Geometric Reinterpretation of Partial Differential Equations and Applications

Abstract

We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In each case, Reduction needs to be combined with a reinterpretation of the underlying geometry. We argue that the geometric meaning assigned to a problem has an influence, positive or negative, on the range of methods envisioned for its solution, and that the Euler–Poisson–Darboux (EPD) equation cannot be properly understood within a single geometric framework. This explains the central position of EPD-like equations.

1. Introduction

We extend the scope of the Fuchsian Reduction method [1] for elliptic equations by reinterpretation of the underlying geometry. Thus, results that appear at face value to be entirely beyond the scope of the method, because the assumptions in its basic theorems are not satisfied, can be handled by such a reinterpretation. Applications include problems on which a considerable literature already exists, such as conformal mapping and the meron problem.

Fuchsian Reduction consists in transforming a partial differential equation (PDE) for an unknown $u(x_1,x_2,\dots )$ (that could have one or several components), defined in a neighborhood of a hypersurface $\Sigma$ having and equation of the form $T:=T(x_1,x_2\dots )=0$, via changes of unknown of the form $u\mapsto v$ with

$$u(x_1,\dots )=S(x_1,\dots )+T^{m}v(x_1,\dots ),$$

where S captures the singular behavior of the solution, and m is a positive exponent. The new unknown may be called a “renormalized unknown” since the singular part has been removed from it. Under rather general assumptions, the equation for v is then of Fuchsian type, as defined in Section 2 below. Moreover, any further changes of unknown of the same type, such as

$$v(x_1,\dots )=S_1(x_1,\dots )+T^{m_1}{v}_1(x_1,\dots ),$$

also yields a Fuchsian equation. Here, $T^m{S}_1$ represents higher-order (smoother) corrections to the leading behavior S, and the exponent $m_1$ is positive. Therefore, the reduction of a Fuchsian problem is also Fuchsian, justifying the emphasis on this class of PDEs. Another reason comes from history: it seems that the first PDE to be solved is of this class (Section 3 below).

The Reduction approach has proved useful for PDEs of widely different types because there is a general and systematic way to choose the variable T, the function S, and the positive exponent1 m, so that the resulting Fuchsian equation could be solved, in situations where no other approach gives comparable results. A guide for systematic applications was outlined in the form of an algorithm in Section 1.4 of [1]. For a discussion of the drawbacks of other approaches, see the introduction of [1], and the discussion of [2], Section 6.

In general, the function T is not uniquely determined. All that matters is that T should vanish on the hypersurface $\Sigma$. Not so in the application to the problem of boundary blow-up for elliptic equations on a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ($n\ge 2$), where $\Sigma =\partial \mathrm{\Omega }$ (the boundary of $\mathrm{\Omega }$); T is then usually taken to be the distance $d_{\mathrm{\Omega }}$ to the $\partial \mathrm{\Omega }$. Since $d_{\mathrm{\Omega }}$ is the Euclidean shortest distance from a point $x\in \mathrm{\Omega }$, it appears that the underlying geometry of $\mathrm{\Omega }$ is fixed by the nature of the problem. A similar example is furnished by the equation for axially symmetric solutions of the Laplace equation, where the variable T represents the distance to the axis of symmetry. We give a few examples here showing that a change of geometry is possible and that it yields new information when combined with regularity estimates for Fuchsian PDEs. This approach is natural if we take stock of the earlier stages of the theory of Fuchsian equations. One earlier example is Penrose’s method of conformal compactification [3], which is a mapping $\mathrm{\Phi }$ of Minkowski space $M^4$, into the Einstein static universe (ESU); it allows the transformation of equations that are well-behaved under conformal transformations, such as $u_{tt}-\Delta u=u^3$ into a similar equation in which the time coordinate on the ESU is bounded. The details of the transformation are worked out in [4], pp. 70–75, see also [5]. While this method is usually applied to study the behavior of fields at null infinity [6], we may add an application to the interpretation of blow-up: a solution with blow-up on a hypersurface $\Sigma$ in the ESU may represent a global solution or a solution with blow-up on $M^4$, depending on whether $\Sigma$ intersects $\mathrm{\Phi }\left(M^4\right)$ or not. Thus, the distinction between global existence and blow-up depends on the choice of geometric interpretation. This suggests that allowing changes in the geometric interpretation of a given PDE may yield information that cannot be obtained by other means. We focus here on problems related to Fuchsian PDEs, where clear illustrations are available, but our point is no doubt valid beyond this class of PDEs.

