P. D. Lebedev, A. A. Uspenskii, “Numerical-analytic construction of a generalized solution to the eikonal equation in the plane case”, Sb. Math., 215:9 (2024), 1224

Nonlinear first-order partial differential equations typically fail to have classical (differentiable) solutions on the whole domain under consideration. The concept of a generalized solution (from respective classes of functions) is known to describe adequately the processes modelled in mechanics, geometric optics, optimal control, differential games, seismology, economics and other theoretical and applied fields involving first-order partial differential equations. There are various approaches towards the definition of a generalized solution to a first-order partial differential equation and the verification of its uniqueness [1]–[4].

Kruzhkov [1], with the help of some constructions from functional analysis and using short-wave approximation machinery, introduced a generalized solution to the eikonal equation, which is the basic equation of geometrical optics. His method has extensively been developed, and applications to the constructions of generalized solutions of partial differential equations of various types were found, thereby replenishing the stock of available vanishing viscosity methods [2]. This approach to the definition of generalized solutions to first-order partial differential equations is related both to problems with a small parameter multiplying the highest derivative and to the properties of their solutions and functions obtained by taking the limit with respect to the small parameter. There is also a different approach, based on constructions in the theory of positional differential games [5]. Using this approach, Subbotin [3] gave the definition of a minimax solution for equations of this type. Note that, in contrast to the vanishing viscosity method, this definition does not involve derivatives of high order. It must also be noted that the minimax approach is capable of constructing solutions to first-order partial differential equations with nonsmooth boundary conditions and with singularities in the differential operators. Methods of the theory of singularities of differentiable mappings [6] have proved quite useful for the analytic identification of nonsmooth singularities in generalized solutions to boundary value problems for first-order partial differential equations with smooth boundary conditions. Using the machinery of group analysis classes of eikonal equations are singled out for which one can obtain explicit formulae for wave fronts with point sources [7].

A transition from theoretical constructions to numerical algorithms should involve correct approximations of nonsmooth functions. Numerical methods for the construction of generalized solutions to Hamilton-type equations and the eikonal equation, which use difference schemes for the numerical solution of differential equations of mathematical physics and constructions from the method of characteristics are being developed overseas (see, for example, [8]) and in Russia (see, for example, [9]). In the framework of the minimax approach it proves possible to create numerical and numerical-analytic algorithms for the construction of nonsmooth solutions in the theory of positional differential games and optimal control problems and of solutions to the Hamilton–Jacobi equations, where constructions from subdifferential calculus and nonsmooth analysis are used (see, for example, [10]).

In the present paper we consider the planar Dirichlet problem for the eikonal equation with constant refractive index. We consider the case where the boundary set has a twice differentiable boundary with singular points, at which the third-order derivatives of the coordinate functions make finite jumps. A numerical-analytic approach towards the construction of a Kruzhkov generalized solution is proposed. The constructions developed in the minimax approach to the Hamilton–Jacobi equation [11]–[14] are used to create a new method for constructing a generalized eikonal, which is based, in particular, on the theory of fixed points [15] of smooth mappings. The key ingredient of this method is the measure of nonconvexity of a closed subset of a Euclidean space [16], and the fundamental relations are derived using approximations provided by the theory of jets [17] (we also use some methods of convex analysis [18]). Thus, for boundary sets with twice smooth boundaries whose curvature has points of nonsmooth extremum we have traversed all links in the ‘theory–methods–algorithms–numerical simulation’ research chain.

§ 1. Object of investigation, main definitions

In his famous paper [1] Kruzhkov investigated the existence of a generalized solution to the first-order partial differential equation

After having constructed the nonlocal theory for the general eikonal equation Kruzhkov considered the simpler class of equations

Formula (1.4) has related Hopf formulae (based on extremal operations and duality theory) [19], which define viscosity solutions (and thus minimax solutions, as they coincide with viscosity solutions) of the corresponding boundary value problems for first-order partial differential equations [20]. It is also worth pointing out that formulae of this kind are used as a basis for the development of difference schemes for the numerical construction of minimax (viscosity) solutions to the Hamilton–Jacobi equation [21].

In what follows we create structural elements for the numerical-analytic construction of the solution to the plane boundary value problem with

$u^{(0)}(x)\equiv 0$ . According to [1], in this case the problem is related to the evolution of wave fronts in the domain

$\Omega \subset \mathbb{R}^k$ , where the initial front

$\Gamma=\Gamma (0)$ coincides with the boundary

$\partial \Omega$ and the successive fronts

$\Gamma (\tau)$ , where the nonnegative parameter

$\tau$ plays the role of time, are constructed from the envelops of families of ovaloids.

We consider the Dirichlet boundary value problem for the eikonal equation on the plane of constant refractive index

The boundary condition in (1.5) is set on

$\Gamma=\partial \Omega$ , the boundary of the closed set

$\Omega \subset \mathbb{R}^2$ , and

$\Gamma$ has no points of self-intersection. We assume that

$\Gamma=\{x=(\gamma_1 (t),\gamma_2 (t))\colon t\in T\}$ , where

$\gamma \colon T\to \mathbb{R}^2$ is a continuous mapping of an interval

$T=(\check{t},\widehat{t}\,)$ ,

$-\infty\leqslant\check{t}<\widehat{t}\leqslant\infty$ , to the plane. We will also consider contours — curves defined on

$T=[\check{t},\widehat{t}\,]$ ,

$-\infty<\check{t}<\widehat{t}<\infty$ , such that

$\gamma(\check{t})=\gamma(\widehat{t})$ .

Emphasizing the relationship between two different approaches (the minimax approach and that of Kruzhkov’s) to the definition of a generalized solution to a first-order partial differential equation, we note that

$\widetilde{u}(x)=\rho (x,M)$ is a minimax solution to the Dirichlet problem for the Hamilton–Jacobi equation (see [11])

In what follows

$M\subset \mathbb{R}^2$ is called the boundary set of problem (1.5), and the closure of its complement

$\Omega \subset \mathbb{R}^2$ is called the domain of definition of a generalized solution to (1.5). In view of the above definitions these sets have the common boundary

$\Gamma=\partial \Omega=\partial M$ . The structure of the singular set of the generalized solution to (1.5) depends substantially on the differential properties of

$\Gamma$ . Let us indicate some properties of the curve

$\Gamma$ describing the boundary set

$M$ . By

$\det(a,b)$ we denote the determinant of the

$2\times 2$ matrix whose rows are the vectors

$a=(a_1,a_2)$ and

$b=(b_1,b_2)$ . We assume in what follows that the boundary of the boundary set satisfies the following conditions:

(

$\Gamma1$ )

$\gamma (t)=(\gamma_1 (t),\gamma_2 (t))$ is twice continuously differentiable on

$T$ ;

(

$\Gamma3$ ) there exists a finite set

$T^0\subset T$ of points

$t_0\in T^0$ at which the one-sided third-order derivatives are finite, and at least one of the equalities

$\gamma_1'''(t_0 -0)=\gamma_1''' (t_0+0)$ and

$\gamma_2''' (t_0-0)=\gamma_2''' (t_0+0)$ fails;

$\{\Gamma \}_T$ we denote the set of curves

$\Gamma$ (without self-intersections) with the above differential properties (

$\Gamma1$ )–(

$\Gamma4$ ).

Now we give the definition of the main constructive elements of the approach we develop (see also [11]–[14]).

$(\text{A}1)$

$t_2((t_0 -\delta_1,t_0))=(t_0,t_0+\delta_2)$ ,

$\delta_1$ ,

$\delta_2 >0$ ;

Accordingly, we say that the local diffeomorphism

$t_1=t_1(t_2)$ defined by (1.8) is right-semicontinuous at

$t_1=t_0$ and maps the right half-neighbourhood of

$t_2=t_0$ to a right half-neighbourhood if (similarly to

$(\text{A}1)$ and

$(\text{A}2)$ ):

$(\underline{\text{A}1})$

$t_1((t_0,t_0+\delta_2))=(t_0 -\delta_1,t_0)$ ,

$\delta_1 , \delta_2 >0$ ;

We fix

$t_0\in T$ such that

$k(t_0)\neq 0$ (see [23]) and select two arbitrary time instants

$t_1$ and

$t_2$ ,

$t_1<t_0 <t_2$ , from the neighbourhood

$O(t_0, \delta)=(t_0-\delta, t_0+\delta)$ ,

$\delta >0$ . We draw the tangent lines through the points

$\gamma (t_1)$ and

$\gamma (t_2)$ , and construct the two-parameter family of solutions

$x^*=x^*(t_1, t_2)$ of the corresponding system of equations

Geometrically, a pseudovertex is a point of intersection of the curve

$\Gamma$ with the closure of the line of quasisymmetry. For conditions of the existence of a pseudovertex on a twice differentiable curve, see [23].

