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The inhomogeneous pLaplacian equation with Neumann boundary conditions in the limit \(p\to \infty \)
Advances in Continuous and Discrete Models volume 2023, Article number: 8 (2023)
Abstract
We investigate the limiting behavior of solutions to the inhomogeneous pLaplacian equation \(\Delta _{p} u = \mu _{p}\) subject to Neumann boundary conditions. For righthand sides, which are arbitrary signed measures, we show that solutions converge to a Kantorovich potential associated with the geodesic Wasserstein1 distance. In the regular case with continuous righthand sides, we characterize the limit as viscosity solution to an infinity Laplacian / eikonal type equation.
1 Introduction
The purpose of this paper is to study the behavior of solutions of the inhomogeneous pLaplacian equation with Neumann boundary conditions as \(p\to \infty \). The precise equation we consider is
where \(\Omega \subset \mathbb{R}^{d}\) is a Lipschitz domain, and the righthand side \(\mu _{p}\in \mathcal {M}(\overline{\Omega })\) is a signed Radon measure which satisfies the compatibility condition \(\mu _{p}(\overline{\Omega })=0\). We index the righthand side by p to include the case that it varies with p. In the rest of the paper we will refer to (1.1) as the pPoisson equation since for \(p=2\) it obviously coincides with the standard Poisson equation.
We prove two convergence results stated in Sect. 2.4 below. The first one is purely variational and states that, if the righthand sides \(\mu _{p}\) converge weakstar to a measure \(\mu \in \mathcal {M}(\overline{\Omega })\) as \(p\to \infty \), then weak solutions \(u_{p}\) of (1.1) converge (up to a subsequence) to a Kantorovich potential \(u_{\infty}\), which realizes the maximum in the following version of the Wasserstein1 distance between the positive part \(\mu ^{+}\) and the negative part \(\mu ^{}\) of μ:
The second result uses techniques from viscosity solutions to prove that for continuous data \(\mu _{p}\in \mathrm{C}(\overline{\Omega })\), converging uniformly to \(\mu \in \mathrm{C}(\overline{\Omega })\), solutions \(u_{p}\) converge to a viscosity solution of the following infinity Laplacian / eikonal type partial differential equation (PDE):
Consequently, the only information on μ, which “survives” the limit \(p\to \infty \) in the pPoisson problem (1.1), is the support of its positive and negative part.
Similar results have already been established for several related problems associated with the pLaplace operator. In [1], the limit of pPoisson equations with nonnegative righthand side and Dirichlet boundary conditions was related to a PDE similar to (1.3). In [2] the asymptotics of the homogeneous pLaplacian equation with nonhomogeneous Neumann boundary conditions was investigated and related to an optimal transport problem and a viscosity PDE of infinity Laplacian type. Furthermore, in [3] a vectorvalued version of (1.1) with righthand side independent of p was studied. Solutions were shown to converge to a Kantorovich potential and to solve a PDE in divergence form with measure coefficients. Similar results were established in [4, 5], however, imposing stricter regularity conditions on the righthand side in (1.1). Furthermore, in [6] the case of mixed boundary conditions and regular fixed righthand sides was related to optimal transport through a window on the boundary. Infinity Laplacian eigenvalue problems, their approximation with pLaplacian problems, and their relation to optimal transport were investigated in [7–11].
Apart from the theoretical interest in understanding the limiting behavior of solutions to (1.1), our investigations are also driven by recent developments in data science. In [12] it was proposed to utilize the pPoisson equation to solve semisupervised learning tasks. To this end, one assumes to have access to labels \(g:\mathcal{O}\to \mathbb{R}\) of a closed subset \(\mathcal{O}\subset \overline{\Omega }\) of the domain, in particular, \(\mathcal{O}\) could be a finite collection of points. For extending these labels from a discrete set \(\mathcal{O}=\{x_{i}: i=1,\dots,m\}\) with \(m\in \mathbb{N}\) to the whole domain Ω̅, it was suggested in [12] to solve (1.1) with the righthand side given by
where \(\delta _{x}\in \mathcal {M}(\overline{\Omega })\) denotes the Dirac measure located at \(x\in \overline{\Omega }\). While this method, termed “Poisson learning”, performs very well in practice, a full analysis is still pending. In particular, a rigorous convergence proof of the finitedimensional approximation of Poisson learning on weighted graphs—which is used in applications—would be desirable.
The results of the present article apply to the continuum description of Poisson learning and, in particular, address the asymptotics as \(p\to \infty \). For the balanced case of two labelled classes with equal size, i.e., \(g:\mathcal{O}\to \{\pm 1\}\) and \(\overline{g}=0\), our main results can be interpreted as follows: The labelling function u arising as limit of solutions to Poisson learning as \(p\to \infty \) is directly connected to the solution of the optimal transport problem, which transports the empirical measure \(\sum_{i: g(x_{i})=+1}\delta _{x_{i}}\) of the points with label +1 to the empirical measure \(\sum_{i: g(x_{i})=1}\delta _{x_{i}}\) of the points with label −1.
The plan of this paper is the following: Sect. 2 reviews some important mathematical background and states our main results which are proved in Sect. 3. In more detail, Sect. 3.1 proves compactness of solutions of (1.1) as \(p\to \infty \), Sect. 3.2 is devoted to the optimal transport characterization of cluster points, and Sect. 3.3 relates them to the limiting PDE (1.3).
