaystyle \sigma \in [0,\epsilon ]}$ (or ${\displaystyle \sigma \in [-\epsilon ,0]}$).

Define a solution for (CP) (see $(42)$ and $(43)$) by

$${\displaystyle y_i(x)={\hat{y}}_i(\hat{\sigma }(x);\hat{\psi }(x)),i\in \{1,\dots ,m\},x\in B(x_0,\rho )\subseteq D,}$$

(52)

where ${\displaystyle {\hat{y}}_i(\sigma ,\lambda ),i\in \{1,\dots ,m\}}$, fulfil $(44)$ and ${\displaystyle \left\{\left(\hat{\sigma }(x),\hat{\psi }(x)\right):x\in B(x_0,\rho )\subseteq D\right\}}$ verifies $(49)$ and $(50)$.

By a direct inspection, the Cauchy conditions $(43)$ are verified (see $(49)$ and $(44)$). We get a solution of $(42)$ relying on $(51)$ and

$${\displaystyle y_i\left(G(\sigma )\left[x\right]\right)={\hat{y}}_i(\sigma +\hat{\sigma }(x);\hat{\psi }(x)),i=\overline{1,m},\sigma \in [0,\epsilon ].}$$

(53)

Applying a direct derivation $\frac{d}{d\sigma }{|}_{\sigma =0}$, from $(53)$, we obtain

$$\begin{array}{c}{\displaystyle ⟨{\partial }_x{y}_i(x),X(x)⟩=y_{i+1}(x),i\in \{1,\dots ,m-1\},}\\ {\displaystyle ⟨{\partial }_x{y}_m(x),X(x)⟩=f(x)+\sum _{i=1}^m{a}_i(x)y_i(x),x\in IntB(x_0,\rho )\subseteq D,}\end{array}$$

(54)

which stands for the first order system $(42)$.

We are able now to provide the main result of this paper. The above given computation will be stated as follows.

Theorem 1.

Let the smooth vector field ${\displaystyle X\in C^m(D\subseteq R^n;R^n)}$ be given such that ${\displaystyle X=(X_1,\dots ,X_n)}$ satisfies ${\displaystyle X_n(x_0)\ne 0}$ for some ${\displaystyle x_0=({\hat{x}}_0,x_{0n})\in D}$. Consider $(44)$ and its solution

$${\displaystyle \left\{{\hat{y}}_1(\sigma ,\lambda ),\dots ,{\hat{y}}_m(\sigma ,\lambda ):\sigma \in [-\alpha ,\alpha ],\lambda \in B({\hat{x}}_0,2\rho )\subseteq R^{n-1}\right\}.}$$

Define ${\displaystyle y_i(x)={\hat{y}}_i(\hat{\sigma }(x);\hat{\psi }(x)),i\in \{1,\dots ,m\},x\in B(x_0,\rho )\subseteq D}$, where ${\displaystyle \lambda =\hat{\psi }(x)\in B({\hat{x}}_0,2\rho )\subseteq R^{n-1}}$ and ${\displaystyle \hat{\sigma }(x)\in [-\alpha ,\alpha ]}$ is the unique solution of $(48)$ fulfilling $(49)$ and $(51)$. Then ${\displaystyle \{(y_1(x),\dots ,y_m(x))\in R^m:x\in B(x_0,\rho )\subseteq D\}}$ satisfies the first order system of PDEs given in $(42)$, with Cauchy conditions $(43)$ and ${\displaystyle y_1\in Ker{P}_m{|}_{\hat{B}}}$ (see $(41)$).

Proof. 

Taking into account the aforementioned computations (see relations $(53)$ and $(54)$), the proof is immediate and complete. □

Remark 3.

Consider a second order PDE of the form

$${\displaystyle ⟨{\partial }_x^{2}h(x)X(x),X(x)⟩+⟨{\partial }_{x}h(x),\left[{\partial }_{x}X(x)\right]X(x)⟩=a_1(x)h(x),}$$

(55)

where ${\displaystyle a_1\in C^1(D\subseteq R^n)}$ and ${\displaystyle X\in C^2(D\subseteq R^n;R^n)}$ is a smooth vector field satisfying ${\displaystyle X(x_0)\ne 0_{R^n}}$ for some ${\displaystyle x_0\in D}$. Then, there exists a nontrivial solution ${\displaystyle h(·)\in C^2(B(x_0,\rho )\subseteq D)}$ satisfying $(55)$, for any ${\displaystyle x\in IntB(x_0,\rho )}$. It relies on Lemma 1, noticing that $(55)$ can be rewritten as

$${\displaystyle {(\overrightarrow{X})}^2(h)(x)=a_1(x)h(x),x\in B(x_0,\rho )\subseteq D,a_1\in C^1(D),}$$

(56)

where ${\displaystyle \overrightarrow{X}(h)(x):=⟨{\partial }_{x}h(x),X(x)⟩}$. It shows that any ${\displaystyle a_1(·)=\mu \in R}$ is an eigenvalue for the linear differential operator ${\displaystyle {(\overrightarrow{X})}^2}$ acting on the space ${\displaystyle C^2(B(x_0,\rho )\subseteq D)}$. In particular, for ${\displaystyle a_1(·)=0\in R}$ and ${\displaystyle X(x_0)\ne 0_{R^n}}$, there exists a nontrivial solution (${\displaystyle \overrightarrow{X}(h)\ne 0,h\in C^2(B(x_0,\rho )\subseteq D)}$) fulfilling a second order PDE

$${\displaystyle {(\overrightarrow{X})}^2(h)(x)=0,x\in B(x_0,\rho )\subseteq D.}$$

(57)

For the case ${\displaystyle n\ge 2}$, a nontrivial solution of $(57)$ can be found such that ${\displaystyle \overrightarrow{X}(h)(x)\ne const.}$, for ${\displaystyle x\in B(x_0,\rho )\subseteq D}$. It comes from the existence of a nontrivial first integral associated with ODE

$${\displaystyle \frac{dG}{d\sigma }(\sigma ,\lambda )=X\left(G(\sigma ,\lambda )\right),G(0,\lambda )=\lambda \in B(0,\rho )\subseteq D,\sigma \in [-a,a].}$$

(58)

4. Conclusions and Further Developments

In this paper, we have investigated the kernel of a polynomial differential equation involving a derivation in ${\displaystyle R^n}$ by solving the Cauchy Problem for the corresponding first order system of PDEs. Moreover, we have proved that the kernel representation has a special significance on the space of solutions to corresponding system of PDEs.

The mathematical framework developed in this work can be easily extended for the study of some higher-order hyperbolic, parabolic or Hamilton–Jacobi equations involving a finite set of derivations.