1 Introduction

I have been interested in image processing for a long time. Let \(\varOmega\) be a plane rectangle and \(x\in \varOmega\). If \(u_0(x)\) denotes the gray scale of a perturbed black and white image, e.g. a medical sonogram, how can one reconstruct the original image?

There is a diffusion approach: \(u_0\) is the initial value to an evolution problem. Hopefully essential features of the image will evolve and noise is reduced. In 1987 Perona and Malik suggested taking the noisy image \(u_0\) as initial datum for a diffusion equation such as

$$\begin{aligned} u_t-\mathrm{div}\ \left( {\frac{\nabla u}{(1+|\nabla u|^2)^2}}\right) =0, \end{aligned}$$
(1)

or, more generally

$$\begin{aligned} u_t-\mathrm{div}\ \left( a(|\nabla u|^2)\nabla u\right) =0, \end{aligned}$$
(2)

with a(s) positive and decreasing to zero as \(s\rightarrow \infty\), and under no-flux boundary conditions.

Small diffusion near discontinuities in \(u_0\) was supposed to lead to edge preservation, while large diffusion elsewhere would somehow mollify the brightness function and remove noise. If \(a(s)=s^{-1/2}\), then formally Eq. (2) becomes

$$\begin{aligned} u_t-\mathrm{div}\ \left( {\frac{\nabla u}{|\nabla u|}}\right) =0. \end{aligned}$$
(3)

This is commonly called total variation flow, TV-flow for short. Equation 2 can be rephrased as

$$\begin{aligned} u_t-\mathrm{div}(a(|\nabla u|^2)\nabla u)=u_t-a(|\nabla u|^2)\varDelta u-2a'(|\nabla u|^2)\nabla u D^2u\nabla u=0, \end{aligned}$$
(4)

with \(D^2u\) denoting the Hessian matrix of second derivatives.

For each \(t>0\) let us express the second spatial derivatives in intrinsic coordinates normal and parallel to level surfaces of u and set \(\nu =-\frac{\nabla u}{|\nabla u|}\). Then

$$\begin{aligned} \varDelta u=u_{\nu \nu }+(n-1)H u_\nu =u_{\nu \nu }+\varDelta _{n-1}u, \end{aligned}$$

where H is the mean curvature of a level surface of u, or where \(\varDelta _{n-1}\) is the (n-1)-dimensional Laplace operator tangent to this level surface. Hence \(u_t-a(|\nabla u|^2)\varDelta u-2a'(|\nabla u|^2)|\nabla u|^2 u_{\nu \nu }=0,\) or in short

$$\begin{aligned} u_t-b(|\nabla u|^2)u_{\nu \nu }-a(|\nabla u|^2)\varDelta _{n-1}u=0. \end{aligned}$$
(5)

Thus, for \(a\not =b\) the diffusion is anisotropic, and often b is not positive. For TV-flow it is identically zero and for Perona–Malik flow it becomes negative, if the modulus of the spatial gradient is above a certain threshold. Then there is a backward diffusion effect which leads to a steepening of profiles in direction of the gradient of u, but at the same time there is forward diffusion along level surfaces of u.

2 Anisotropic diffusion

Little is known about such diffusion equations. But if we assume no flux on the boundary \(\partial \varOmega\), then for any natural number \(k>1\ldots\)

$$\begin{aligned} \frac{\partial }{\partial t} ||u||_k= & {} ||u||^{1-k}_k\int _\varOmega |u|^{k-2}uu_t\ dx=||u||^{1-k}_k\int _\varOmega |u|^{k-2}u\,\mathrm{div} (a\,\nabla u)\ dx \\= & {} -||u||^{1-k}_k\int _\varOmega |u|^{k-2}\,|\nabla u|^2\ a(|\nabla u|^2)\,dx <0, \end{aligned}$$

u does not increase in \(L^k(\varOmega )\) and thus in \(L^\infty (\varOmega )\).

This constitutes a sort of maximum principle [26] that had previously been dismissed as hopeless because of the backward nature of (5) for negative b. There was also an abundance of numerical experiments that revealed many interesting effects, see e.g [8, 13]. Positive data with black objects against a white background would vanish in finite time with a speed depending on their shape and size, for instance.