2 Mathematical preliminaries and main results
2.1 Weak solution to the pLaplacian equation
The pLaplacian for \(p\in [1,\infty )\) is defined as
For \(\mathrm{C}^{2}\)functions u, it admits the decomposition formula
where \(\Delta u = \operatorname{div}(\nabla u)\) denotes the Laplacian and \(\Delta _{\infty }u:=\langle \nabla u,D^{2}u\nabla u\rangle \) is called the infinity Laplacian.
Since we are interested in the case \(p\to \infty \) anyway, we assume in the whole article that \(p>d\), in which case the Sobolev embedding \(\mathrm{W}^{1,p}(\Omega )\hookrightarrow \mathrm{C}^{0,1\frac{d}{p}}( \overline{\Omega })\) makes sure that the following concept of weak solutions to (1.1) is well defined.
Definition 2.1
Let \(p>d\). A function \(u\in \mathrm{W}^{1,p}(\Omega )\) is called a weak solution to (1.1) if it satisfies \(\int _{\Omega} \vert u \vert ^{p2}u\,\mathrm{d}x= 0\) and
It is obvious that weak solutions in the sense of Definition 2.1 coincide with solutions of the variational problem
since the Euler–Lagrange equations of this problem precisely coincide with (2.3), cf. [13]. Using standard arguments from calculus of variations, it can be shown that this problem admits a unique solution for every \(p>1\). Apart from guaranteeing existence and uniqueness, this variational characterization will be essential for deriving the optimal transport characterization of the limit \(\lim_{p\to \infty}u_{p}\) of weak solutions \(u_{p}\in \mathrm{W}^{1,p}(\Omega )\). For higher regularity statements for solutions of the pPoisson equation, we refer the interested reader to [14].
2.2 Geodesic geometry
As it turns out, the correct metric on Ω when working with (1.1) (or (2.4)) and its limit as \(p\to \infty \) is not the Euclidean one but the geodesic distance. It is defined as
and turns \((\Omega,d_{\Omega})\) into a length space. The geodesic distance measures the length of the shortest curve in Ω connecting two points. If Ω is convex, then the curve \(\gamma (t)=(1t)x+ty\) shows \(d_{\Omega}(x,y)= \vert xy \vert \), but in general it holds \(d_{\Omega}(x,y)\geq \vert xy \vert \). A derived quantity, which appears naturally in the context of the Neumann problem (1.1), is the geodesic diameter of Ω, defined as
The geodesic diameter appears in the optimal constant in the inequality
and in the first nontrivial Neumann eigenvalue of the infinity Laplacian [8, 15], given by
One can use the geodesic distance to define the geodesic Lipschitz constant of \(u\in \mathrm{C}(\overline{\Omega })\) as
With this at hand, one can introduce a geodesic version of the Wasserstein1 distance:
Note that, as stated in [16, page 269], any function \(u\in \mathrm{W}^{1,\infty}(\Omega )\) has a continuous representative (denoted by the same symbol), and it holds
This shows that \(\operatorname{Lip}_{\Omega}(u)\leq \operatorname{ess\,sup}_{\Omega} \vert \nabla u \vert \). Furthermore, since for points \(x,y\) that lie in a ball that is fully contained in Ω it holds \(d_{\Omega}(x,y)= \vert xy \vert \), it is easily seen (see [1, page 23]) that in fact
2.3 Weakstar convergence of measures
For measuring the convergence of the righthand side measures \(\mu _{p}\) in (1.1) as \(p\to \infty \), we utilize weakstar convergence of measures.
Definition 2.2
(Weakstar convergence of measures)
As sequence of Radon measures \((\mu _{n})_{n\in \mathbb{N}}\subset \mathcal {M}(\overline{\Omega })\) is said to converge weakstar to \(\mu \in \mathcal {M}(\overline{\Omega })\) (written \(\mu _{n}\overset{\ast }{\rightharpoonup }\mu \)) if
Remark 2.1
(Smooth approximation)
It is easy to see that any Radon measure can be approximated in the weakstar topology by convolving it with a mollifier.
2.4 Main results
The following are our main results. The proof of Theorem 1 can be found in Sect. 3.2 and the one of Theorem 2, along with precise definitions of the notion of viscosity solutions and some corollaries, in Sect. 3.3.
Theorem 1
Assume that \(\mu _{p}\overset{\ast }{\rightharpoonup }\mu \) in \(\mathcal {M}(\overline{\Omega })\) as \(p\to \infty \) and let \(\mu =\mu ^{+}  \mu ^{}\), with nonnegative measures \(\mu ^{\pm}\in \mathcal {M}(\overline{\Omega })\), be the Jordan decomposition of μ. Then (up to a subsequence) weak solutions \(u_{p}\in \mathrm{W}^{1,p}(\Omega )\) of (1.1) uniformly converge to a function \(u_{\infty}\in \mathrm{W}^{1,\infty}(\Omega )\), which satisfies
Theorem 2
If \(\mu _{p}\in \mathrm{C}(\overline{\Omega })\) converges uniformly to \(\mu \in \mathrm{C}(\overline{\Omega })\) as \(p\to \infty \), then (up to a subsequence) weak solutions \(u_{p}\in \mathrm{W}^{1,p}(\Omega )\) of (1.1) converge uniformly to a function \(u_{\infty}\in \mathrm{W}^{1,\infty}(\Omega )\), which is a viscosity solution of
3 Limit behavior as \(p\to \infty \)
3.1 Convergence of solutions
In this section we show that if the sequence of righthand sides \(\mu _{p}\) in (1.1) has uniformly bounded mass, then the sequence of solutions \((u_{p})_{p>0}\) admits a convergent subsequence.