Section 2 recalls the definition of the class of Fuchsian PDEs, gives some algebraic properties showing that they may be transformed into one another, and recalls basic regularity theorems about them in the elliptic case. Section 3 shows that a class of Fuchsian PDEs, known as the Euler–Poisson–Darboux (EPD) equation, was the first class of PDEs to be solved. The wave and Laplace equations are very special cases of the EPD equation, and this class is invariant under reduction, as just mentioned. This explains why it is advantageous to give a central place to the class of EPD-like equations. We further argue here that geometric reinterpretations help leverage the advantages of the Reduction method to obtain new results. Two examples illustrate this: Section 4 contains a very simple proof of a theorem by Brezis and Lions on the meron problem. Section 5 shows how to obtain precise estimates on the conformal radius of a domain with corners as a consequence of results on a smoother domain. Both rely on reinterpretations of the underlying geometry.

2. Background on Fuchsian Reduction

Despite their name, the Fuchsian equations arising in Fuchsian Reduction are treated in a spirit very different from the one envisioned by Fuchs. That’s because they should be related to an older tradition, that goes back to the study what we now call the Euler–Poisson–Darboux (EPD) equation. We therefore recall the theory of Fuchsian ODEs as it is taught now (Section 2.1), then show how the EPD equation arose, and was later reinterpreted as a perturbation of Fuchsian ODEs (Section 2.2). We then recall some background results on general Fuchsian PDEs.

2.1. Fuchsian ODEs

ODEs of Fuchs–Frobenius type2 were singled out in the late nineteenth century as those ODEs in one complex variable, with analytic coefficients, of the form

$${\displaystyle \frac{d^{n}u}{d{z}^n}}\left(z\right)+a_1\left(z\right){\displaystyle \frac{d^{n-1}u}{d{z}^{n-1}}}\left(z\right)+\dots +a_n\left(z\right)u\left(z\right)=0$$

for which formal series solutions, near places where the coefficients have poles, always converge. If this property holds only near one pole, say $z_0\in \mathbb{C}$, one usually says that $z_0$ is a regular singular point. Such a series may then be constructed provided the coefficients $a_k$ admit at $z_0$ a pole of order k at most. Thus, Bessel’s equation has a regular singularity at the origin, but not at infinity.3 While most of the special functions of Mathematical Physics solve equations of which some or all singularities are of Fuchs type, they constitute a special class if we remain in the class of analytic solutions and coefficients. However, by allowing non-analytic changes of variables, it is possible to reduce non-Fuchsian equations to Fuchsian equations with non-analytic coefficients [7]. In the sequel “Fuchsian” equations usually have coefficients of limited regularity.

2.2. EPD Equations as the Source of Fuchsian PDEs

A further generalization leading to what has been called Fuchsian PDEs in the last fifty years or so is an outgrowth of the study of the Euler–Poisson–Darboux (EPD) equation

$$E_{\lambda ,\mu }u:=u_{tt}+{\displaystyle \frac{\lambda }{t}}{u}_t+{\displaystyle \frac{\mu }{t^2}}u+ϵ\Delta u=0,$$

(1)

with $ϵ=\pm 1$, $\Delta ={\sum }_{k=1}^n\partial /\partial x_k$, where $\lambda$ and $\mu$ are constant parameters.4 The wave equation is recovered for $ϵ=-1$, $\lambda =\mu =0$. This line of thought is older than, and unrelated to that leading to Fuchsian ODEs; the similarity and resulting unification came after the fact.5 In fact, as explained in Section 3 below, the EPD was actually the first PDE to have been solved, although not the first one to have been written.6 The equations arising from Fuchsian Reduction include variable-coefficient generalizations of the EPD equation. The Laplace-Beltrami operator in a space of coordinates $(t,x_1,\dots ,x_n)$ and metric $t^m(d{t}^2+d{x}_1^2+\dots +x_n^2)$ is of this form.

More generally, a hyperbolic PDE is said to be Fuchsian if it may be cast, by introducing derivatives of v as new variables, in the form of a system of the general form

$$(T{\partial }_T+A)\overrightarrow{U}=T^{ϵ}F\left[\overrightarrow{U}\right],$$

where F may depend on T, other variables $x_j$, the components of $\overrightarrow{U}$, and their first-order derivatives. The EPD is of this form if we take u, $t{u}_t$ and the $t{u}_j$ as components of $\overrightarrow{U}$.

For elliptic equations of the second order, it is often preferable to take as reduced equation a second-order problem, and to define as Fuchsian any problem $Au=F\left[u\right]$ where A is a Fuchsian operator of type (I) or (II) as defined in [9] (Section 5); we recall these definitions as well as three basic theorems proved there.

An operator A is said to be of type (I) on a given domain $\mathrm{\Omega }$ if it can be written

$$A={\partial }_i\left(d^2{a}^{ij}{\partial }_j\right)+d{b}^i{\partial }_i+c,$$

with $\left(a^{ij}\right)$ uniformly elliptic and of class $C^{\alpha }$, and $b^i$, c bounded, and d is the (Euclidean shortest) distance to the boundary of $\mathrm{\Omega }$.

Remark 1.

One can also allow terms of the type ${\partial }_i\left(b^{\prime i}u\right)$ in $Au$, if $b^{\prime i}$ is of class $C^{\alpha }$, but this refinement will not be needed here.