In an attempt to understand this I started playing with the equation. In 2004 at a conference in honour of Gunnar Aronsson in Linköping I presented the following considerations. Consider an evolution equation of type \(w_t-A_pw=0\). The Ansatz \(w(t,x)=T(t)u(x)\) leads to \(T'(t)u(x)-A_p(T(t)u(x))\)=0, and if \(A_p\) is homogeneous of degree d, to \(T'(t) u(x)-T(t)^d A_p(u(x))=0\). So separation of variables gives \(T^dT'(t)=-\lambda\) and \(A_pu+\lambda u=0.\) For esthetic reasons the “eigenfunction” for \(A_p\) should only be called eigenfunction if \(A_p\) is homogeneous of degree 1. Otherwise a multiple of the eigenfunction would no longer be one. In fact, only in this case does T decay exponentially in time and preserves shape independent of size. For \(d<1\) as in the TV case, it decays linearly to zero in finite time. Therefore I defined

$$\begin{aligned} A_pu:=\frac{1}{p}|\nabla u|^{2-p}\varDelta _p u \end{aligned}$$

Note that \(A_pu= \frac{1}{p}A_1(u)+\frac{p-1}{p}A_\infty u\) is a convex combination of the limits \(A_1\) and \(A_\infty\). The operator \(A_p\) can be rewritten as \(\sum _{i,j}a_{ij}(x)u_{x_i x_j}(x)\) with coefficients

$$\begin{aligned} a_{ij}=\frac{1}{p}\left( \delta _{ij}+(p-2)\frac{u_{x_i} u_{x_j}}{|\nabla u|^2}\right) . \end{aligned}$$

The coefficient matrix \(a:=\frac{1}{p}I+\frac{p-2}{p} \frac{\nabla u\otimes \nabla u}{|\nabla u|^2}\) is positive definite and bounded for every \(p\in (1,\infty )\). Its eigenvalues are bounded below by \(\min \{\frac{p-1}{p},{1}\}\) and above by \(\max \{\frac{p-1}{p},{1}\}\). In this sense the normalized p-Laplacian \(\varDelta _p^N=A_p\) is more benign than the degenerate or singular p-Laplacian \(\varDelta _p\). I also pointed out that the eigenvalue problem \(A_\infty u+\lambda u=0\) had \(\lambda =1\) and \(u=\cos (|x|)\) as a solution.

The equation \(u_t-\varDelta _p^Nu=0\) was studied

  • for \(p=1\) by Evans and Spruck in [15] and in a game-theoretical context by Kohn and Serfaty in [35],

  • for \(p=\infty\) by Juutinen, and myself in [23] and in game-theoretical context by Peres, Schramm, Sheffield and Wilson in [40],

  • and for \(p\in (1,\infty )\) by Does, who started her dissertation in 2006 and published the results in [14], and in terms of game-theory by Peres and Sheffield in [41] .

In fact, Does studied existence, time asymptotics and spatial Lipschitz continuity for viscosity solutions to

$$\begin{aligned} u_t-\varDelta _p^N u&=0 \quad&\quad \text{ in } (0,\infty )\times \varOmega \\ \frac{\partial u}{\partial \nu }&=0 \quad&\quad \text{ on } (0,\infty )\times \partial \varOmega \\ u(0,x)&=u_0(x)&\quad \text{ initially }, \end{aligned}$$

for \(p\in (1,\infty )\), and she did numerical simulations for various p. More regularity was later found by Banerjee and Garofalo [3], Jin and Silvestre [19], Attouchi and Parviainen [2] and Høeg and Lindqvist [18]. Spatial gradients are indeed not only bounded but locally Hölder-continuous, even for inhomogeneous equations, and \(u_t\) and \(D^2u\) are in \(L^2_\mathrm{loc}(\varOmega _T)\) (at least for \(|p- 2|<0.8\)).

3 Elliptic problems

To avoid confusion let me point out that another 1-homogeneous version of \(A_p\) is \(A_p(u):=|u|^{2-p}\varDelta _p u\). The first results on the parabolic version \(w_t-|w|^{2-p}\varDelta _pw=0\) can be found in [34]. The Ansatz \(w(t,x)=T(t)u(x)\) leads to an eigenvalue problem which is widely studied. It transforms into

$$\begin{aligned} \varDelta _pu+\lambda |u|^{p-2}u=0. \end{aligned}$$
(6)

For finite p its first eigenvalue minimizes the Rayleigh quotient \(||\nabla v||_p/||v||_p\) on \(W^{1,p}_0(\varOmega )\). For \(p\rightarrow \infty\) there is the seminal paper of Juutinen, Lindqvist and Manfredi [24], while for \(p\rightarrow 1\) the first Dirichlet eigenfunction leads to so-called Cheeger sets \(\varOmega _C\). A Cheeger set of \(\varOmega\) minimizes \(|\partial D| / |D|\) among all \(D\subset \varOmega\), see [16].