To this end, we first derive an upper bound for the pDirichlet energy \(\int _{\Omega} \vert \nabla u \vert ^{p}\,\mathrm{d}x\) in terms of the data, which will then allow us to deduce convergence.
Proposition 3.1
Let \(u_{p}\in \mathrm{W}^{1,p}(\Omega )\) be a weak solution of (1.1) with data \(\mu _{p}\in \mathcal {M}(\overline{\Omega })\). Then it holds
Proof
Choosing \(\phi =u_{p}\) in (2.3) and using Hölder’s and Morrey’s inequalities yields
where the optimal constant for the Morrey inequality is defined as
Using that \(p\mapsto \sqrt[p]{\sigma _{p}}\) converges to the value \(\frac{2}{\operatorname{diam}(\Omega )}\in (0,\infty )\), which is the first nontrivial Neumann eigenvalue of the infinity Laplacian [8, 15], concludes the proof. □
Before proving the convergence theorem, we need the following important lemma.
Lemma 3.1
Let \(u_{p}\subset \mathrm{L}^{p}(\Omega )\) converge to \(u_{\infty}\in L^{\infty}(\Omega )\) uniformly on Ω. Then, for every \(0\leq k \leq p1\), it holds
Proof
Let \(\varepsilon >0\) be given. Then, for p sufficiently large, it holds \(\operatorname{ess\,sup}\vert u_{p}u_{\infty } \vert \leq \varepsilon \). Consequently, using Minkowski’s inequality,
and hence
Using the reverse triangle inequality, one analogously obtains
Combining these two inequalities and using that \(\varepsilon >0\) was arbitrary concludes the proof. □
Now we can prove that the sequence of solutions of (1.1) has a convergent subsequence.
Proposition 3.2
Let \(u_{p}\in \mathrm{W}^{1,p}(\Omega )\) be a weak solution of (1.1) with data \(\mu _{p}\in \mathcal {M}(\overline{\Omega })\) and assume that the data satisfy
Then there exists a function \(u_{\infty}\in \mathrm{W}^{1,\infty}(\Omega )\) such that as \(p\to \infty \) (up to a subsequence) the functions \(u_{p}\) converge to \(u_{\infty}\) uniformly and weakly in \(\mathrm{W}^{1,m}(\Omega )\) for any \(m>1\). Furthermore, it holds
Proof
We follow the strategy from [1]. For \(p>m\), Hölder’s inequality yields
Consequently, using Proposition 3.1 it follows
Introducing the first nonzero eigenvalue of the pLaplacian operator [8, 15]
it holds
and therefore we can estimate
Using Proposition 3.1 together with the fact that according to [8] it holds \(\sqrt[p]{\lambda _{p}}\to \lambda _{\infty}:= \frac{2}{\operatorname{diam}(\Omega )}\in (0,\infty )\) as \(p\to \infty \), we obtain
Thanks to (3.3) and (3.6), the sequence \(u_{p}\) has uniformly bounded \(\mathrm{W}^{1,m}\)norms, and hence (up to a subsequence) converges weakly to a function \(u_{\infty}\) in \(\mathrm{W}^{1,m}(\Omega )\). Furthermore, for \(m>d\), one has the compact embedding [17] of \(\mathrm{W}^{1,m}(\Omega )\) into \(\mathrm{C}^{0,1\frac{d}{m}}(\overline{\Omega })\), which (after another round of subsequence refinement) proves the uniform convergence.
It remains to argue that \(u_{\infty}\in \mathrm{W}^{1,\infty}(\Omega )\) and to prove (3.2). Using the weak lower semicontinuity of the \(\mathrm{L}^{m}\)norm, we obtain from (3.3) that
Taking the mth root and sending \(m\to \infty \) an application of Lemma 3.1 shows
Using again lower semicontinuity and (3.6) yields
Taking the mth root and applying Lemma 3.1 with \(p=m\) and \(k=0\) yields
Hence, we have established all inequalities in (3.2). □
Example 3.1
If \(\mathrm{d}\mu _{p}(x)=\operatorname{sign}(x)\,\mathrm{d}x\) for all \(p>1\), then the explicit solution of the pLaplacian equation (1.1) is given by \(u_{p}(x)=\frac{p1}{p}\operatorname{sign}(x) [1(1 \vert x \vert )^{\frac{p}{p1}} ]\). As \(p\to \infty \), the functions \(u_{p}\) converge uniformly to \(u_{\infty}(x)=x\), see Fig. 1. Note that the Neumann boundary conditions get lost in the limit \(p\to \infty \), see also Sect. 3.3.
3.2 Optimal transport characterization
The main theorem in this section characterizes the limit \(u_{\infty}\) as optimal transport potential. We assume that the data measures \(\mu _{p}\) converge in the weakstar sense of measures. This makes sure that one can pass to the limit in duality products where both factors converge, as the following lemma shows.
