The Perron solution for vector-valued equations

Abstract

Given a continuous function on the boundary of a bounded open set in $\mathbb {R}^d$ there exists a unique bounded harmonic function, called the Perron solution, taking the prescribed boundary values at least at all regular points (in the sense of Wiener) of the boundary. We extend this result to vector-valued functions and consider several methods of constructing the Perron solution which are classical in the real-valued case. Special emphasis is on the case where the codomain is a Banach lattice. In this case we investigate Perron’s classical construction via sub- and supersolutions. We also apply our results to solve elliptic and parabolic boundary value problems of vector-valued functions.

Appendix A: the vector lattice \(\mathcal {H}_b\)

An ordered vector space which has the interpolation property is automatically an order complete vector lattice, i.e. every set bounded from above has a supremum. To see this let A be a set bounded from above and B be the set of its upper bounds. Since $a\le b$ for all $a\in A, b\in B$ the interpolation property yields an element w satisfying $a\le w\le b$ for all $a\in A, b\in B$, i.e. w is the least upper bound of A. In particular in the setting of Lemma 2.16 this applies to $\mathcal {H}_b(\Omega ,X)$ with respect to the order on $C_b(\Omega ,X)$. Note however that $\mathcal {H}_b(\Omega ,X)$ is not a sublattice of $C(\Omega ,X)$ since the pointwise supremum of two harmonic functions is not harmonic in general. For two vectors $u,v\in \mathcal {H}_b(\Omega ,\mathbb {R})$ we denote their supremum in the vector lattice $\mathcal {H}_b(\Omega ,\mathbb {R})$ by $u\vee _\mathcal {H}v$. In the case $X=\mathbb {R}$, the fact that $\mathcal {H}_b(\Omega ,\mathbb {R})$ is an order complete vector lattice can also be found in a more general form in [12, Part IV, Theorem 11]. Moreover:

Theorem A.1

$\mathcal {H}_b(\Omega ,\mathbb {R})$ is an order complete Banach lattice.

For the proof we will need

Lemma A.2

Let $u,v\in \mathcal {H}_b(\Omega ,\mathbb {R})$. Choose $\omega _n\subset \subset \omega _{n+1}\subset \subset \Omega$ such that $\bigcup _n\omega _n=\Omega$ and such that $\omega _n$ is regular for each $n\in \mathbb {N}$. For all $\xi \in \partial \omega _n$ set

$$\begin{aligned} f_n(\xi ):=u(\xi )\vee _\mathbb {R}v(\xi ), \end{aligned}$$

where $u(\xi )\vee _\mathbb {R}v(\xi )$ denotes the supremum in $\mathbb {R}$. Let $h_n$ be the solution to the Dirichlet problem in $\omega _n$ with boundary data $f_n$. Then $h_n\rightarrow u\vee _\mathcal {H}v$ uniformly on compact sets.

Proof

First let $K\subset \Omega$ be compact. We may without loss of generality assume that $K\subset \omega _1$. For $n\in \mathbb {N}$ and $\xi \in \partial \omega _n$ we have

$$\begin{aligned} h_n(\xi )=u(\xi )\vee _\mathbb {R}v(\xi )\le \left( u\vee _\mathcal {H}v\right) (\xi ). \end{aligned}$$

Note that $h_n$ and $u\vee _\mathcal {H}v$ are harmonic functions while $u(\cdot )\vee _\mathbb {R}v(\cdot )$ is subharmonic. The maximum principle implies that

$$\begin{aligned} u(\xi )\vee _\mathbb {R}v(\xi )\le h_n(\xi )\le \left( u\vee _\mathcal {H}v\right) (\xi ) \end{aligned}$$

for all $\xi \in \omega _n$. In particular this holds for $\xi \in K$. The Poisson integral formula implies that $h_n$ is equicontinuous, hence the Arzela–Ascoli Theorem yields a subsequence of $h_n$ which converges uniformly on K to a function $h\in \mathcal {H}_b(K^\circ ,\mathbb {R})$.

Next take $K_n:=\overline{\omega _n}$ then using a diagonal argument we find a subsequence of $h_n$ which converges to $h\in \mathcal {H}_b(\Omega ,\mathbb {R})$ uniformly on every $K_n$. The above inequality shows that $h=u\vee _\mathcal {H}v$. A subsequence argument shows that $h_n\rightarrow u\vee _\mathcal {H}v$. $\square$

One can easily use Lemma A.2 to show that this construction works analogously for other lattice operations:

Corollary A.3

In the formulation of Lemma A.2 choose

$$\begin{aligned} f_n(\xi ):=|u(\xi )|_\mathbb {R}, \end{aligned}$$

where $|u(\xi )|_\mathbb {R}$ denotes the absolute value in $\mathbb {R}$. Then $h_n\rightarrow |u|_\mathcal {H}$ uniformly on compact sets, where $|u|_\mathcal {H}$ denotes the absolute value in the lattice $\mathcal {H}_b(\Omega ,\mathbb {R})$.

Proof of Theorem A.1

It remains to show that $\Vert \,|u|_\mathcal {H}\,\Vert _\infty =\Vert u\Vert _\infty$. The estimate ,,$\ge$“ is immediate. On the other hand if $h_n$ is chosen as in Corollary A.3, then the maximum principle implies that

$$\begin{aligned} 0\le h_n(\xi )\le \max _{\xi \in \partial \omega _n}|u(\xi )|_\mathbb {R}\le \Vert u\Vert _\infty . \end{aligned}$$

Since $h_n(\xi )$ converges to $|u(\xi )|_\mathcal {H}$ the estimate ,,$\le$“ follows. $\square$