Let

$P_M(x)$ be the metric projection operator onto

$M$ , which associates with a point

$x\in \operatorname{int}\Omega$ (

$\operatorname{int}\Omega=\Omega \setminus \Gamma$ ) the set of its nearest points in

$M$ . If the operator

$P_M(x)$ ,

$x\in \operatorname{int}\Omega$ , is single valued (that is,

$\operatorname{card}P_M(x)=1$ ), then the solution to the eikonal equation is differentiable in the domain

$\Omega$ (see [24]). In problems under consideration here this is the case of a convex boundary set

$M$ . In this case

$M$ is a sun in the sense of [25], and the eikonal is a smooth function, which can be constructed by classical methods. The metric projection onto a nonconvex set

$M$ is not single valued (that is, there are points at which

$\operatorname{card}P_M(x)>1$ ).

For problem (1.5) the bisector

$L$ of

$Z=M$ consists of the points that have at least two nearest elements (orthogonal projections) in

$M$ . So, the bisector

$L$ of

$M\subset \mathbb{R}^k$ consists of the points at which the Euclidean distance function is not smooth, that is,

$L$ is the singular set of the eikonal. Any bisector is a symmetry set, whose topological properties are considered in the theory of singularities of smooth mappings (see, for example, [26]).

Pseudovertices and branches of a bisector are the main structural elements in the problem of construction of the singular sets of problem (1.5).

The significant part of (1.10) is symmetric, so if the local diffeomorphism

${t_2=t_2 (t_1)}$ ,

$t_1\in(t_0 -\delta_1,t_0)$ , is a solution to (1.10), then so is the inverse local diffeomorphism

$t_1=t_1(t_2)$ ,

$t_2\in(t_0,t_0+\delta_2)$ (see [12]). In addition, the conditions of lower semicontinuity

$(\text{A}2)$ and

$(\underline{\text{A}2})$ imply that the graphs of these diffeomorphisms have a common limit point

$(t_1,t_2)=(t_0,t_0)$ . In this case the one-sided markers are reciprocal,

§ 2. The main theoretical result

Let us find the ‘place’ of one-sided markers in the spectrum

$\Lambda=[-\infty,0]$ in the case of the nonsmooth curvature of the boundary of the boundary set in problem (1.5). The main result is formulated for the left-hand marker of a pseudovertex; the value of the right-hand marker is reciprocal to that of the left-hand marker.

Proof. In the proof of (2.2) we use the transversality conditions (see [12])

$\begin{equation} \lim_{t_1 \to t_0 -0} \biggl(\frac{\partial G(t_1,t_2 (t_1))}{\partial t_1} +t_2' \, \frac{\partial G(t_1,t_2(t_1))}{\partial t_2}\biggr)=0 \end{equation} \tag{2.3}$

at the points

$(t_1,t_2)=(t_1,t_2(t_1))$ of the graph of the local diffeomorphism

${t_2=t_2 (t_1)}$ ,

$t_1 \in(t_0 -\delta_1,t_0)$ ,

$\delta_1 >0$ , which generates the pseudovertex

$x^{(0)}=(\gamma_1(t_0), \gamma_2(t_0))$ . This local diffeomorphism

$t_2=t_2 (t_1)$ is a solution to the equation

$\begin{equation} G(t_1,t_2)\triangleq \frac{\gamma_2(t_2)-\gamma_2(t_1)}{\gamma_1(t_2)-\gamma_1(t_1)} -\frac{-\gamma_1'(t_1)\gamma_1'(t_2)+\gamma_2'(t_1)\gamma_2'(t_2)+s(t_1)s(t_2)} {\gamma_2'(t_1)\gamma_1'(t_2)+\gamma_2'(t_2)\gamma_1'(t_1)}=0, \end{equation} \tag{2.4}$

to which the original equation (1.10) can be reduced (for a justification, see [13]). Here

$s(t)=\sqrt {(\gamma_1' (t))^{2}+(\gamma_2' (t))^{2}}$ is the length of the tangent vector, which is positive in view of Condition

$(\Gamma2)$ . Effectively, (2.3) describes the condition of transversal intersection of the closure of the graph of the local diffeomorphism

$t_2=t_2 (t_1)$ with the graph of the identity diffeomorphism

$t_2=t_1$ at the common limit point

$(t_1,t_2)=(t_0,t_0)$ . By the symmetry of (2.4) the identity diffeomorphism is a solution to this equation (see [12]); however, it does not satisfy conditions

$(\text{A}1)$ and

$(\text{A}2)$ . The function

$t_2=t_1$ must be considered as an ‘extraneous’ solution to (1.10). Note that the presence of this solutions in the constructions under consideration hinders considerably the use of classical theorems of analysis: for example, one cannot use here the theorem on the existence and uniqueness of an implicitly defined function.

By the assumptions of the theorem the left-hand marker exits and is finite. Hence by (2.3) it can be evaluated as

$\begin{equation} t_2' (t_0 -0)=-\lim_{t_1 \to t_0 -0} \biggl(\frac{\partial G(t_1,t_2 (t_1))}{\partial t_1} \cdot\biggl(\frac{\partial G(t_1,t_2 (t_1))}{\partial t_2}\biggr)^{-1}\biggr). \end{equation} \tag{2.5}$

Next, using essentially the theory of jets [17], we find approximations of the partial derivatives in (2.5) taken along the diffeomorphism

$t_2=t_2 (t_1)$ .

To make the presentation more compact, we adhere to the following conventions. In all approximation expansions involving a one-sided direction, the point under consideration is fixed and we have $t_2=t_0$ or $t_1=t_0$ (depending on whether we expand to the left or right). For brevity we omit the value $t_0$ of the arguments $t_2$ and $t_1$ . First we find approximations for arbitrary positive increments $\Delta_1=t_0-t_1$ and $\Delta_2=t_2-t_0$ , and then we approximate for constrained increments conditioned by the local diffeomorphism $t_2=t_2 (t_1)$ . It is worth pointing out that if a triple of points $t_1,t_0,t_2$ in $T$ , where $t_1 <t_0 <t_2$ , is related by $t_2=t_2 (t_1)$ , then the increment $\Delta_2=\Delta_2(\Delta_1)=t_2(t_1)-t_0$ (that is, $\Delta_2$ ) depends on $\Delta_1$ , and

$\begin{equation} \Delta_2 \,{=}\,\Delta_2(\Delta_1)\,{=}\,t_2(t_1)-t_0\,{=}\,t_2'(t_0-0)(t_1-t_0)+o (t_1-t_0) \,{=}\,-\lambda\Delta_1+o(\Delta_1). \end{equation} \tag{2.6}$

This implies, in particular, that

$\begin{equation} \Delta \,{=}\,\Delta_1+\Delta_2(\Delta_1)\,{=}\,\Delta_1-\lambda\Delta_1+o(\Delta_1) \,{=}\,(1-\lambda)\Delta_1+o(\Delta_1),\quad\text{where } 1-\lambda\neq0. \end{equation} \tag{2.7}$

Here

$o(\Delta_1^k)$ ,

$k=1,2,3$ , denotes the class of functions of higher order of smallness than the argument to the left of the point under consideration, that is,

$\begin{equation*} \lim_{\Delta_1\downarrow0} \frac{o(\Delta_1^k)}{\Delta_1^k}=0. \end{equation*} \notag$

We denote by

$\varepsilon(\Delta_1)$ an infinitesimal function in the left half-neighbourhood of the point under consideration; here

$\begin{equation*} \lim_{\Delta_1\downarrow0}\varepsilon(\Delta_1)=0. \end{equation*} \notag$

By definition, two scalar functions

$y=q(t)$ and

$y=g(t)$ are equivalent to the left of a point

$t=t_0\in \mathbb{R}$ if

$\begin{equation*} \lim_{t \to t_0-0}\frac{q(t)}{g(t)}=1. \end{equation*} \notag$

In this case we write

$q(t) \sim g(t)$ ,

$t \to t_0-0$ .