An operator is said to be of type (II) if it can be written as

$$A=d^2{a}^{ij}{\partial }_{ij}+d{b}^i{\partial }_i+c,$$

with $\left(a^{ij}\right)$ uniformly elliptic and $a^{ij}$, $b^i$, c of class $C^{\alpha }$.

The basic results for type (I) operators are as follows. To stress that they are local, they are stated in some open neighborhood ${\mathrm{\Omega }}^{\prime }$ of a point on the boundary $\partial \mathrm{\Omega }$.

Theorem 1.

If $Ag=f$, where f et g are bounded and A is of type (I) on ${\mathrm{\Omega }}^{\prime }$, then $d\nabla g$ is bounded, and $dg$ and $d^2\nabla g$ belong to $C^{\alpha }({\mathrm{\Omega }}^{\prime }\cup \partial \mathrm{\Omega })$.

Theorem 2.

If $Ag=df$, where f and g are bounded, $g=O\left(d^{\alpha }\right)$, and A is of type (I) on ${\mathrm{\Omega }}^{\prime }$, then $g\in C^{\alpha }({\mathrm{\Omega }}^{\prime }\cup \partial \mathrm{\Omega })$ and $dg\in C^{1+\alpha }({\mathrm{\Omega }}^{\prime }\cup \partial \mathrm{\Omega })$.

The main result for type (II) operators is as follows:

Theorem 3.

If $Ag=df$, where $f\in C^{\alpha }({\mathrm{\Omega }}^{\prime }\cup \partial \mathrm{\Omega })$, $g=O\left(d^{\alpha }\right)$, and A is of type (II) on ${\mathrm{\Omega }}^{\prime }$, then $d^{2}g$ belongs to $C^{2+\alpha }({\mathrm{\Omega }}^{\prime }\cup \partial \mathrm{\Omega })$.

Our focus in this paper is to show that this set-up actually provides information in cases that appear to lie beyond its scope, namely for domains with corners.

2.3. The Reduction Method

Fuchsian Reduction, as defined above, introduces a new element with respect to the theory of Fuchsian PDEs: non-Fuchsian equations may be transformed into Fuchsian equations by changes of variables and unknowns, if we do not demand that the coefficients remain meromorphic through the transformation. Although we focus here on PDEs, this approach gives new results for ODEs as well [7]. This considerably widens the relevance of Fuchsian equations.

The reason why reduction is effective comes from three facts:

• The Fuchsian form is invariant under further transformations of v, such as $v=\tilde{S}+T\tilde{v}$.
• There are very general existence, uniqueness, and regularity results for Fuchsian PDEs.
• There are systematic procedures for finding the transformation formulae that achieve the reduction.

Reduction has led to the solution of problems, such as the ambient metric problem proposed by Schouten and Haantjes and Fefferman, the conformal radius problem that goes back to Bieberbach and, in higher dimensions, Loewner-Nirenberg, or the blow-up problem for nonlinear wave equations, including those arising from Einstein’s equations, see [1] for the solutions to all of the above. Reduction also subsumes the theory of the Cauchy problem. Indeed, the Cauchy problem for, say, the wave equation in 3-space, for a function $u(x,y,z,t)$ with data prescribed on $t=0$:

$$u(x,y,z,0)=u_0(x,y,z);u_t(x,y,z,0)=u_1(x,y,z)$$

may be reduced by introducing v such that

$$u(x,y,z,t)=u_0(x,y,z)+t{u}_1(x,y,z)+t^{2}v(x,y,z,t).$$

Thus, reduction subsumes the Cauchy problem, and has the advantage of incorporating all the arbitrary functions in the equation rather than prescribing them as side conditions. The condition that v remains bounded as $t\to 0$ singles out the solution u that automatically satisfies the initial conditions.

In a nutshell, the solution to Fuchsian PDEs takes the form of an expansion, possibly truncated after a few terms, containing non-integral powers and logarithms, that may or may not converge but still determines the general solution. For completely integrable systems, the Reduction method yields the convergence of the so-called generalized Painlevé expansions. However, it remains valid for non-integrable problems, the latter being almost always characterized by the presence of logarithmic terms at all orders in the expansion [4], Section 10.5.

Thus, it is advantageous to reduce as much as possible PDEs of interest to a form in which the leading part is EPD-like.

3. The EPD as the First PDE to Have Been Solved

The EPD Equation (1) is elliptic or hyperbolic depending on whether $ϵ=+1$ or $-1$. One may reduce the problem to one with $\mu =0$ by a change of unknown of the form $u(t,x)=t^{\sigma }v(t,x)$ (with $x=(x_1,\dots ,x_n)$), where $\sigma$ is one of the roots of the equation ${\sigma }^2+(\lambda -1)\sigma +\mu =0$.