Lemma 3.2
If \(\mu _{n}\overset{\ast }{\rightharpoonup }\mu \) in \(\mathcal {M}(\overline{\Omega })\) and \(u_{n}\to u\) uniformly in \(\mathrm{C}(\overline{\Omega })\), then it holds
Proof
With the abbreviation \(\langle \mu,u\rangle:=\int _{\Omega }u\,\mathrm{d}\mu \), we can compute
The Banach–Steinhaus theorem (or the uniform boundedness principle) [16, Sect. 2.2] makes sure that \(\sup_{n} \vert \mu _{n} \vert (\overline{\Omega })< \infty \). Together with the uniform convergence of \(u_{n}\) and the weakstar convergence of \(\mu _{n}\), this implies that the righthand side goes to zero when taking the lim sup as \(n\to \infty \). □
To set the scene for the optimal transport characterization, we remind the reader of the usual Wasserstein1 distance \(W_{1}(\mu ^{+},\mu ^{})\) of the two measures \(\mu ^{\pm}\), defined as
where the Lipschitz constant \(\operatorname{Lip}( u)\) in (3.8) is
Functions \(u\in \mathrm{C}(\overline{\Omega })\), which attain the supremum in (3.8), are typically referred to as Kantorovich potentials.
The Lipschitz constant, and hence also the Wasserstein1 distance, is defined with respect to the Euclidean metric on \(\mathbb{R}^{d}\). This is, however, not the most natural metric to consider on the (possibly nonconvex) domain Ω. Indeed it can happen that two points in Ω have a small Euclidean distance although transporting two measures concentrated on these points onto each other within Ω requires a long transportation path. To overcome this, one can use the geodesic distance on Ω, defined in (2.5). Correspondingly, one can also introduce the geodesic Lipschitz constant (2.9) and geodesic Wasserstein1 distance (2.10).
As Theorem 1 states, this geodesic transport distance (2.10) arises naturally in the limiting problem of the pPoisson equation (1.1). We now give the proof of this statement.
Proof of Theorem 1
First we note that the weakstar convergence of \(\mu _{p}\) together with the Banach–Steinhaus theorem in particular implies that
such that Proposition 3.2 assures the existence of a subsequential uniform limit \(u_{\infty}\).
Let \(u\in \mathrm{W}^{1,\infty}(\Omega )\) with \(\operatorname{ess\,sup}_{\Omega} \vert \nabla u \vert \leq 1\) be arbitrary. Without loss of generality we can assume that \(\int _{\Omega} \vert u \vert ^{p2} u\,\mathrm{d}x=0\). Since \(u_{p}\) in particular solves (2.4), it holds
We can rearrange this inequality to
where \(\mu _{p}=\mu _{p}^{+}  \mu _{p}^{}\), with nonnegative measures \(\mu _{p}^{\pm}\in \mathcal {M}(\overline{\Omega })\), is the Jordan decomposition of \(\mu _{p}\). Obviously, it holds \(\mu _{p}^{\pm}\overset{\ast }{\rightharpoonup }\mu ^{\pm}\) as \(p\to \infty \) since the measures \(\mu _{p}^{\pm}\) are mutually singular. Now we use Lemma 3.2 together with the fact that the first term is nonnegative and \(\vert \nabla u \vert \leq 1\) a.e. in Ω to obtain
Since by (2.12) and (3.2) the function \(u_{\infty}\) is feasible for the optimization problem in (2.10), taking the supremum over u shows the assertion. □
Since according to Proposition 3.2 the limit \(u_{\infty}\) also satisfies \(\operatorname{ess\,sup}_{\Omega} \vert u_{\infty } \vert \leq \frac{\operatorname{diam}(\Omega )}{2}\), one could also have the idea to include a boundedness condition in the optimization problem in (2.10). This is motivated by the socalled Kantorovich–Rubinstein (KR) norm of the measure \(\mu =\mu ^{+}\mu ^{}\) on the length space \((\Omega,d_{\Omega})\), which is defined as
The reason why the KR norm does not appear naturally in our context is that for measures with zero mass it is equivalent (and for suitably scaled domains even equal) to the socalled dual Lipschitz norm. This norm coincides with the geodesic Wasserstein distance of the positive and negative part of the measure and is defined as
For completeness, the equivalence is stated in the following proposition.
Proposition 3.3
Let \(\mu \in \mathcal {M}(\overline{\Omega })\) satisfy \(\mu (\overline{\Omega })=0\). Then it holds
Proof
The proof works just as in [9, Proposition 4.3], see also [18, Lemma 2.1]. By omitting the constraint \(\operatorname{ess\,sup}_{\Omega} \vert u \vert \leq 1\), we obtain the first inequality \(\Vert \mu \Vert _{\mathrm{KR}(\overline{\Omega })}\leq \Vert \mu \Vert _{\operatorname{Lip}_{\Omega}^{\ast}( \overline{\Omega })}\).