For the functions considered below and, in particular, for the coordinate functions $\gamma_1(t)$ and $\gamma_2(t)$ , the curvature $k(t)$ , the length of the tangent vector $s(t)$ , and for their derivatives evaluated at the central node $t=t_0$ , we drop the argument; in addition, for one-sided derivatives, in place of the notation $t_0-0$ and $t_0+0$ , we lower the plus or minus sign to the subscript:

$\begin{equation*} \gamma_i'=\gamma_i'(t_0), \quad \gamma_i''=\gamma_i''(t_0), \quad \gamma_{i,-}'''=\gamma_i'''(t_0-0), \quad \gamma_{i,+}'''=\gamma_i'''(t_0+0), \qquad i=1,2, \end{equation*} \notag$

and so on. Expanding the partial derivative in the first variable, we have

$\begin{equation*} \begin{aligned} \, &\frac{\partial G(t_1,t_2)}{\partial t_1} =\frac{\det (\gamma'(t_1),\gamma (t_2)-\gamma (t_1))} {(\gamma_1 (t_2)-\gamma_1 (t_1))^2} -\frac{s^2(t_2)\det (\gamma' (t_1),\gamma''(t_1))}{(\gamma_2'(t_1)\gamma_1'(t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2} \\ &\quad+\frac{s(t_2)[ {s(t_1)(\gamma_2'' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1'' (t_1)) -s'(t_1)(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1'(t_1))}]} {(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2}. \end{aligned} \end{equation*} \notag$

We begin with the last term. We have

$\begin{equation*} s'(t)=\frac{\langle {\gamma'(t),\gamma''(t)} \rangle}{s(t)}=\frac{\gamma_1'(t)\gamma_1''(t)+\gamma_2' (t)\gamma_2'' (t)}{s(t)} \end{equation*} \notag$

and, after rearranging, find that

$\begin{equation*} \begin{aligned} \, & \frac{s(t_2)[ s(t_1)(\gamma_2'' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1'' (t_1)) -s'(t_1)(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))]} {(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2} \\ &\qquad =\frac{s(t_2)}{s(t_1)}\cdot \biggl(\frac{((\gamma_1' (t_1))^2+(\gamma_2'(t_1))^2) (\gamma_2'' (t_1)\gamma_1' (t_2 )+\gamma_2' (t_2)\gamma_1'' (t_1))}{(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2} \\ &\qquad\qquad -\frac{(\gamma_1' (t_1)\gamma_1'' (t_1)+\gamma_2' (t_1)\gamma_2'' (t_1)) (\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))}{(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2}\biggr) \\ &\qquad =\frac{s(t_2)}{s(t_1)} \cdot\biggl(\frac{((\gamma_1' (t_1))^2\gamma_1' (t_2)\gamma_2'' (t_1)-\gamma_1' (t_1)\gamma_2' (t_1)\gamma_1' (t_2)\gamma_1'' (t_1))}{(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2} \\ &\qquad\qquad +\frac{((\gamma_2' (t_1))^2\gamma_2' (t_2)\gamma_1'' (t_1)-\gamma_2' (t_1)\gamma_2' (t_2)\gamma_1' (t_1)\gamma_2'' (t_1))}{(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2}\biggr) \\ &\qquad =\frac{s(t_2)}{s(t_1)}\cdot\biggl( \frac{\gamma_1' (t_1)\gamma_1' (t_2)(\gamma_1' (t_1)\gamma_2'' (t_1)-\gamma_2' (t_1)\gamma_1'' (t_1))}{(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2} \\ &\qquad\qquad +\frac{\gamma_2' (t_1)\gamma_2' (t_2)(\gamma_2' (t_1)\gamma_1'' (t_1)-\gamma_1' (t_1)\gamma_2'' (t_1))}{(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2}\biggr) \\ &\qquad =\frac{s(t_2)}{s(t_1)}\cdot \frac{\gamma_1' (t_1)\gamma_1' (t_2) \det (\gamma' (t_1),\gamma'' (t_1))-\gamma_2'(t_1)\gamma_2' (t_2)\det (\gamma' (t_1),\gamma'' (t_1))}{(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2} \\ &\qquad =\frac{s(t_2)\det (\gamma' (t_1),\gamma'' (t_1))(\gamma_1' (t_1)\gamma_1' (t_2)-\gamma_2' (t_1)\gamma_2' (t_2))} {s(t_1)(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2}. \end{aligned} \end{equation*} \notag$

Therefore,

$\begin{equation*} \begin{aligned} \, \frac{\partial G(t_1,t_2)}{\partial t_1} &=\frac{\det (\gamma'(t_1),\gamma (t_2)-\gamma (t_1))}{(\gamma_1 (t_2)-\gamma_1 (t_1))^2} -\frac{s^2(t_2)s^3(t_1)k(t_1)}{(\gamma_2'(t_1)\gamma_1'(t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2} \\ &\qquad+\frac{s(t_2)s^2(t_1)k(t_1)(\gamma_1' (t_1)\gamma_1' (t_2)-\gamma_2' (t_1)\gamma_2' (t_2))} {(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2} \\ &=\frac{\det (\gamma' (t_1),\gamma (t_2)-\gamma (t_1))}{(\gamma_1 (t_2)-\gamma_1 (t_1))^2} \\ &\qquad+\frac{s(t_2)s^2(t_1)k(t_1)(\gamma_1' (t_1)\gamma_1' (t_2)-\gamma_2'(t_1)\gamma_2' (t_2)-s(t_2)s(t_1))} {(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))^2}. \end{aligned} \end{equation*} \notag$

From now on, we assume that the arguments of the functions involved are related by the local diffeomorphism

$t_2=t_2 (t_1)$ , which generates the pseudovertex; here,

$t_1 \in (t_0 -\delta_1,t_0)$ ,

$\delta >0$ . In other words, by (2.4) these functions satisfy the equality

$\begin{equation*} \frac{\gamma_1' (t_1)\gamma_1' (t_2)-\gamma_2' (t_1)\gamma_2'(t_2)-s(t_2)s(t_1)} {\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1)} =-\frac{\gamma_2 (t_2)-\gamma_2 (t_1)}{\gamma_1(t_2)-\gamma_1 (t_1)}. \end{equation*} \notag$

Using this equality and the formula for the curvature

$\begin{equation*} k(t)=\frac{\det (\gamma' (t),\gamma''(t))}{s^3(t)}, \end{equation*} \notag$

we simplify the partial derivative as follows:

$\begin{equation} \begin{aligned} \, \notag \frac{\partial G(t_1,t_2)}{\partial t_2} &=\frac{\det (\gamma'(t_2),\gamma (t_1)-\gamma (t_2))}{(\gamma_1 (t_2)-\gamma_1 (t_1))^2} \\ &\qquad -\frac{(\gamma_2 (t_2)-\gamma_2 (t_1))s(t_1)\det (\gamma' (t_2),\gamma''(t_2))} {(\gamma_1 (t_2)-\gamma_1 (t_1))s(t_2)(\gamma_2' (t_2)\gamma_1' (t_1)+\gamma_2' (t_1)\gamma_1' (t_2))}. \end{aligned} \end{equation} \tag{2.8}$

Now, to make the calculations more compact, we use a moving (Frenet) frame on the plane. We use the natural parameter $l\geqslant 0$ (the arc length of the curve), where

$\begin{equation} dl=|\gamma'(t)|\,dt \triangleq s(t)\,dt. \end{equation} \tag{2.9}$

We set

$\overline \gamma (l)\triangleq\gamma (t(l))$ . Now the vectors

$v=\overline \gamma'(l)$ and

$w=v'(l)$ , as calculated at

$l=l_0 \triangleq l(t_0)$ , form the orthonormal moving frame attached to the point

$\overline\gamma (l_0)=\gamma (t_0)$ .