Because of the extensive theories for solving the wave and Laplace equations that are now available, one would tend to view the EPD as a perturbation of them. However, it is the other way around. Not only are the wave and Laplace equations the special cases of (1) with $\lambda =\mu =0$, the EPD Equation (1) with $n=1$ and $ϵ=-1$ was actually the first partial differential equation to have been solved, before first-order PDEs were studied systematically [10] (p. 43, translations are ours):

The notion of partial derivative was known at the end of the seventeenth century, but the first partial differential equations do not appear before 1740; incidentally, because they appear in relation to problems in Mechanics […], they are equations of order at least 2. Partial differential equations of the first order started being studied quite generally only around 1770.

The first PDE to have been solved seems to be an equation equivalent to the EPD in d’Alembert’s treatise [11]; indeed, as stated in [10] (p. 46):

It is only in 1743 that d’Alembert writes the first partial differential equation of Mechanics, for the oscillations of a heavy chain [of length ℓ] near its vertical equilibrium position

$${\displaystyle \frac{{\partial }^{2}y}{\partial t^2}}={\displaystyle \frac{\partial y}{\partial t^2}}+(\ell -s){\displaystyle \frac{{\partial }^{2}y}{\partial s}}.$$

(2)

For details on d’Alembert’s argument, see Demidov [12] (pp. 5–6).

This equation may be reduced to an EPD equation as follows. Let $\ell -s=r^2$, so that ${\partial }_s=-{\displaystyle \frac{1}{2r}}{\partial }_r$. Define h by $h(r,t)=y(s,t)$. Equation (2) the becomes

$$h_{tt}-r^2\left({\displaystyle \frac{1}{2r}}{\displaystyle \frac{\partial }{\partial r}}\right)\left({\displaystyle \frac{1}{2r}}{\displaystyle \frac{\partial }{\partial r}}\right)h-{\displaystyle \frac{1}{2r}}{h}_r=0,$$

where subscripts again denote partial derivatives. Expanding and rearranging, we obtain the following:

$$h_{tt}-{\displaystyle \frac{1}{4}}\left(h_{rr}+{\displaystyle \frac{1}{r}}{h}_r\right)=0.$$

Up to a scaling of variables, this is an EPD equation in two dimensions. It may even be reduced to the wave equation in two space dimensions: define a function g of three variables by $g(u,v,t)=h(\sqrt{u^2+v^2},t)$. Then g satisfies the following:

$$g_{tt}-{\displaystyle \frac{1}{4}}\left(g_{uu}+g_{vv}\right)=0.$$

As the passage from h to g suggests, special cases of the EPD equation also arise naturally from axial symmetry. Indeed, the three-dimensional Laplacian, applied to functions u of z and $r=\sqrt{x^2+y^2}$ takes the form

$$\Delta u=u_{rr}+{\displaystyle \frac{1}{r}}{u}_r+u_{zz},$$

where subscripts again denote derivatives. The ubiquitousness of the EPD equation, in relation to problems with axial symmetry, was stressed by A. Weinstein in a series of papers [13,14,15]. The main tools were based on transmutation formulae between EDP equations with different parameters, but these methods do not apply to more general variable-coefficient equations of the same structure. For a recent survey on transmutation methods with numerous references, see [16].

As a simple example in higher dimensions, let u be a function of $n+m$ variables $(x_1,\dots ,x_n,y_1,\dots ,y_m)$ that solves the Laplace equation (or elliptic form of Asgeirsson’s equation)

$$({\Delta }_x+{\Delta }_y)u=0,$$

where ${\Delta }_x={\sum }_{j=1}^n\partial /\partial x_j$ and ${\Delta }_y={\sum }_{j=1}^m\partial /\partial y_j$. Let $r=\sqrt{x_1^2+\dots +x_n^2}$. Then, if u is radial in the x variables, so that $u(x,y)$ may be written as a function $v(r,y)$ of r and y alone, then one checks by direct calculation that v solves the equation:

$${\partial }_{rr}u+{\displaystyle \frac{n-1}{r}}{\partial }_{r}u+{\Delta }_{y}u=0.$$

Another related interpretation of the EPD equation is as follows:

The hyperbolic form of the same equation with $m=n$ (with ${\Delta }_{y}u$ replaced by $-{\Delta }_{y}u$) is satisfied by the mean of a function of m variables over the sphere of radius r; this observation is at the basis of explicit formulae for the solution of the wave equation in any number of space dimensions (see [17], pp. 88–89, 96–100; [18]; [4], pp. 17–18).

The EPD and other Fuchsian PDEs also arise from Laplace–Beltrami operators on rank 1 symmetric spaces, see [19], Section 7.4 and [20], Chapter 2.

Thus, the EPD equation runs through most of the history of PDEs, there is no single geometric interpretation that captures all of its mathematical properties. It is therefore of advantage to view it as a fundamental mathematical object. There is an extensive literature on Fuchsian PDEs but the idea of systematically reducing a non-Fuchsian PDE to an EPD-like equation is the new element introduced by Fuchsian Reduction. The latter, in turn, has also led to new results about Fuchsian PDEs, that we extend next.