For the other inequality, we argue as follows: Since μ has zero mass, we can without loss of generality assume that the supremum in (3.11) is taken over functions that satisfy \(\operatorname{ess\,sup}_{\Omega }u +\operatorname{ess\,inf}_{\Omega }u = 0\) by replacing u with \(uc\) for a suitable constant. Then, using (2.7), we get for all \(u\in \mathrm{C}(\overline{\Omega })\) with \(\operatorname{Lip}_{\Omega}(u)\leq 1\) that
Letting \(t:=1\vee \frac{\operatorname{diam}(\Omega )}{2}\geq 1\), it holds
□
3.3 PDE characterization
Now we also give a PDE characterization of the limit \(u_{\infty}\), which we have shown to be a Kantorovich potential in the previous section. Note that Kantorovich potentials are typically not unique, which is why it is interesting to verify that the limiting procedure \(p\to \infty \) selects a more regular potential. Since \(u_{\infty}\) turns out to solve an infinity Laplacian type PDE in the viscosity sense, we also have to work with viscosity solutions for finite p. However, for that we have to assume that the data \(\mu _{p}\) are continuous and converge uniformly.
Let us first define what it means to be a viscosity solution to the pPoisson equation (1.1). In particular, one has to interpret the Neumann boundary conditions in the viscosity sense, see also [8, 19]. As explained in [20], the proper way to understand boundary conditions for boundary value problems of the form
is to demand that subsolutions satisfy
and supersolutions satisfy the converse inequality with a max in place of the min.
Definition 3.1
(Viscosity solutions of the pPoisson equation)
Let \(\mu _{p}\in \mathrm{C}(\overline{\Omega })\). An upper semicontinuous function \(u:\overline{\Omega }\to \mathbb{R}\) is called viscosity subsolution of (1.1) if

for all \(x_{0}\in \Omega \) and \(\phi \in C^{2}(\Omega )\) such that \(u\phi \) has a local maximum at \(x_{0}\), it holds
$$\begin{aligned} \Delta _{p} \phi (x_{0})\mu _{p}(x_{0}) \leq 0; \end{aligned}$$ 
for all \(x_{0}\in \partial \Omega \) and \(\phi \in C^{2}(\overline{\Omega })\) such that \(u\phi \) has a local maximum at \(x_{0}\), it holds
$$\begin{aligned} \min \biggl\lbrace \bigl\vert \nabla \phi (x_{0}) \bigr\vert ^{p2} \frac{\partial \phi}{\partial \nu}(x_{0}), \Delta _{p}\phi (x_{0})  \mu _{p}(x_{0}) \biggr\rbrace \leq 0; \end{aligned}$$ 
it holds \(\int _{\Omega} \vert u \vert ^{p2}u\,\mathrm{d}x\leq 0\).
A lower semicontinuous function \(u:\overline{\Omega }\to \mathbb{R}\) is called viscosity supersolution of (1.1) if

for all \(x_{0}\in \Omega \) and \(\phi \in C^{2}(\Omega )\) such that \(u\phi \) has a local minimum at \(x_{0}\), it holds
$$\begin{aligned} \Delta _{p} \phi (x_{0})\mu _{p}(x_{0}) \geq 0; \end{aligned}$$ 
for all \(x_{0}\in \partial \Omega \) and \(\phi \in C^{2}(\overline{\Omega })\) such that \(u\phi \) has a local maximum at \(x_{0}\), it holds
$$\begin{aligned} \max \biggl\lbrace \bigl\vert \nabla \phi (x_{0}) \bigr\vert ^{p2} \frac{\partial \phi}{\partial \nu}(x_{0}), \Delta _{p}\phi (x_{0})  \mu _{p}(x_{0}) \biggr\rbrace \geq 0; \end{aligned}$$ 
it holds \(\int _{\Omega} \vert u \vert ^{p2}u\,\mathrm{d}x \geq 0\).
A function \(u\in \mathrm{C}(\overline{\Omega })\) is called a viscosity solution of (1.1) if it is both a sub and supersolution.
We need the following wellknown statement, which asserts that weak solutions to the pPoisson equation are also viscosity solutions.
Lemma 3.3
Let \(\mu _{p}\in \mathrm{C}(\overline{\Omega })\). A continuous weak solution to (1.1) in the sense of Definition 2.1is also a viscosity solution in the sense of Definition 3.1.
Proof
This statement can be found in [19, Proposition 4.8]. We remark that the full proof for the PDE on Ω can be found in [21, Theorem 1.8]. The boundary conditions are treated in precisely the same way as for the pLaplacian eigenvalue problem (which can be regarded as a pPoisson equation), see [8, Lemma 2] and [19, Proposition 3.2]. □
Before we turn to the limiting PDE, we recall that the statement of Proposition 3.2, which states that \(\vert \nabla u_{\infty } \vert \leq 1\) almost everywhere in Ω, can be converted into the viscosity framework.
Proposition 3.4
If \(\mu _{p}\in \mathrm{C}(\overline{\Omega })\) converges uniformly to \(\mu \in \mathrm{C}(\overline{\Omega })\) as \(p\to \infty \), then (up to a subsequence) viscosity solutions \(u_{p}\in \mathrm{W}^{1,p}(\Omega )\) of (1.1) converge uniformly to a function \(u_{\infty}\in \mathrm{W}^{1,\infty}(\Omega )\), which is a viscosity solution of \(\vert \nabla u \vert 1 \leq 0\) and \(1 \vert \nabla u \vert \geq 0\).
Proof
As in [19, Proposition 4.7], which solely relies on [1, Part III, Lemma 1.1]. □
It is important to remark that in the viscosity sense the inequality \(\vert \nabla u \vert 1\leq 0\) is not equivalent to \(1 \vert \nabla u \vert \geq 0\), which is why we make the distinction explicit.