Expanding the vector function in a Taylor series about this point up to a third-order term, we have

$\begin{equation*} \begin{aligned} \, \overline \gamma (l_0 -\Delta l_1) &=\overline \gamma (l_0)-\overline \gamma'(l_0)\Delta l_1 \\ &\qquad+\frac{1}{2}\overline \gamma''(l_0)\Delta l_1^2 -\frac{1}{6}\overline \gamma_-''' (l_0)\Delta l_1^3+o(\Delta l_1^3), \qquad \Delta l_1 >0, \\ \text{and} \qquad \overline \gamma (l_0+\Delta l_2) &=\overline \gamma (l_0)+\overline \gamma'(l_0)\Delta l_2 \\ &\qquad+\frac{1}{2}\overline \gamma''(l_0)\Delta l_2^2 +\frac{1}{6}\overline \gamma_+''' (l_0)\Delta l_2^3 +o(\Delta l_2^3), \qquad \Delta l_2>0. \end{aligned} \end{equation*} \notag$

Hence in the moving base

$v_0=\overline \gamma'(l_0)$ ,

$w_0=v'(l_0)$ with origin

$\overline \gamma (l_0)$ the expansions assume the following form (see (5.9) in [27]):

$\begin{equation} \nonumber \overline \gamma (l_0 -\Delta l_1) =\overline \gamma (l_0)+\biggl(-\Delta l_1+\frac{1}{6}\overline k^2(l_0)\Delta l_1^3+o(\Delta l_1^3)\biggr)v_0 \end{equation} \notag$

$\begin{equation} \qquad +\biggl(\frac{1}{2}\overline k (l_0)\Delta l_1^2-\frac{1}{6}\overline k_-' (l_0)\Delta l_1^3+o(\Delta l_1^3)\biggr)w_0 \end{equation} \tag{2.10}$

and

$\begin{equation} \nonumber \overline \gamma (l_0+\Delta l_2) =\overline \gamma (l_0)+\biggl(\Delta l_2 -\frac{1}{6}\overline k^2(l_0)\Delta l_2^3+o(\Delta l_2^3)\biggr)v_0 \end{equation} \notag$

$\begin{equation} \qquad +\biggl(\frac{1}{2}\overline k (l_0)\Delta l_2^2+\frac{1}{6}\overline k_+' (l_0)\Delta l_2^3+o(\Delta l_2^3)\biggr)w_0 . \end{equation} \tag{2.11}$

Here

$\overline k (l)$ is the curvature of the curve at the corresponding point.

In view of (2.9)–(2.11) the original parametrization of the vector function has the following expansions at the extreme points:

$\begin{equation} \begin{aligned} \, \gamma (t_1) &=\gamma+\biggl(-s\Delta_1+\frac{1}{6}k^2s^3\Delta_1^3+o(\Delta_1^3)\biggr)v_0 \notag \\ &\qquad+\biggl(\frac{1}{2}ks^2\Delta_1^2 -\frac{1}{6}k_-' s^3\Delta_1^3+o(\Delta_1^3)\biggr)w_0 \end{aligned} \end{equation} \tag{2.12}$

and

$\begin{equation} \begin{aligned} \, \gamma (t_2) &=\gamma+\biggl(s\Delta_2-\frac{1}{6}k^2s^3\Delta_2^3+o(\Delta_2^3)\biggr)v_0 \notag \\ &\qquad+\biggl(\frac{1}{2}ks^2\Delta_2^2+\frac{1}{6}k_+' s^3\Delta_2^3+o(\Delta_2^3)\biggr)w_0. \end{aligned} \end{equation} \tag{2.13}$

Here

$\Delta l_i=l'(t_0)\Delta_i=s\Delta_i$ ,

$i=1,2$ , and

$\Delta_i \to 0$ as

$\Delta l_i \to 0$ . At these nodes the derivatives up to order 2 have the form

$\begin{equation} \gamma'(t_1)=\biggl(s-\frac{1}{2}k^2s^3\Delta_1^2+o(\Delta_1^2)\biggr)v_0 +\biggl(-ks^2\Delta_1^+\frac{1}{2}k_-' s^3\Delta_1^2+o(\Delta_1^2)\biggr)w_0, \end{equation} \tag{2.14}$

$\begin{equation} \gamma'(t_2)=\biggl(s-\frac{1}{2}k^2s^3\Delta_2^2+o(\Delta_2^2)\biggr)v_0 +\biggl(ks^2\Delta_2+\frac{1}{2}k_+' s^3\Delta_2^2+o(\Delta_2^2)\biggr)w_0, \end{equation} \tag{2.15}$

$\begin{equation} \gamma''(t_1)=(k^2s^3\Delta_1+o(\Delta_1))v_0 +(ks^2-k_-' s^3\Delta_1+o(\Delta_1))w_0 \end{equation} \tag{2.16}$

and

$\begin{equation} \gamma''(t_2)=(-k^2s^3\Delta_2+o(\Delta_2))v_0+(ks^2+k_+' s^3\Delta_2+o(\Delta_2))w_0 . \end{equation} \tag{2.17}$

Here we have used that

$\Delta_1'=(t_0 -t_1)'=-1$ and

$\Delta_2'=(t_2 -t_0)'=1$ .

Using (2.10)–(2.12) we expand the determinant as follows:

$\begin{equation*} \begin{aligned} \, &\det (\gamma' (t_1),\gamma (t_2)-\gamma (t_1)) \\ &{=}{\kern1pt}\Delta {\begin{vmatrix} s-\dfrac{1}{2}k^2s^3\Delta_1^2+o(\Delta_1^2) &-ks^2\Delta_1+\dfrac{1}{2}k_-' s^3\Delta_1^2+o(\Delta_1^2) \\ s\!-\!\dfrac{1}{6}k^2 s^3(\Delta_2^2\!-\!\Delta_1 \Delta_2\!+\!\Delta_1^2)\!+\!o(\Delta_2^2) &\dfrac{1}{2}ks^2(\Delta_2\!-\!\Delta_1)\!+\!\dfrac{1}{6} k_+'s^3\dfrac{\Delta_2^3}{\Delta}\!+\!\dfrac{1}{6}k_-' s^3 \dfrac{\Delta_1^3}{\Delta}\!+\!o(\Delta_2^2) \end{vmatrix}} \\ &{=}\,\Delta \biggl(\frac{1}{2}ks^3(\Delta_2 -\Delta_1)+\frac{1}{6}k_+' s^4\frac{\Delta_2^3}{\Delta} +\frac{1}{6}k_-'s^4\frac{\Delta_1^3}{\Delta}+ks^3\Delta_1 -\frac{1}{2}k_-' s^4\Delta_1^2+o(\Delta_{12}^2)\biggr) \\ &{=}\,s^3\Delta \biggl(\frac{1}{2}k(\Delta_2+\Delta_1)+\frac{1}{6}k_+' s\frac{\Delta_2^3}{\Delta} +\frac{1}{6}k_-'s\frac{\Delta_1^3}{\Delta}-\frac{1}{2}k_-' s\Delta_1^2+o(\Delta_{12}^2)\biggr) \\ &{=}\,s^3\Delta^2\biggl(\frac{1}{2}k+\frac{1}{6}k_+' s\frac{\Delta_2^3}{\Delta^2} +\frac{1}{6}k_-' s\frac{\Delta_1^3}{\Delta^2}-\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta}+o(\Delta_{12})\biggr). \end{aligned} \end{equation*} \notag$

The difference is transformed using (2.12) and (2.13):

$\begin{equation} \begin{aligned} \, \notag \gamma_1 (t_2)-\gamma_1 (t_1) &=\gamma_1+s\Delta_2 -\frac{1}{6}k^2s^3\Delta_2^3 -\gamma_1+s\Delta_1-\frac{1}{6}k^2s^3\Delta_1^3+o(\Delta_{12}^3) \\ \notag &=s(\Delta_2+\Delta_1)-\frac{1}{6}k^2s^3\Delta_2^3 -\frac{1}{6}k^2s^3\Delta_1^3+o(\Delta_{12}^3) \\ \notag &=s\Delta -\frac{1}{6}k^2s^3(\Delta_2^3+\Delta_1^3)+o(\Delta_{12}^3) \\ &=s\Delta \biggl(1-\frac{1}{6}k^2s^2(\Delta_2^2 -\Delta_1 \Delta_2+\Delta_1^2) +o(\Delta_{12}^2)\biggr) \notag \\ &=s\Delta(1+o(\Delta_{12})) . \end{aligned} \end{equation} \tag{2.18}$