For \(p\in (1,\infty )\) the first eigenfunction is known to be simple, see [5] and [29]. Many isoperimetric inequalities for \(p=2\) carry over to \(p\in (1,\infty )\) and a survey on this eigenvalue problem can be found in [33]. Let me just remark in passing that many results for the Laplacian have been extended to the p-Laplacian and its limits as \(p\rightarrow \infty\) or \(p\rightarrow 1\), these include

  • maximum principles

  • simplicity of the first Dirichlet eigenfunction

  • symmetry for overdetermined bvps and \(\varDelta _p u=-1\)

  • concavity properties of u for convex domains and positive eigenfunctions or for torsion functions, i.e., solutions to \(-\varDelta _p=1\). I will not elaborate on those because the focus of this paper is on the normalized p-Laplacian.

4 Mean values

Suppose u is locally of class \(C^3\) and \(D\subset \mathbb {R}^n\) is point-symmetric, i.e. \(D=-D\). For \(h\in D\) the Taylor expansion around \(x\in \varOmega \subset \mathbb {R}^n\) says

$$\begin{aligned} u(x)=u(x+ h)- \sum _{i=1}^n u_i(x) \cdot h_i-\frac{1}{2} \sum _{i,j=1}^n u_{ij}(x)h_ih_j+ o(h^2)\ , \end{aligned}$$
(7)

and integration with respect to h over D yields

$$\begin{aligned} u(x)=\frac{1}{|D|}\int _{\ D}u(x+h)\, dh-\sum _{i,j=1}^na_{ij}u_{ij}(x)+o(h^2) \end{aligned}$$
(8)

with \(a_{ij}=\frac{1}{2}\int _Dh_ih_j\, dh\). If D is a ball, then \(a_{ij}=\delta _{ij}h^2\), and for harmonic functions the quadratic term in h vanishes.

If u solves another elliptic equation \(\sum _{i,j}a_{ij}u_{x_ix_j}=0\) with symmetric coefficients \(a_{ij}=a_{ji}\) one can replace the ball, for instance, with an appropriate ellipse or parallelepiped. The direction of the axes are then determined by the eigenvectors of the coefficient matrix. This proves a pointwise asymptotic mean value property for solutions to elliptic equations with constant and variable coefficients [17].

If u is just a viscosity solution (even of a degenerate quasilinear elliptic equation), the asymptotic mean value property continues to hold in the viscosity sense. Moreover, it continues to hold if the Lebesgue measure on D is mixed with or replaced by point measures. Ton give an example, think of approximating \(\varDelta u(x)\) in 2d by the five point stencil \(u(x\pm h_i)-4u(x)\) with \(h_1=(1,0)\) and \(h_2=(0,1)\).

There are numerous papers on asymptotic mean value properties for \(\varDelta _p^Nu=0\), starting with Le Gruyer [37] and Manfredi, Parviainen and Rossi [39], first for \(p\ge 2\) only. This was partly due to the notation of \(p\varDelta _p^Nu=\varDelta u +(p-2)\varDelta _\infty ^Nu\) as a sum of two elliptic operators that turned into a difference for \(p\in (1,2)\).

Rewriting \(\varDelta _p^Nu=\frac{1}{p} \varDelta _1^N u +\frac{p-1}{p} \varDelta _\infty ^N\) as a convex combination of elliptic operators made things conceptually easier and led to the paper [31] on \(p\in [1,\infty ]\), where D is a ball \(B_\varepsilon\), and the average is taken over a combination of

  1. (a)

    an equatorial \((n-1)\)-dimensional plane orthogonal to \(\nabla u(x)\) and

  2. (b)

    the midrange of u on \(B_\varepsilon\).

5 Eigenfunctions

The existence of eigenfunctions, i.e. solutions to

$$\begin{aligned} \varDelta _p^Nu+\lambda u=0\hbox { in }\varOmega ,\quad u=0\hbox { on }\partial \varOmega , \end{aligned}$$
(9)

was established for \(p=\infty\) by Juutinen [20] and for general \(p\in (1,\infty )\) by Birindelli and Demengel [6]. Nothing seems to be known about the case \(p=1\). An aphorism attributed to Ennio De Giorgi says “Chi cerca, trova. Chi ricerca, ritrova.” So Juutinen apparently rediscovered cos(|x|) as radial eigenfunctions.

The simplicity of the first eigenfunction for \(p\in (1,\infty )\) was shown only recently [12], and for \(p=\infty\) and \(p=1\) it appears to be open. Also the validity of the Faber–Krahn inequality is open for any \(p\not =2\).

What about higher eigenfunctions? Can we find some (even only radial ones) at least for a ball? If \(u(x)=v(|x|)\) is an eigenfunction of \(\varDelta _p^Nu\), the eigenvalue problem

$$\begin{aligned} \varDelta _p^Nu+\lambda u=0\;\;\hbox { in }B_R(0),\quad u=0\;\;\hbox { on }\partial B_R(0), \end{aligned}$$
(10)

transforms into

$$\begin{aligned} v''(r)+\frac{n-1}{p-1}\frac{1}{r}v'(r)+\frac{p}{p-1}\lambda v(r)=0 \quad \hbox {in }(0,R) \end{aligned}$$
(11)

with boundary conditions

$$\begin{aligned} v'(0)=0=v(R). \end{aligned}$$
(12)

This is a Bessel-type equation in a fractional dimension, and it has a complete orthonormal system of eigenfunctions in the weighted space \(L^2_{r^{(2n-2)/(p-1)}}(0,R)\), see [32]. Nothing seems to be known about nonradial eigenfunctions for \(p\not =2\).