Let us now turn to the limiting PDE (1.3) satisfied by \(u_{\infty}\) for which we assume that the limiting data \(\mu \in \mathrm{C}(\overline{\Omega })\) are continuous. We prove that every limit \(u_{\infty}\) of solutions to the pPoisson equation (1.1) as \(p\to \infty \) is a viscosity solution of (1.3), which we restate here for convenience:
Note that this PDE does not contain any boundary conditions and it also does not specify the behavior on the closed set \(\overline{\Omega }\setminus ({\{\mu >0\}}\cup{\{\mu <0\}}\cup{ \overline{\{\mu \neq 0\}}^{c}} )\). Note that even the weak boundary conditions in the viscosity sense, introduced before Definition 3.1, do not carry over to the limiting problem, which is consistent with the findings in [1, 19]. Regarding the behavior outside the three sets that occur in (3.13), one should remark that the PDE is discontinuous there. Using lower and upper semicontinuous envelopes of this discontinuous function, one can make sense of a weaker form of the PDE on the whole of Ω, see [22, Remark 4.3] for a similar problem and [20, Remark 6.3] for a general statement.
In contrast, for Neumann eigenvalue problems of the pLaplacian, it is possible to formulate boundary conditions and obtain a limiting PDE on the whole of Ω̅, see [8].
Let us now define what precisely we mean by viscosity solutions to equation (3.13).
Definition 3.2
(Viscosity solutions of the limiting equation)
Let \(\mu \in \mathrm{C}(\overline{\Omega })\). An upper semicontinuous function \(u:\overline{\Omega }\to \mathbb{R}\) is called a viscosity subsolution of (3.13) if

for all \(x_{0}\in \Omega \) and \(\phi \in C^{2}(\Omega )\) such that \(u\phi \) has a local maximum at \(x_{0}\), it holds
$$\begin{aligned} \textstyle\begin{cases} \min \lbrace \vert \nabla \phi (x_{0}) \vert 1, \Delta _{\infty }\phi (x_{0}) \rbrace \leq 0& \text{if }x_{0} \in {\{\mu >0\}}, \\ \Delta _{\infty }\phi (x_{0}) \leq 0& \text{if }x_{0}\in { \overline{\{\mu \neq 0\}}^{c}}, \\ \max \lbrace 1 \vert \nabla \phi (x_{0}) \vert , \Delta _{\infty }\phi (x_{0}) \rbrace \leq 0& \text{if }x_{0} \in {\{\mu < 0\}}, \end{cases}\displaystyle \end{aligned}$$ 
it holds \(\max_{\overline{\Omega }} u + \operatorname{ess\,inf}_{\Omega }u \leq 0\).
A lower semicontinuous function \(u:\overline{\Omega }\to \mathbb{R}\) is called a viscosity supersolution if

for all \(x_{0}\in \Omega \) and \(\phi \in C^{2}(\Omega )\) such that \(u\phi \) has a local minimum at \(x_{0}\), it holds
$$\begin{aligned} \textstyle\begin{cases} \min \lbrace \vert \nabla \phi (x_{0}) \vert 1, \Delta _{\infty }\phi (x_{0}) \rbrace \geq 0& \text{if }x_{0} \in {\{\mu >0\}}, \\ \Delta _{\infty }\phi (x_{0}) \geq 0& \text{if }x_{0}\in { \overline{\{\mu \neq 0\}}^{c}}, \\ \max \lbrace 1 \vert \nabla \phi (x_{0}) \vert , \Delta _{\infty }\phi (x_{0}) \rbrace \geq 0& \text{if }x_{0} \in {\{\mu < 0\}}, \end{cases}\displaystyle \end{aligned}$$ 
it holds \(\operatorname{ess\,sup}_{\Omega }u + \min_{\overline{\Omega }}u \geq 0\).
A function \(u\in \mathrm{C}(\overline{\Omega })\) is called viscosity solution it is both a sub and supersolution.
Now we can prove the main theorem of this section.
Proof of Theorem 2
The conditions of Proposition 3.2 are trivially fulfilled, which guarantees the existence of a (subsequential) uniform limit \(u_{\infty}\in \mathrm{C}(\overline{\Omega })\). We only show the subsolution property, the supersolution property can be shown analogously.
Let \(x_{0}\in \Omega \) and \(\phi \in \mathrm{C}^{2}(\Omega )\) such that \(u_{\infty}\phi \) has a local maximum at \(x_{0}\). Choose a sequence \((p_{i})_{i\in \mathbb{N}}\subset (d,\infty )\) converging to ∞ such that \(u_{p_{i}}\to u_{\infty}\) uniformly. Then there exists a sequence of points \((x_{i})_{i\in \mathbb{N}}\subset \Omega \) converging to \(x_{0}\in \Omega \) such that \(u_{p_{i}}\phi \) has a local maximum in \(x_{i}\) for all \(i\in \mathbb{N}\). Since \(u_{p_{i}}\) is a viscosity solution of (1.1), by (2.2) it holds
Case 1, \(x_{0}\in {\{\mu >0\}}\): We have to show that
In fact, for showing this, we will not even have to use that \(\mu (x_{0})>0\), but (3.15) is true for all \(x\in \Omega \). The condition \(\mu (x_{0})>0\) will only be relevant for showing the converse inequality for supersolutions.