In view of the last two equalities, the minuend in (2.8) assumes the form

$\begin{equation} \begin{aligned} \, \notag &\frac{\det (\gamma' (t_1),\gamma (t_2)-\gamma (t_1))}{(\gamma_1 (t_2)-\gamma_1 (t_1))^2} \\ \notag &\qquad =\frac{s^3\Delta^2(\frac{1}{2}k+\frac{1}{6}k_+'s\frac{\Delta_2^3}{\Delta^2} +\frac{1}{6}k_-' s\frac{\Delta_1^3}{\Delta^2}-\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta}+o(\Delta_{12}))} {s^2\Delta^2(1+o(\Delta_{12}))} \\ &\qquad = s\biggl(\frac{1}{2}k+\frac{1}{6}k_+' s\frac{\Delta_2^3}{\Delta^2} +\frac{1}{6}k_-' s\frac{\Delta_1^3}{\Delta^2}-\frac{1}{2}k_-'s\frac{\Delta_1^2}{\Delta}+o(\Delta_{12})\biggr) . \end{aligned} \end{equation} \tag{2.19}$

Let us now consider the subtrahend in (2.8). Expanding in succession the factors in its numerator and denominator and using (2.12)–(2.17), we find that

$\begin{equation*} \begin{gathered} \, \begin{aligned} \, &\gamma_2 (t_2)-\gamma_2 (t_1) \\ &\qquad=\Delta\biggl(\frac{1}{2}ks^2(\Delta_2 -\Delta_1)+\frac{1}{6}k_+' s^3\frac{\Delta_2^3}{\Delta} +\frac{1}{6}k_-'s^3\frac{\Delta_1^3}{\Delta}+o(\Delta_{12}^2)\biggr) \\ &\qquad=s^2\Delta^2\biggl(\frac{1}{2}k\frac{\Delta_2 -\Delta_1}{\Delta} +\frac{1}{6}k_+' s\frac{\Delta_2^3}{\Delta^2}+\frac{1}{6}k_-'s\frac{\Delta_1^3}{\Delta^2}+o(\Delta_{12})\biggr), \end{aligned} \\ \begin{aligned} \, &\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1) \\ &\qquad =\biggl(-ks^2\Delta_1+\frac{1}{2}k_-' s^3\Delta_1^2+o(\Delta_1^2)\biggr) \biggl(s-\frac{1}{2}k^2s^3\Delta_2^2+o(\Delta_2^2)\biggr) \\ &\qquad\qquad +\biggl(ks^2\Delta_2+\frac{1}{2}k_+' s^3\Delta_2^2+o(\Delta_2^2)\biggr)\biggl(s-\frac{1}{2}k^2s^3\Delta_1^2+o(\Delta_1^2)\biggr) \\ &\qquad =-ks^3\Delta_1^+\frac{1}{2}k_-' s^4\Delta_1^2+ks^3\Delta_2 +\frac{1}{2}k_+' s^4\Delta_2^2+o(\Delta_{12}^2) \\ &\qquad =ks^3(\Delta_2 -\Delta_1)+\frac{1}{2}k_-' s^4\Delta_1^2+\frac{1}{2}k_+' s^4\Delta_2^2+o(\Delta_{12}^2), \end{aligned} \\ \begin{aligned} \, \det (\gamma' (t_1),\gamma''(t_1)) &=\begin{vmatrix} s-\dfrac{1}{2}k^2s^3\Delta_1^2+o(\Delta_1^2) &-ks^2\Delta_1+\dfrac{1}{2}k_-' s^3\Delta_1^2+o(\Delta_1^2) \\ k^2s^3\Delta_1+o(\Delta_1) &ks^2-k_-'s^3\Delta_1+o(\Delta_1) \end{vmatrix} \\ &=ks^3-k_-' s^4\Delta_1+o(\Delta_1) =s^3(k-k_-' s\Delta_1+o(\Delta_1)), \end{aligned} \\ s(t_2)=s+\Delta_2 s'+o(\Delta_2)\quad\text{and} \quad s(t_1)=s-s'\Delta_1+o(\Delta_1). \end{gathered} \end{equation*} \notag$

Using these expansions and employing (2.18), for the minuend in (2.8) we have

$\begin{equation*} \begin{aligned} \, &\frac{(\gamma_2 (t_2)-\gamma_2 (t_1))s(t_2)\det (\gamma' (t_1),\gamma''(t_1))} {(\gamma_1 (t_2)-\gamma_1 (t_1))s(t_1)(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))} \\ &=s^2\Delta^2\biggl(\frac{1}{2}k\frac{\Delta_2 -\Delta_1}{\Delta} +\frac{1}{6}k_+' s\frac{\Delta_2^3}{\Delta^2}+\frac{1}{6}k_-'s\frac{\Delta_1^3}{\Delta^2}+o(\Delta_{12})\biggr) \\ &\quad \times \frac{s^3(k-k_-' s\Delta_1+o(\Delta_1))}{s\Delta(1-\frac{1}{6}k^2s^2(\Delta_2^2 -\Delta_1\Delta_2+\Delta_1^2)+o(\Delta_{12}^2))} \\ &\quad\times\frac{s+\Delta_2 s'+o(\Delta_2)}{(ks^3(\Delta_2 -\Delta_1)\,{+}\,\frac{1}{2}k_-'s^4\Delta_1^2\,{+}\,\frac{1}{2}k_+' s^4\Delta_2^2\,{+}\,o(\Delta_{12}^2))(s\,{-}\,s'\Delta_1\,{+}\,\frac{1}{2}s''\Delta_1^2\,{+}\,o(\Delta_1^2))} \\ &=s^5\Delta^2 \biggl(\frac{1}{2}k^2\frac{\Delta_2 \,{-}\,\Delta_1}{\Delta}\,{+}\,\frac{1}{6}kk_+' s\frac{\Delta_2^3}{\Delta^2} \,{+}\,\frac{1}{6}kk_-' s\frac{\Delta_1^3}{\Delta^2}\,{-}\,\frac{1}{2}kk_-'s\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}\,{+}\,o(\Delta_{12})\!\biggr) \\ &\quad \times \frac{s+\Delta_2 s'+o (\Delta_2)}{s^4\Delta (k(\Delta_2 -\Delta_1)+\frac{1}{2}k_-' s\Delta_1^2+\frac{1}{2}k_+' s\Delta_2^2\,{+}\,o(\Delta_{12}^2)) (s\,{-}\,s'\Delta_1\,{+}\,\frac{1}{2}s''\Delta_1^2\,{+}\,o(\Delta_1^2))} \\ &=s\Delta \biggl(\frac{1}{2}k^2s\frac{\Delta_2 -\Delta_1}{\Delta}+\frac{1}{6}kk_+' s^2\frac{\Delta_2^3}{\Delta^2} +\frac{1}{6}kk_-'s^2\frac{\Delta_1^3}{\Delta^2} \\ &\quad \qquad -\frac{1}{2}kk_-' s^2\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta} +\frac{1}{2}k^2ss'\frac{(\Delta_2 -\Delta_1)\Delta_2}{\Delta}+o (\Delta_{12}) \biggr) \\ &\quad \times\frac{1}{ks (\Delta_2 -\Delta_1)+\frac{1}{2}k_-' s^2\Delta_1^2+\frac{1}{2}k_+' s^2\Delta_2^2 -kss' (\Delta_2 -\Delta_1)\Delta_1+o (\Delta_{12}^2)} \\ &=\biggl(\frac{1}{2}k^2s\frac{\Delta_2 -\Delta_1}{\Delta}+\frac{1}{6}kk_+' s^2\frac{\Delta_2^3}{\Delta^2} +\frac{1}{6}kk_-'s^2\frac{\Delta_1^3}{\Delta^2} \\ &\quad \qquad -\frac{1}{2}kk_-' s^2\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta} +\frac{1}{2}k^2ss'\frac{(\Delta_2 -\Delta_1)\Delta_2}{\Delta}+o(\Delta_{12}) \biggr) \\ &\quad \times\frac{1}{k\frac{(\Delta_2 -\Delta_1)}{\Delta}+\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o(\Delta_{12})}. \end{aligned} \end{equation*} \notag$