6 Dirichlet problems

Important contributions to the Dirichlet problem

$$\begin{aligned} -\varDelta _p^Nu=f\hbox { in }\varOmega ,\quad u=0\hbox { on }\partial \varOmega , \end{aligned}$$
(13)

were given by Lu and Wang [38]. They proved existence and uniqueness (if f does not change sign). And only a few years ago Attouchi, Parviainen and Ruosteenoja [1] proved \(C^{1,\alpha }\) regularity.

6.1 Overdetermined Serrin type problem

In the groundbreaking papers [42, 43] and for \(p=2\) Serrin and Weinberger proved that the overdetermined problem

$$\begin{aligned} -\varDelta _p^Nu=1\hbox { in }\varOmega ,\quad u=0\quad \hbox {and}\quad \frac{\partial u}{\partial \nu }=\text {const.} \hbox { on }\partial \varOmega , \end{aligned}$$
(14)

has only a solution if \(\varOmega\) is a ball. Does this result carry over to general p?

  • For \(p=\infty\) a stadium, i.e. the union of a rectangle and two half discs, was discovered in [7] as a counterexample. Its boundary \(\partial \varOmega\) is of class \(C^{1,1}\).

  • Crasta and Fragalà [9] could show that for \(p=\infty\) only stadium-like domains were possible as convex \(C^{1,1}\) domains. Among those only balls are \(C^2\).

  • Finally for \(p\in (1,\infty )\) it was shown in [4] that \(\partial \varOmega\) had to be a sphere, provided it was \(C^{2,\alpha }\).

6.2 Concavity results on convex \(\varOmega\)

In the 1980s it was shown that the solution to \(-\varDelta u=1\) in \(\varOmega\), \(u=0\) on \(\partial \varOmega\), is \(\frac{1}{2}\)-powerconcave, i.e. \(v=\sqrt{u}\) is concave. [25] contains an extensive description and list of references on this and related issues. What about \(-\varDelta _p^N u=1\) for \(p\not =2\)?

  • For \(p=\infty\) Crasta and Fragalà [10] recovered this result.

  • For \(p\in (2,\infty )\) I gave it to Kühn, a Ph.D.-student, who proved it 2017.

  • It is in fact true for any \(p\in (1,\infty ]\), this can be derived from an earlier paper of Juutinen [21]. One has to note that u is a viscosity solution to \(-\varDelta _p^Nu=1\) iff \(v=-\sqrt{u}\) is a viscosity solution to

    $$\begin{aligned} \varDelta _p^Nv=-\frac{1}{v}\left( |\nabla v|^2+\frac{1}{2}\right) \quad \hbox { in }\varOmega . \end{aligned}$$
    (15)

    I find it remarkable that the right hand side is independent of p and that its harmonic concavity with respect to v proves to be helpful, just like in the classical proof in [25].

In the same spirit it was shown that a positive solution to \(-\varDelta u=\lambda u\) in \(\varOmega\), \(u=0\) on \(\partial \varOmega\) is log-concave, i.e. \(v=\ln {u}\) is concave. What can be said in this respect about solutions to \(-\varDelta _p^N u=\lambda u\) for \(p\not =2\)?

in 2010 Juutinen tried to prove the log-concavity of u for \(p=\infty\) and he almost succeeded. In fact he did so three years later in [22] for any \(p\in (1,\infty ]\) without noticing it. To realize this one has to notice that \(v=-\ln (u)\) satisfies the equation

$$\begin{aligned} -\varDelta _p^N v+ |\nabla v|^2 +\lambda =0 \hbox { in }\varOmega , \end{aligned}$$
(16)

and \(v(x)\rightarrow \infty\) as \(x\rightarrow \partial \varOmega\). This equation satisfies all the structural assumptions that Juutinen uses in [22]. For \(p\in (1,\infty )\) also Kühn published a proof [36], but unfortunately it contains a gap. That the result is true for any \(p\in (1,\infty )\) was finally shown by Crasta and Fragalà [11]. So all is well that ends well. Or: In the end everything will be fine. Otherwise it is not the end yet.

For both equations \(-\varDelta _N^pu=1\) and \(-\varDelta _p^Nu=\lambda u\) the case \(p=1\) seems totally open and needs further investigation.