If \(\vert \nabla \phi (x_{0}) \vert =0\), then we have by definition that \(\Delta _{\infty}\phi (x_{0})=0\). In the case that \(\vert \nabla \phi (x_{0}) \vert >0\), we get for \(i\in \mathbb{N}\) sufficiently large that \(\vert \nabla \phi (x_{i}) \vert >0\) and can divide by this quantity to get
Then either \(\vert \nabla \phi (x_{0}) \vert 1\leq 0\) or \(\vert \nabla \phi (x_{0}) \vert 1 > 0\), and in the latter case we obtain \(\Delta _{\infty}\phi (x_{0})\leq 0\) by taking the limit \(i\to \infty \) and using the uniform boundedness of \(\mu _{p_{i}}\). Combining all those cases yields (3.15).
Case 2, \(x_{0}\in {\overline{\{\mu \neq 0\}}^{c}}\): We have to show that
If \(\vert \nabla \phi (x_{0}) \vert =0\), then we have by definition that \(\Delta _{\infty}\phi (x_{0})=0\). If \(\vert \nabla \phi (x_{0}) \vert >0\), then (3.16), the openness of \({\overline{\{\mu \neq 0\}}^{c}}\), and the uniform convergence of \(\mu _{p_{i}}\) imply that \(\Delta _{\infty }\phi (x_{0})\leq 0\). In either case, we obtain (3.17).
Case 3, \(x_{0}\in {\{\mu <0\}}\): We have to show that
Since the functions \(\mu _{p_{i}}\) converge uniformly to μ, the set \({\{\mu <0\}}\) is open, and \(\mu (x_{0})\) is strictly negative, (3.14) can only be satisfied if \(\vert \nabla \phi (x_{i}) \vert >0\) for \(i\in \mathbb{N}\) sufficiently large. Dividing by \((p_{i}2) \vert \nabla \phi (x_{i}) \vert ^{p_{i}4}>0\) then again yields (3.16). If \(1 \vert \nabla \phi (x_{0}) \vert \leq 0\), then in the limit \(i\to \infty \) and using the uniform boundedness of \(\mu _{p_{i}}\) we get \(\Delta _{\infty }\phi (x_{0}) \leq 0\) If, however, \(1 \vert \nabla \phi (x_{0}) \vert >0\), one gets \(\Delta _{\infty}\phi (x_{0})\leq \infty \) as \(i\to \infty \), which is impossible since \(\phi \in \mathrm{C}^{2}(\Omega )\). Combining these two cases, we obtain (3.18).
Mean value: Finally, let us turn to the mean value condition. Letting \(u^{\pm }:= \max (\pm u,0)\) denote the positive/negative part of a function \(u:\Omega \to \mathbb{R}\) for all \(i\in \mathbb{N}\), it holds
which is equivalent to
Applying Lemma 3.1 with \(k=1\) then yields
which by the upper semicontinuity of \(u_{\infty}\) is equivalent to \(\max_{\overline{\Omega }} u_{\infty }+ \operatorname{ess\,inf}_{\Omega }u_{\infty } \leq 0\). □
Next we show that viscosity solutions of (3.13) have the intriguing property that they solve the eikonal equation \(\vert \nabla u \vert =1\) on the interior of the support of the data μ and are infinity harmonic elsewhere. But even more is true: namely that u is an infinity superharmonic solution of the eikonal equation on \({\{\mu >0\}}\) and analogously a subharmonic one on \({\{\mu <0\}}\). This is formalized in the following corollary.
Corollary 3.1
The function \(u_{\infty}\) is a viscosity solution of
Proof
Let \(x_{0}\in {\{\mu >0\}}\) and \(\phi \in \mathrm{C}^{2}(\Omega )\) such that \(u_{\infty}\phi \) has a local minimum at \(x_{0}\). Since \(u_{\infty}\) is in particular a supersolution of (3.13), it follows that
Furthermore, Proposition 3.4 shows that \(\vert \nabla u_{\infty } \vert 1\leq 0\) in the viscosity sense, which shows the claim. If \(x_{0}\in {\{\mu <0\}}\), then one analogously uses the subsolution property of \(u_{\infty}\) to infer that \(1 \vert \nabla u_{\infty } \vert \leq 0\) and \(\Delta _{\infty }u_{\infty}\leq 0\) in the viscosity sense and again utilizes Proposition 3.4 to conclude. □
It is important to remark that the limiting PDE (3.13) does not admit unique solutions. This is illustrated in the following example.
Example 3.2
(Nonuniqueness of the limiting PDE)
Let us consider the situation \(\Omega =(1,1)\subset \mathbb{R}\) and μ being an arbitrary continuous function with \(\{\mu <0\}=(1,0)\) and \(\{\mu >0\}=(0,1)\). We claim that the following family of functions (see Fig. 2)
is a viscosity solution of (3.13). Indeed, it is trivial to see that \(u_{t}\) is even a classical solution of (3.13) on \((2,2)\setminus \{\pm t\}\). So we just have to check the two corner points at ±t. For \(x_{0}=t\) and \(\phi \in \mathrm{C}^{2}(\Omega )\) touching u from above in \(x_{0}\), it is obvious that \(\vert \phi '(x_{0}) \vert \leq 1\), and hence \(\min \lbrace \vert \phi (x_{0}) \vert 1,\Delta _{ \infty}\phi (x_{0}) \rbrace \leq 0\). Furthermore, there is no \(\phi \in \mathrm{C}^{2}(\Omega )\) touching u from below in \(x_{0}\). Similarly, one can argue for \(x_{0}=t\) and obtain that \(u_{t}\) is a viscosity solution of (3.13). Note that none of the functions \(u_{t}\) has homogeneous Neumann boundary conditions.