Note that

$\begin{equation*} \begin{aligned} \, s(t_1)s'(t_1) &=\frac{s(t_1)\langle \gamma'(t_1),\gamma''(t_1)\rangle}{s(t_1)} =\langle \gamma'(t_1),\gamma''(t_1) \rangle \\ &=\biggl(s-\frac{1}{2}k^2s^3\Delta_1^2+o(\Delta_1^2)\biggr)(k^2s^3\Delta_1+o(\Delta_1)) \\ &\qquad+\biggl(-ks^2\Delta_1+\frac{1}{2}k_-' s^3\Delta_1^2+o(\Delta_1^2)\biggr)(ks^2-k_-' s^3\Delta_1+o(\Delta_1)) \\ &=k^2s^4\Delta_1 -k^2s^4\Delta_1+o(\Delta_1)=o(\Delta_1). \end{aligned} \end{equation*} \notag$

By the mean value theorem,

$\begin{equation*} s(t_1)s'(t_1)=((s'(\vartheta))^2+s(\vartheta) s''(\vartheta))\Delta_1,\qquad t_0<\vartheta<t_1. \end{equation*} \notag$

Hence

$ss'=C\Delta_1+o(\Delta_1)$ ,

$C=-((s'(\vartheta))^2+s(\vartheta) s''(\vartheta))$ . Therefore, the numerator of the fraction under consideration contains the infinitely small quantity

$\begin{equation*} \frac{k^2s s'}{2}\frac{(\Delta_2 -\Delta_1)\Delta_2}{\Delta}=o (\Delta_{12}^2) \end{equation*} \notag$

of higher order with respect to

$o (\Delta_1)$ . Discarding this quantity, the fraction simplifies as

$\begin{equation} \begin{aligned} \, \notag &\frac{(\gamma_2 (t_2)-\gamma_2 (t_1))s(t_2)\det (\gamma'(t_1),\gamma''(t_1))} {(\gamma_1 (t_2)-\gamma_1 (t_1))s(t_1)(\gamma_2' (t_1)\gamma_1' (t_2)+\gamma_2' (t_2)\gamma_1' (t_1))} \\ &\ =\frac{\frac{1}{2}k^2s\frac{\Delta_2-\Delta_1}{\Delta} +\frac{1}{6}kk_+' s^2\frac{\Delta_2^3}{\Delta^2}+\frac{1}{6}kk_-'s^2\frac{\Delta_1^3}{\Delta^2} -\frac{1}{2}kk_-' s^2\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o(\Delta_{12})} {k\frac{(\Delta_2 -\Delta_1)}{\Delta}+\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1+o(\Delta_{12})}{\Delta}+o(\Delta_{12})}. \end{aligned} \end{equation} \tag{2.20}$

Using (2.19) and (2.20) and applying some algebraic transformations, we can write the partial derivative

$\dfrac{\partial G(t_1,t_2)}{\partial t_1}$ for

$t_2=t_2 (t_1)$ as follows:

$\begin{equation} \begin{aligned} \, \notag &\frac{\partial G(t_1,t_2 (t_1))}{\partial t_1} =\biggl(\frac{1}{2}ks+\frac{1}{6}k_+' s^2\frac{\Delta_2^3}{\Delta^2}+\frac{1}{6}k_-' s^2\frac{\Delta_1^3}{\Delta^2} -\frac{1}{2}k_-'s^2\frac{\Delta_1^2}{\Delta}+o(\Delta_{12})\biggr) \\ \notag &\qquad\qquad -\frac{\frac{1}{2}k^2s\frac{\Delta_2 -\Delta_1}{\Delta}+\frac{1}{6}kk_+' s^2\frac{\Delta_2^3}{\Delta^2} +\frac{1}{6}kk_-'s^2\frac{\Delta_1^3}{\Delta^2}-\frac{1}{2}kk_-' s^2\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta} +o(\Delta_{12})}{k\frac{(\Delta_2 -\Delta_1)}{\Delta}+\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o (\Delta_{12})} \\ \notag &=\biggl(\frac{1}{2}ks+\frac{1}{6}k_+' s^2\frac{\Delta_2^3}{\Delta^2} +\frac{1}{6}k_-' s^2\frac{\Delta_1^3}{\Delta^2}-\frac{1}{2}k_-'s^2\frac{\Delta_1^2}{\Delta}+o (\Delta_{12})\biggr) \\ \notag &\qquad \times \frac{k\frac{(\Delta_2 -\Delta_1)}{\Delta}+\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o (\Delta_{12})} {k\frac{(\Delta_2 -\Delta_1)}{\Delta}+\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o (\Delta_{12})} \\ \notag &\qquad\qquad -\frac{\frac{1}{2}k^2s\frac{\Delta_2 -\Delta_1}{\Delta}+\frac{1}{6}kk_+' s^2\frac{\Delta_2^3}{\Delta^2} +\frac{1}{6}kk_-'s^2\frac{\Delta_1^3}{\Delta^2}-\frac{1}{2}kk_-' s^2\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta} +o(\Delta_{12})}{k\frac{(\Delta_2 -\Delta_1)}{\Delta}+\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o(\Delta_{12})} \\ \notag &=\frac{\frac{1}{6}kk_-' s^2(\frac{3\Delta_1^2}{2\Delta}+\frac{\Delta_1^3 (\Delta_2 -\Delta_1)}{\Delta^3} -\frac{3\Delta_1^2 (\Delta_2 -\Delta_1)}{\Delta^2}-\frac{\Delta_1^3}{\Delta^2} +\frac{3(\Delta_2 -\Delta_1)\Delta_1}{\Delta})}{k\frac{(\Delta_2 -\Delta_1)}{\Delta} +\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+' s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o(\Delta_{12})} \\ &\qquad\qquad +\frac{\frac{1}{6}kk_+' s^2 (\frac{3\Delta_2^2}{2\Delta}+\frac{\Delta_2^3 (\Delta_2 -\Delta_1)}{\Delta^3} -\frac{\Delta_2^3}{\Delta^2})+o(\Delta_{12})} {k\frac{(\Delta_2-\Delta_1)}{\Delta} +\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta}+\frac{1}{2}k_+' s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta} +o (\Delta_{12})}. \end{aligned} \end{equation} \tag{2.21}$

In the expansion (2.21) of the partial derivative

$\dfrac{\partial G(t_1,t_2)}{\partial t_1}$ along the diffeomorphism

$t_2=t_2 (t_1)$ , we factor out

$\Delta_1=t_0 -t_1$ and evaluate, using (2.6) and (2.7), the limit of the fraction in brackets, cancelling out the common factor

$k(t_0)\ne 0$ in the numerator and denominator (see (2.1)). As a result, we have

$\begin{equation} \begin{aligned} \, \notag &\lim_{\Delta_1 \downarrow 0}\biggl[\frac{\frac{1}{6}kk_-'s^2(\frac{3\Delta_1}{2\Delta} +\frac{\Delta_1^2 (\Delta_2 -\Delta_1)}{\Delta^3}-\frac{3\Delta_1 (\Delta_2-\Delta_1)}{\Delta^2} -\frac{\Delta_1^2}{\Delta^2}+\frac{3 (\Delta_2 -\Delta_1)}{\Delta})} {k\frac{(\Delta_2 -\Delta_1)}{\Delta} +\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta}+\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta} -ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o (\Delta_{12})} \\ \notag &\qquad\qquad +\frac{\frac{1}{6}kk_+' s^2(\frac{\Delta_2^3 (\Delta_2-\Delta_1)}{\Delta_1 \Delta^3} +\frac{3\Delta_2^2}{2\Delta\Delta_1}-\frac{\Delta_2^3}{\Delta_1 \Delta^2})+\varepsilon(\Delta_{12})} {k\frac{(\Delta_2 -\Delta_1)}{\Delta}+\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2 -\Delta_1)\Delta_1}{\Delta}+o(\Delta_{12})}\biggr] \\ \notag &\qquad =\frac{\frac{1}{6}kk_-' s^2(\frac{3}{2}\frac{1}{1-\lambda}-\frac{1+\lambda}{(1-\lambda)^3} +\frac{3+3\lambda}{(1-\lambda)^2}-\frac{1}{(1-\lambda)^2}-\frac{3+3\lambda}{(1-\lambda)})}{-\frac{1+\lambda}{1-\lambda}k} \\ \notag &\qquad\qquad +\frac{\frac{1}{6}kk_+' s^2(\frac{\lambda^4+\lambda^3}{(1-\lambda)^3}+\frac{3}{2}\frac{\lambda^2}{(1-\lambda)} +\frac{\lambda^3}{(1-\lambda)^2})}{-\frac{1+\lambda}{1-\lambda}k} \\ \notag &\qquad =\frac{\frac{1}{6}k_-' s^2(\frac{3}{2}-\frac{1+\lambda}{(1-\lambda)^2}+\frac{2+3\lambda}{(1-\lambda)}-3-3\lambda) l+\frac{1}{6}k_+' s^2(\frac{\lambda^4+\lambda^3}{(1-\lambda)^2}+\frac{3}{2}\lambda^2+\frac{\lambda^3}{(1-\lambda)})}{-(1+\lambda)} \\ &\qquad =-\frac{1}{12}k_-' s^2\frac{-6\lambda^3+3\lambda^2-1}{(1-\lambda)^2(1+\lambda)} -\frac{1}{12}k_+' s^2\lambda^2\frac{3\lambda^2-2\lambda+3}{(1-\lambda)^2(1+\lambda)} . \end{aligned} \end{equation} \tag{2.22}$