4. Boundary Regularity for a Nonlinear Elliptic PDE

Consider the following problem, for a function $u(x,y)$ of two variables, in a domain of the form ${\mathrm{\Omega }}_a:=\{(x,y)\mid 0<x<a\mathrm{and}|y|<a\}$:

$$(-x^2\Delta +2)u=u^{2}g\left(u\right).$$

(3)

Here $\Delta :={\partial }_{xx}+{\partial }_{yy}$ and the function g is of class $C^{\infty }$ for simplicity. Bounded solutions are of class $C^{\infty }$ in ${\mathrm{\Omega }}_a$ by interior regularity. We are interested in their behavior at the boundary $(x=0)$. This problem arose in the theory of multimeron solutions of Yang–Mills equations [21], which leads to the equation

$$(-x^2\Delta -1)\psi +{\psi }^3=0\mathrm{with}\psi (0,y)=\pm 1.$$

(4)

Thus, it is assumed that there are finitely many points $a_0<a_1<\dots$ on the axis $(x=0)$ such that $\psi =+1$ or $\psi =-1$ in each of the intervals determined by the $a_j$. Letting $u=\psi \mp 1$ leads to (3) with $g\left(u\right)=-u\pm 3$, if we assume (by translation) that none of the $a_j$ is zero, and a is small enough so that no $a_j$ lies in $(-a,a)$. In this case, solutions exist [22,23], and, if u is continuous up to $x=0$, then are $\mathcal{O}\left(x^2\right)$. We now prove the following.

Theorem 4.

If $u=\mathcal{O}\left(x^2\right)$ is a solution of (3), then $u(x,y)/x^2$ is of class $C^{\infty }$ up to $x=0$.

Proof. 

Let $D=x{\partial }_x$. Equation (3) takes the form

$$-\left[(D-2)(D+1)+x^2{\partial }_y^2\right]u=u^{2}g\left(u\right).$$

(5)

Let us substitute $u=x^{2}v$ into this equation, using twice the general relation

$$(D-\rho )x^{2}v=x^2(D-\rho +2)v,$$

valid for any $\rho$, and dividing through by $x^2$, we obtain the following equation for v.

$$-\left[D(D+3)+x^2{\partial }_y^2\right]v=x^2{v}^{2}g\left(x^{2}v\right).$$

(6)

Let us introduce new variables $x_1,\dots ,x_p$ such that $x^2=x_1^2+\dots +x_p^2$. Define a new function w of $p+1$ variables by $w(x_1,\dots ,x_p,y):=v(\sqrt{x_1^2+\dots +x_p^2},y)$. Let $L_p:={\partial }_{x_1}^2+\dots +{\partial }_{x_p}^2$ be the Laplace operator in p dimensions. When acting on radial functions (that only depend on x which is the radial variable in the space of coordinates $x_1,\dots ,x_p$), we obtain

$$x^2{L}_{p}w=x^2\left(w_{xx}+{\displaystyle \frac{p-1}{x}}{w}_x\right)=D(D+p-2)v$$

since v is simply equal to w considered as a function of $(x,y)$. Let us now take $p=5$ in order to make $D(D+p-2)v$ equal to the term $D(D+3)v$ in (6). Equation (6) then becomes

$$-\left[x^2{L}_5+x^2{\partial }_y^2\right]w=x^2{w}^{2}g\left(x^{2}w\right).$$

(7)

Since $L_6:=L_5+{\partial }_y^2$ is the Laplace operator in the six variables $(x_1,\dots ,x_5,y)$, we obtain, after dividing (7) by $x^3$,

$$-L_{6}w=w^{2}g\left(x^{2}w\right).$$

(8)

Now, w is defined when $(x_1,\dots ,x_p,y)$ lies in a full neighborhood of the origin in ${\mathbb{R}}^6$ and, by assumption, $v=u/x^2\in L^{\infty }\left({\mathrm{\Omega }}_0\right)$ so that w is bounded. Therefore, $L_{6}w$ is bounded, and, therefore, by interior regularity estimates, w is of class $C^{1+\alpha }$ for any $\alpha \in (0,1)$. But then $w^{2}g\left(x^{2}w\right)$ is of class $C^{1+\alpha }$, hence w is $C^{3+\alpha }$, and so forth. Since $u(x,y)=x^{2}w(x,0,\dots ,0,y)$ for any $x>0$, the result follows. □

A proof of this result, which strongly uses special features of the equation, is given in [24]. The above proof, based on the recognition of the Fuchsian structure of the problem, is simpler and has a wider scope; it applies to general nonlinearities g. We now turn to a second problem in which Fuchsian Reduction was used to advantage. After recalling earlier results, we show that they may be significantly improved by a geometric reinterpretation.