Since the concept of viscosity solutions heavily relies on continuity and is not compatible with discontinuous or even measure data μ, we have to use approximation techniques if we want to make sense of (3.13) if μ is a measure. In particular, it seems natural to replace the open set \({\overline{\{\mu \neq 0\}}^{c}}\) with the open set \(\Omega \setminus \operatorname{supp}\mu \). However, one cannot just replace \(\{\mu \gtrless \}\) with \(\operatorname{supp}\mu ^{\pm}\) since the latter sets are not open and might even have empty interior. For an arbitrary measure data \(\mu \in \mathcal {M}(\overline{\Omega })\), which we extend to zero outside Ω̅, we consider the mollifications
where \(\phi \in \mathrm{C}^{\infty}_{c}(\mathbb{R}^{d})\) is a smooth kernel with \(\operatorname{supp}\phi \in B_{1}(0)\) and \(\phi _{\varepsilon }(x)=\varepsilon ^{d}\phi (x/\varepsilon )\). It is obvious from the definition of \(\mu _{\varepsilon }\) that if \(x\in \Omega \setminus \operatorname{supp}\mu \) then \(x\in \Omega \setminus \operatorname{supp}\mu ^{\varepsilon }\) for all \(\varepsilon >0\) small enough. Furthermore \(\mu ^{\varepsilon }\overset{\ast}{\rightharpoonup }\mu \) as \(\varepsilon \downarrow 0\). Using the techniques from the proof of Theorem 2, we immediately get the following result.
Corollary 3.2
Let \(\mu \in \mathcal {M}(\overline{\Omega })\) and , where \(\lim_{p\to \infty}\varepsilon _{p}=0\) and \(\mu ^{\varepsilon _{p}}\) is defined as in (3.20). Let, furthermore, \(u_{p}\in \mathrm{W}^{1,p}(\Omega )\) be viscosity solutions of (1.1) with data \(\mu _{p}\in \mathrm{C}(\overline{\Omega })\). Then the function \(u_{\infty}\in \mathrm{W}^{1,\infty}(\Omega )\) is a viscosity solution of
Proof
Let \(x_{0}\in \Omega \setminus \operatorname{supp}\mu \) and \(\phi \in \mathrm{C}^{2}(\Omega )\) such that \(u_{\infty}\phi \) has a local maximum at \(x_{0}\). Choose a sequence \((p_{i})_{i\in \mathbb{N}}\subset (d,\infty )\) converging to ∞ such that \(u_{p_{i}}\to u_{\infty}\) uniformly. As always, there exists a sequence of points \((x_{i})_{i\in \mathbb{N}}\subset \Omega \) converging to \(x_{0}\in \Omega \) such that \(u_{p_{i}}\phi \) has a local maximum in \(x_{i}\) for all \(i\in \mathbb{N}\). For all sufficiently large \(i\in \mathbb{N}\), it holds \(x_{0}\in \Omega \setminus \operatorname{supp}\mu _{p_{i}}\), and hence \(\mu _{p_{i}}(x_{0})=0\). As in Case 2 of the proof of Theorem 2, we can conclude that \(\Delta \phi (x_{0})\leq 0\). The supersolution property is shown analogously. □
4 Conclusion
In this article we have investigated limits of the pLaplace equation with measurevalued righthand side as \(p\to \infty \). We proved the existence of (subsequential) limits and characterized them as Kantorovich potentials for the optimal transport problem of transporting the positive part of the righthand side onto the negative one. For continuous data, we also proved that such limits are viscosity solutions of a degenerate PDE, involving the infinity Laplacian and the eikonal equation. It will be interesting to investigate in which sense the limiting PDE can be interpreted for measurevalued data, which have a support with empty interior. Here, lower / upper semicontinuous relaxations as in [20, 22] might be promising tools.
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Acknowledgements
The author would like to thank Jeff Calder and Simone Di Marino for enlightening discussions. This work was partially done while the author was visiting the Simons Institute for the Theory of Computing to participate in the program “Geometric Methods in Optimization and Sampling” during the fall of 2021, and the author is very grateful for the hospitality of the institute.
Funding
This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—GZ 2047/1, ProjektID 390685813. Parts of this work were also done while the author was in residence at Institut MittagLeffler in Djursholm, Sweden during the semester on Geometric Aspects of Nonlinear Partial Differential Equations in 2022, supported by the Swedish Research Council under grant no. 201606596. Open Access funding enabled and organized by Projekt DEAL.
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Bungert, L. The inhomogeneous pLaplacian equation with Neumann boundary conditions in the limit \(p\to \infty \). Adv Cont Discr Mod 2023, 8 (2023). https://doi.org/10.1186/s13662023037548
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DOI: https://doi.org/10.1186/s13662023037548
MSC
 35D30
 35D40
 49Q22
Keywords
 pLaplacian
 Infinity Laplacian
 Wasserstein distance
 Optimal transport
 Viscosity solution