This gives us the following linear asymptotics of the function

$\dfrac{\partial G(t_1,t_2(t_1))}{\partial t_1}$ to the left of

$t_0\in \mathbb{R}$ :

$\begin{equation} \frac{\partial G(t_1,t_2 (t_1))}{\partial t_1}\sim C_1 (\lambda)(t_0 -t_1), \qquad t_1 \to t_0-0. \end{equation} \tag{2.23}$

Here the coefficient in the asymptotic expression is equal to the limit of (2.22),

$\begin{equation*} C_1 (\lambda)=-\frac{1}{12}k_-' s^2\frac{-6\lambda^3+3\lambda^2-1}{(1-\lambda)^2(1+\lambda)} -\frac{1}{12}k_+' s^2\lambda^2\frac{3\lambda^2-2\lambda+3}{(1-\lambda)^2(1+\lambda)}. \end{equation*} \notag$

Recalling the symmetry consideration, in the expansion of

$\dfrac{\partial G(t_1,t_2 (t_1))}{\partial t_2}$ we use the corresponding components of

$\dfrac{\partial G(t_1,t_2 (t_1))}{\partial t_1}$ , where we replace the increment

$\Delta_{2}$ by

$-\Delta_{1}$ ,

$\Delta_{1}$ by

$-\Delta_{2}$ ,

$\Delta$ by

$-\Delta$ ,

$k_-'$ by

$k_+'$ , and

$k_+'$ by

$k_-'$ . Now the denominator of the fraction in (2.5) assumes the form

$\begin{equation} \begin{aligned} \, \notag \frac{\partial G(t_1,t_2 (t_1))}{\partial t_2} &=\frac{\frac{1}{6}kk_+' s^2(-\frac{3\Delta_3^2}{2\Delta} +\frac{\Delta_2^3 (\Delta_2-\Delta_1)}{\Delta^3} -\frac{3\Delta_2^2 (\Delta_2-\Delta_1)}{\Delta^2} +\frac{\Delta_2^3}{\Delta^2}+\frac{3(\Delta_2-\Delta_1)\Delta_2}{\Delta})} {k\frac{(\Delta_2-\Delta_1)}{\Delta}+\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta} +\frac{1}{2}k_+' s\frac{\Delta_2^2}{\Delta}-ks'\frac{(\Delta_2-\Delta_1)\Delta_2}{\Delta}+o(\Delta_{12})} \\ &\qquad + \frac{\frac{1}{6}kk_-' s^2(\frac{\Delta_1^3 (\Delta_2-\Delta_1)}{\Delta^3} -\frac{3\Delta_1^2}{2\Delta}+\frac{\Delta_1^3}{\Delta^2})+o(\Delta_{12})}{-k\frac{(\Delta_2-\Delta_1)}{\Delta} -\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta}-\frac{1}{2}k_+' s\frac{\Delta_2^2}{\Delta} -ks'\frac{(\Delta_2-\Delta_1)\Delta_2}{\Delta}+o(\Delta_{12})}. \end{aligned} \end{equation} \tag{2.24}$

Proceeding similarly, we factor out

$\Delta_1=t_0 -t_1$ in the expansion (2.24) of the partial derivative

$\dfrac{\partial G(t_1,t_2)}{\partial t_2}$ along the diffeomorphism

$t_2=t_2 (t_1)$ and take the limit in the fraction as this increment tends to zero, with the use of condition (2.1) and the limit relations (2.6) and (2.7). As a result, we obtain

$\begin{equation*} \begin{aligned} \, & \lim_{\Delta_1 \downarrow 0} \biggl[ \frac{\frac{1}{6}kk_+'s^2(-\frac{3\Delta_3^2}{2\Delta \Delta_1} +\frac{\Delta_2^3(\Delta_2-\Delta_1)}{\Delta_1 \Delta^3} -\frac{3\Delta_2^2 (\Delta_2-\Delta_1)}{\Delta_1\Delta^2} +\frac{\Delta_2^3}{\Delta_1 \Delta^2} +\frac{3(\Delta_2-\Delta_1)\Delta_2}{\Delta_1 \Delta})}{-k\frac{(\Delta_2-\Delta_1)}{\Delta} -\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta}-\frac{1}{2}k_+'s\frac{\Delta_2^2}{\Delta} -ks'\frac{(\Delta_2-\Delta_1)\Delta_2}{\Delta}+o(\Delta_{12})} \\ &\qquad\qquad +\frac{\frac{1}{6}kk_-' s^2(\frac{\Delta_1^2 (\Delta_2-\Delta_1)}{\Delta^3}-\frac{3\Delta_1}{2\Delta} +\frac{\Delta_1^2}{\Delta^2})+\varepsilon(\Delta_{12})}{-k\frac{(\Delta_2-\Delta_1)}{\Delta} -\frac{1}{2}k_-' s\frac{\Delta_1^2}{\Delta}-\frac{1}{2}k_+' s\frac{\Delta_2^2}{\Delta} -ks'\frac{(\Delta_2-\Delta_1)\Delta_2}{\Delta}+o(\Delta_{12})}\biggr] \\ &\qquad =\frac{\frac{1}{6}kk_+' s^2(-\frac{3}{2}\frac{\lambda^2}{1-\lambda} +\frac{\lambda^3(1+\lambda)}{(1-\lambda)^3}+3\frac{\lambda^2(1+\lambda)}{(1-\lambda)^2} -\frac{\lambda^3}{(1-\lambda)^2}+\frac{3\lambda^2+3\lambda}{1-\lambda})}{\frac{1+\lambda}{1-\lambda}k} \\ &\qquad\qquad +\frac{\frac{1}{6}kk_-' s^2(-\frac{(1+\lambda)}{(1-\lambda)^3}-\frac{3}{2}\frac{1}{(1-\lambda)} +\frac{1}{(1-\lambda)^2})}{\frac{1+\lambda}{1-\lambda}k} \\ &\qquad =\frac{1}{12}k_+' s^2\frac{\lambda^4-3\lambda^2+6\lambda}{(1-\lambda)^2(1+\lambda)} +\frac{1}{12}k_-'s^2\frac{-3\lambda^2+2\lambda -3}{(1-\lambda)^2(1+\lambda)}. \end{aligned} \end{equation*} \notag$

This gives us the linear asymptotics of the function

$\dfrac{\partial G(t_1,t_2(t_1))}{\partial t_2}$ to the left of

${t_0\in \mathbb{R}}$ :

$\begin{equation} \frac{\partial G(t_1,t_2 (t_1))}{\partial t_2}\sim C_2 (\lambda)(t_0 -t_1), \qquad t_1 \to t_0-0. \end{equation} \tag{2.25}$