5. Conformal Radius for Non-Smooth Domains

In this section, we show that results on the boundary regularity of the conformal radius [25,26,27] also yield information for domains with corners, using properties of the Riemann mapping function. The results in [27] were obtained by reducing the elliptic Liouville equation (Equation (10) below), on a $C^{2+\alpha }$ domain, to an EPD-like equation where the expansion variable called T above was the distance to the boundary. We now extend the asymptotics obtained in that paper to the case of domains with corners by changing the underlying geometry.

Let $\mathrm{\Omega }$ be an open, simply connected, bounded domain in $\mathbb{C}$. The plane ${\mathbb{R}}^2$ will be identified with $\mathbb{C}$, the points of which will be written interchangeably $(x,y)$ or $z=x+iy$. It follows from the Riemann mapping theorem that there is a conformal mapping $f_{\mathrm{\Omega }}$, unique up to fractional-linear transformations of the unit disk B, that maps $\mathrm{\Omega }$ onto B. Rather than prescribing the target, one can normalize the map by its behavior at one point of $\mathrm{\Omega }$. This leads to the notion of conformal radius. First, for every point $z_0\in \mathrm{\Omega }$, there is a unique conformal map $g_{\mathrm{\Omega },z_0}$ that (i) sends $\mathrm{\Omega }$ one-to-one onto a ball of some radius $v_{\mathrm{\Omega }}\left(z_0\right)$ and satisfies (ii) $g_{\mathrm{\Omega },z_0}\left(z_0\right)=0$ and (iii) $d{g}_{\mathrm{\Omega },z_0}/dz\left(z_0\right)=1$. The conformal radius7 of $\mathrm{\Omega }$ is defined as the function $v_{\mathrm{\Omega }}:\mathrm{\Omega }\to {\mathbb{R}}_{+}^{*}$, $z_0\mapsto v_{\mathrm{\Omega }}\left(z_0\right)$. Now, define $u_{\mathrm{\Omega }}$ through the relation

$$exp(-u_{\mathrm{\Omega }})=v_{\mathrm{\Omega }}.$$

(9)

Then, $u_{\mathrm{\Omega }}\left(P\right)$ satisfies

$$-\Delta u_{\mathrm{\Omega }}+4e^{2u_{\mathrm{\Omega }}}=0,$$

(10)

and tends to $+\infty$ as the point $P=(x,y)$ tends to the boundary $\partial \mathrm{\Omega }$. The conformal radius is related to the mapping function $f_{\mathrm{\Omega }}$ through the following relation, where $z=x+iy$.8

$$u_{\mathrm{\Omega }}(x,y)=ln{\displaystyle \frac{|f_{\mathrm{\Omega }}^{\prime }\left(z\right)|}{1-|f_{\mathrm{\Omega }}{\left(z\right)|}^2}}$$

(11)

Of course, this function is smooth inside $\mathrm{\Omega }$. Our focus in this paper is on its boundary regularity.

Let us now assume the boundary $\partial \mathrm{\Omega }$ of $\mathrm{\Omega }$ to be of class $C^{2+\alpha }$ where $0<\alpha <1$. Let $d_{\mathrm{\Omega }}\left(P\right)$ be the (shortest Euclidean) distance of P to the boundary. The Kellogg–Warschawski theorem [28,30,31] (th. 3.6), ensures that $f_{\mathrm{\Omega }}$ has the same regularity as the boundary, namely $C^{2+\alpha }$. All we can conclude from Equation (11) is that $u_{\mathrm{\Omega }}$ is of class $C^{1+\alpha }\left(\overline{\mathrm{\Omega }}\right)$. Fuchsian Reduction ([27], see also [1] (Chapter 9)) shows that $v_{\mathrm{\Omega }}$ is actually of class $C^{2+\alpha }$ up to the boundary:

Theorem 5.

If Ω is of class $C^{2+\alpha }$, then $v_{\mathrm{\Omega }}=exp(-u_{\mathrm{\Omega }})\in C^{2+\alpha }\left(\overline{\mathrm{\Omega }}\right)$. The following asymptotics hold:

$$v_{\mathrm{\Omega }}\left(P\right)=2d_{\mathrm{\Omega }}\left(P\right)-d_{\mathrm{\Omega }}^2\left(P\right)(\kappa \left(P\right)+\mathcal{O}\left(d_{\mathrm{\Omega }}^{\alpha }\left(P\right)\right))$$

(12)

where $d_{\mathrm{\Omega }}\left(P\right)$ is the distance of P to the boundary, and $\kappa \left(P\right)$ is the curvature of the boundary at the boundary point closest to P.9

This somewhat paradoxical result has a counterpart in the hyperbolic case: the arbitrary functions in the closed form solution of the Liouville equation are actually more regular than the solution [1] (th. 10.29).

From the relation (9), we obtain immediately

$$\begin{array}{ccc}\hfill u_{\mathrm{\Omega }}\left(P\right)& =& -ln\left\{2d_{\mathrm{\Omega }}\left(P\right)\left[1-{\displaystyle \frac{1}{2}}\kappa \left(P\right)d_{\mathrm{\Omega }}\left(P\right)+\mathcal{O}\left(d_{\mathrm{\Omega }}^{1+\alpha }\left(P\right)\right)\right]\right\}\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill & =& -ln\left(2d_{\mathrm{\Omega }}\left(P\right)\right)+{\displaystyle \frac{1}{2}}\kappa \left(P\right)d_{\mathrm{\Omega }}\left(P\right)+\mathcal{O}\left(d_{\mathrm{\Omega }}^{1+\alpha }\left(P\right)\right).\hfill \end{array}$$

(14)

It would appear that the above results, while optimal for such domains, do not give any information for, say, domains with corners. However, this is not the case because it is possible to map a $C^{2+\alpha }$ domain on a domain with a corner by simple transformations such as $z\mapsto z^{\mu }$ where $\mu$ is not necessarily integral or even fractional.

To show this, consider a domain $\mathrm{\Gamma }$ and a conformal homeomorphism $h:\mathrm{\Gamma }\to \mathrm{\Omega }$. Since $f_{\mathrm{\Omega }}$ maps $\mathrm{\Omega }$ conformally onto the unit disk B, we have a conformal map $f_{\mathrm{\Gamma }}:=f_{\mathrm{\Omega }}\circ h:\mathrm{\Gamma }\to B$. Therefore, applying (11) to $\mathrm{\Gamma }$, we obtain10 a formula for the hyperbolic radius function for $\mathrm{\Gamma }$.

$$u_{\mathrm{\Gamma }}\left(z\right)=ln{\displaystyle \frac{|f_{\mathrm{\Gamma }}^{\prime }\left(z\right)|}{1-|f_{\mathrm{\Gamma }}{\left(z\right)|}^2}}$$

Since $f_{\mathrm{\Gamma }}^{\prime }\left(z\right)=f_{\mathrm{\Omega }}^{\prime }\left(h\left(z\right)\right)h^{\prime }\left(z\right)$, it follows that

$$u_{\mathrm{\Gamma }}\left(z\right)=u_{\mathrm{\Omega }}\left(h\left(z\right)\right)+ln\left|h^{\prime }\left(z\right)\right|.$$

Here is a simple result that seems difficult to obtain by any other means, in the case when $\mathrm{\Gamma }$ has an angular point such that it may be transformed into a $C^{2+\alpha }$ domain $\mathrm{\Omega }$ via the transformation $h:z\mapsto z^{\mu }$, where $\mu \in {\mathbb{R}}_{+}\ast$. We assume $\mathrm{\Gamma }\subset \mathbb{C}\setminus {\mathbb{R}}_{-}$ and take the principal determination of $z^{\mu }$ for definiteness. Observe that it would be quite cumbersome to describe directly the regularity of $\partial \mathrm{\Gamma }$. The above formula then gives the simple result

$$u_{\mathrm{\Gamma }}\left(z\right)=u_{\mathrm{\Omega }}\left(z^{\mu }\right)+ln\left|\mu z^{\mu -1}\right|.$$

Equation (14) now yields a very precise expansion of $u_{\mathrm{\Gamma }}$ that seems difficult to obtain in any other way.

Another noteworthy point is that the distance function $d_{\mathrm{\Gamma }}$ may be controlled using $d_{\mathrm{\Omega }}$, even though, given $P\in \mathrm{\Gamma }$, the closest point to P on the boundary $\partial \mathrm{\Gamma }$ is not necessarily unique, unlike what happens for $\mathrm{\Omega }$.

Theorem 6.

For any $z\in \mathrm{\Gamma }$,

$${\displaystyle \frac{1}{4}}|h^{\prime }\left(z\right)\le {\displaystyle \frac{d_{\mathrm{\Omega }}\left(h\left(z\right)\right)}{d_{\mathrm{\Gamma }}\left(z\right)}}\le 4\left|h^{\prime }\left(z\right)\right|.$$

(15)

Proof. 

The argument relies on the following result [28] (Cor. 1.4).11

Theorem 7.

Let f map the unit disk B conformally into $\mathbb{C}$. Define $B^{\prime }:=f\left(B\right)$. The distance function $d_{B^{\prime }}$ to the boundary of the image of B satisfies the inequality

$${\displaystyle \frac{1}{4}}{(1-|z|}^2\left)\right|f^{\prime }\left(z\right)|\le d_{B^{\prime }}\left(f\left(z\right)\right)\le {(1-|z|}^2\left)\right|f^{\prime }\left(z\right)|.$$

(16)