In view of (2.23) and (2.25) equality (2.5) assumes the form

$\lambda=\dfrac{C_1(\lambda)}{C_2 (\lambda)}$ , that is,

$\begin{equation*} \lambda=-\frac{-\frac{1}{12}k_-' s^2\frac{-6\lambda^3+3\lambda^2-1}{(1-\lambda)^2(1+\lambda)} -\frac{1}{12}k_+' s^2\lambda^2\frac{3\lambda^2-2\lambda+3}{(1-\lambda)^2(1+\lambda)}} {\frac{1}{12}k_+' s^2\frac{\lambda^4-3\lambda^2+6\lambda}{(1-\lambda)^2(1+\lambda)} +\frac{1}{12}k_-' s^2\frac{-3\lambda^2+2\lambda -3}{(1-\lambda)^2(1+\lambda)}}. \end{equation*} \notag$

The structure of this equality is typical for fixed points. Cancelling the numerator and denominator by

$\frac{1}{12}s^2\ne 0$ and performing some algebraic transformations, we single out the zeros of the polynomials in the numerators and denominators:

$\begin{equation*} \begin{aligned} \, & k_+' \frac{\lambda^5-3\lambda^3+6\lambda^2}{(1-\lambda)^2(1+\lambda)} +k_-' \frac{-3\lambda^3+2\lambda^2-3\lambda}{(1-\lambda)^2(1+\lambda)} \\ &\qquad\qquad -k_-' \frac{-6\lambda^3+3\lambda^2-1}{(1-\lambda)^2(1+\lambda)}-k_+' \frac{3\lambda^4-2\lambda^3 +3\lambda^2}{(1-\lambda)^2(1+\lambda)}=0, \\ & k_+' \frac{\lambda^5-3\lambda^4-\lambda^3+3\lambda^2}{(1-\lambda)^2(1+\lambda)} +k_-' \frac{3\lambda^3-\lambda^2-3\lambda+1}{(1-\lambda)^2(1+\lambda)}=0, \\ & k_+' \frac{\lambda^2(1+\lambda)(\lambda -1)(\lambda -3)}{(1-\lambda)^2 (1+\lambda)}+k_-' \frac{(1+\lambda)(\lambda-1)(3\lambda -1)}{(1-\lambda)^2(1+\lambda)}=0 . \end{aligned} \end{equation*} \notag$

Note that

$1-\lambda >0$ because

$\lambda\in\Lambda=[ {-\infty,0} ]$ . In addition,

$1+\lambda \ne 0$ because the value of the marker

$\lambda=-1$ is realized in the case of a pseudovertex with smooth curvature [13], which we exclude. Canceling the common factors in the numerators and denominators, we have

$\begin{equation*} k_+' \frac{\lambda^2(3-\lambda)}{(1-\lambda)}+k_-' \frac{1-3\lambda}{(1-\lambda)}=0. \end{equation*} \notag$

Dividing both parts by

$(1-\lambda)^2\ne 0$ , we obtain

$\begin{equation} k_+' \frac{\lambda^2(3-\lambda)}{(1-\lambda)^3}+k_-' \frac{1-3\lambda}{(1-\lambda)^3}=0. \end{equation} \tag{2.26}$

Thus,

$\alpha (\lambda)k' (t_0+0)+\beta (\lambda)k' (t_0 -0)=0$ , where

$\begin{equation*} \begin{gathered} \, \alpha (\lambda)=\frac{\lambda^2(3-\lambda)}{(1-\lambda)^3}\quad\text{and} \quad \beta (\lambda)=\frac{1-3\lambda}{(1-\lambda)^3}; \\ \alpha (\lambda)\geqslant 0, \qquad \beta (\lambda)\geqslant 0\quad\ \ \text{and} \quad\ \ \alpha (\lambda)+\beta (\lambda)=1. \end{gathered} \end{equation*} \notag$

Here we have taken into account that, by the binomial formula, the numerators of the corresponding fractional coefficients give in combination a cubed binomial from the denominators of the fractions:

$\begin{equation*} \lambda^2(3-\lambda)+1-3\lambda=1-3\lambda+3\lambda^2-\lambda^3=(1-\lambda)^3. \end{equation*} \notag$

In addition, these numerators are nonnegative because

$\lambda \leqslant 0$ . Hence so are the coefficients themselves, because their common denominator

$(1-\lambda)^3$ is positive.

Now (2.3) follows from (2.26). It is worth pointing out that the coefficients of the convex combination of the one-sided curvatures are found constructively.

This proves the theorem.

§ 3. Formulae for one-sided markers

The formulae for the one-sided markers are of the same type, and there is no significant difference in calculating them. Consider, for example, the problem of finding the left-hand marker. Canceling the normalizing factor

$\dfrac{1}{(1-\lambda)^3}$ in (2.26), we obtain

§ 4. Numerical-analytic modelling of a generalized solution

Consider the Dirichlet problem (1.5) on a closed unbounded convex set

$\Omega \subset \mathbb{R}^2$ with

$C^2$ -boundary

The curve

$\Gamma=\partial \Omega$ has a unique pseudovertex

$x^{(0)}=(\gamma_1(t_0), \gamma_2(t_0))$ corresponding to

$t_0=0$ ; here

$T^0=\{0 \}$ . It is easily seen that the curvature of the curve ceases to be smooth at this point because

$\gamma_{2,-}''' (t_0)=0\ne \gamma_{2,+}''' (t_0)=-\frac{3}{8}$ . The curvature

$k(t_0)$ is

$\frac{16}{17^{3/2}}\ne 0$ . Let us find the ratio of the one-sided curvatures at the pseudovertex and compare it with zero. We have

To the unique pseudovertex of the boundary condition there corresponds a single branch

$L(x^{(0)})$ of the singular set. To construct this branch one has to know at least one diffeomorphism in the pair of reciprocal diffeomorphisms

$t_{2}= t_{2}(t_{1})$ and

$t_{1}=t_{1}(t_{2})$ . In the general case these functions can be found analytically only on rare occasions (see [11]). This is why in our further constructions we have to recourse to numerical methods (see [13]). The graphs (plotted numerically) of the mutually inverse diffeomorphisms

$t_{2}=t_{2}(t_{1})$ and

$t_{1}= t_{1}(t_{2})$ (which generate the pseudovertex) are shown in Figure 1.

This figure illustrates the geometry of one-sided markers in the plane of the parameters

$t_{1}$ and

$t_{2}$ . The left-hand marker

$\lambda$ is the slope to the positive direction of the

$t_{1}$ -axis of the one-sided left-hand tangent to the graph

$t_{2}=t_{2}(t_{1})$ at the gluing point of the graphs of the mutually inverse diffeomorphisms. Correspondingly, the right-hand marker

$\mu=\lambda^{-1}$ is the slope to the positive direction of the

$t_{2}$ -axis of the one-sided right-hand tangent to the graph

$t_{1}=t_{1}(t_{2})$ at the same point.

Next, once both the left-hand marker of the pseudovertex and the local diffeomorphism

$t_{2}=t_{2}(t_{1})$ generating this pseudovertex are known, we construct the branch of the singular set by solving the system of equations (1.11). Note that here we can also follow another approach towards the construction of the branch of a singular set, where we integrate an ordinary differential equation for which the initial conditions are controlled by the markers of the pseudovertex and the dynamics is governed by the local diffeomorphisms generating this pseudovertex (see [14]). These authors developed a software package [29] constructing the level curves of the Euclidean distance function to a closed set

$M$ . The main step of the algorithm consists in finding the locus of points lying on normal vectors to the curve

$\Gamma$ at a fixed distance

$r>0$ from the feet of the normals. Next, we search for the intersection of this set with the singular set

$L$ , after which we exclude the parts lying at distances less than

$r$ to

$M$ .

Figure 2 shows the singular set obtained numerically and the ‘contour map’ of the generalized eikonal (evolution of wave fronts). Given a set of curves

$\Phi$ , the software package returns the table of values of the function

$u(x_1,x_2)$ on a rectangular grid in a prescribed compact subset of

$\Omega$ .

Figure 3 shows an approximation of the graph of the generalized solution to problem (1.5), which loses smoothness at points in the singular set.

Introduction

§ 1. Object of investigation, main definitions

§ 2. The main theoretical result

§ 3. Formulae for one-sided markers

§ 4. Numerical-analytic modelling of a generalized solution

Bibliography