Applying this by taking $f=f_{\mathrm{\Omega }}^{-1}$ (so that $f\left(B\right)=\mathrm{\Omega }$), replacing z in (16) by $f_{\mathrm{\Omega }}\left(z\right)$, and using the relation ${\left(f_{\mathrm{\Omega }}^{-1}\right)}^{\prime }\left(z\right)=1/f_{\mathrm{\Omega }}^{\prime }\left[f_{\mathrm{\Omega }}^{-1}\left(z\right)\right]$, we obtain

$${\displaystyle \frac{1}{4}}(1-|f_{\mathrm{\Omega }}{\left(z\right)|}^2)\le d_{\mathrm{\Omega }}\left(z\right)|f_{\mathrm{\Omega }}^{\prime }\left(z\right)|\le (1-|f_{\mathrm{\Omega }}\left(z\right){|}^2).$$

(17)

Similarly, taking $f=f_{\mathrm{\Gamma }}^{-1}$, and using $f_{\mathrm{\Gamma }}^{\prime }\left(z\right)=h^{\prime }\left(z\right)f_{\mathrm{\Omega }}^{\prime }\left(h\left(z\right)\right)$, we obtain the inequalities

$${\displaystyle \frac{1}{4}}(1-|f_{\mathrm{\Gamma }}{\left(z\right)|}^2)\le d_{\mathrm{\Gamma }}\left(z\right)|f_{\mathrm{\Omega }}^{\prime }\left(h\left(z\right)\right)\left|\right|h^{\prime }\left(z\right)|\le (1-|f_{\mathrm{\Gamma }}\left(z\right){|}^2).$$

(18)

Next, applying (17) at the point $h\left(z\right)$ and using $f_{\mathrm{\Gamma }}\left(z\right)=f_{\mathrm{\Omega }}\left(h\left(z\right)\right)$ yields

$${\displaystyle \frac{1}{4}}(1-|f_{\mathrm{\Gamma }}{\left(z\right)|}^2)\le d_{\mathrm{\Omega }}\left(h\left(z\right)\right)|f_{\mathrm{\Omega }}^{\prime }\left(h\left(z\right)\right)|\le (1-|f_{\mathrm{\Gamma }}\left(z\right){|}^2).$$

(19)

Combining with (18) (at the point z) yields the desired inequalities. □

It follows from Theorem 6 that the singular part of $u_{\mathrm{\Gamma }}$ is controlled by that of $u_{\mathrm{\Omega }}$. Indeed, we have, by taking logarithms in (15),

$$ln|h^{\prime }\left(z\right)|-ln4\le ln\left[2d_{\mathrm{\Omega }}\left(h\left(z\right)\right)\right]-ln\left[2d_{\mathrm{\Gamma }}\left(z\right)\right]\le ln\left|h^{\prime }\left(z\right)\right|+ln4.$$

(20)

Thus, even though the regularity of the conformal mapping in the Riemann mapping theorem is not sufficient to prove the optimal regularity of the conformal radius, combining reduction techniques with estimates for the Riemann map gives new information.

6. Concluding Remarks

The present paper extends the scope of Fuchsian Reduction in two ways.

First, it shows that, somewhat surprisingly, the Euler–Poisson–Darboux equation, rather than the Laplace or wave equations, provided the first incentive for developing analytic methods for solving partial differential equations. Indeed, it was the first PDE to have been solved. Viewing the EDP equation as a perturbation of an ODE is a late, misleading development due to the growth of generalizations of ODEs with a regular singular point or “of Fuchsian type.” Putting EPD-like equations at the center not only makes many manipulations natural, but it also connects back to the source of PDE theory. The theory of the Cauchy problem is merely the case in which the expansions given by Reduction contain only integral powers and no logarithms.

The second point is that one may significantly extend the scope of existing systematic methods of reduction by allowing changes to the underlying geometry. This includes adding space variables or performing non-smooth transformations. In other words, assuming a PDE is naturally written in Euclidean space because its leading part contains the Laplacian sets unnecessary blinkers that can be removed by being aware of other geometrical interpretations. This is achieved here for the meron problem and for the regularity of the conformal radius in non-smooth domains, building on our results on the $C^{2+\alpha }$ case [27], that were already based on Fuchsian Reduction. The main new element in both problems is that the optimal regularity of the solution only becomes clear if the underlying geometry is reinterpreted. The blow-up problem for nonlinear wave equations has already provided one example, since non-analytic solutions involving logarithmic terms such as $TlnT$, $T{(lnT)}^2$,...may be uniformized by treating the $T(lnT)$ as independent variables [1] (§4.3 and Chapter 10), which is already an example of a “method of ascent” into higher-dimensional spaces. For the meron problem, the result is a simple consequence of viewing the problem as derived from a higher-dimensional problem by descent to two dimensions. For the second problem, the problem on a non-smooth domain is viewed as equivalent to another one on a smoother domain via the Riemann mapping function. The optimal regularity of the Riemann function was not sufficient to conclude, which is why Reduction was needed [27]; here, this work is extended to domains with corners using a version of the Schwarz-Pick theorem.

Thus, the geometric interpretation of a PDE determines what manipulations are natural and, therefore, has considerable heuristic value. Given that the tools used here are applicable to many other problems, as indicated throughout the paper, it is expected that further progress is at hand.