On the existence of solutions of the Dirichlet problem for $p$-Laplacian on Riemannian manifolds

TL;DR

This paper establishes a criterion for the existence of solutions to the p-Laplacian Dirichlet problem on complete Riemannian manifolds with boundary, focusing on solutions with bounded Dirichlet integral.

Contribution

It provides a new criterion for the existence of solutions to the p-Laplacian Dirichlet problem on Riemannian manifolds with boundary, extending previous results.

Findings

Derived a criterion for solution existence

Applied to manifolds with boundary

Focused on solutions with bounded Dirichlet integral

Abstract

We obtain a criterion for the existence of solutions of the problem $$ \Delta_p u = 0 \quad \mbox{in } M \setminus \partial M, \quad \left. u \right|_{ \partial M } = h, $$ with the bounded Dirichlet integral, where $M$ is an oriented complete Riemannian manifold with boundary and $h \in W_{p, loc}^1 (M)$, $p > 1$.

Equations

$${\Delta }_{p}u=0\text{mbox}inM\setminus \partial M,{u|}_{\partial M}=h,$$

$${\Delta }_{p}u=0\text{mbox}inM\setminus \partial M,$$

$${u|}_{\partial M}=h,$$

$${\int }_M|\nabla u{|}^{p}dV<\infty .$$

$${\int }_M{g}^{ij}|\nabla u{|}^{p-2}{\nabla }_{j}u{\nabla }_i\phi dV=0$$

$${\phi |}_K=\psi ,$$

$${u|}_K=h$$

$${\psi |}_{\omega }=1.$$

$$\Vert u-h-{\phi }_i{\Vert }_{W_p^1(\omega )}\to 0\text{mbox}asi\to \infty$$

$$\Vert (1-\tau )\psi (u-h-{\phi }_i){\Vert }_{W_p^1(M)}=\Vert (1-\tau )\psi (u-h-{\phi }_i){\Vert }_{W_p^1(\omega )}\to 0\text{mbox}asi\to \infty ;$$

$${\mathrm{cap}}_{\psi }(K,\Omega )=\underset{\phi }{\inf }{\int }_{\Omega }|\nabla \phi {|}^{p}dV,$$

$${\mathrm{cap}}_{\psi }(E,\Omega )=\underset{K}{\sup }{\mathrm{cap}}_{\psi }(K,\Omega ),$$

$${\mathrm{cap}}_{\psi }(K_1,{\Omega }_1)\le {\mathrm{cap}}_{\psi }(K_2,{\Omega }_2).$$

$${\mathrm{cap}}_{\lambda \psi }(K,\Omega )=|\lambda {|}^p{\mathrm{cap}}_{\psi }(K,\Omega ).$$

$${\mathrm{cap}}_{{\psi }_1+{\psi }_2}^{1/p}(K,\Omega )\le {\mathrm{cap}}_{{\psi }_1}^{1/p}(K,\Omega )+{\mathrm{cap}}_{{\psi }_2}^{1/p}(K,\Omega ).$$

$${\frac{\partial u}{\partial \nu }|}_{\partial M}=0$$

$${\mathrm{cap}}_{h-w}(\partial M)<\infty$$

$${\int }_G|u{|}^{p}dV\le C\left({\int }_G|\nabla u{|}^{p}dV+{\left|{\int }_{\omega }udV\right|}^p\right)$$

$${\int }_{\omega }|\phi -\alpha {|}^{p}dV\le C{\int }_{\omega }|\nabla \phi {|}^{p}dV,$$

$$\alpha =\frac{1}{\mathrm{mes}\omega }{\int }_{\omega }\phi dV$$

$${\int }_M|\nabla (u-{\phi }_i){|}^{p}dV\to \underset{\phi \in C_0^{\infty }(M)}{\inf }{\int }_M|\nabla (u-\phi ){|}^{p}dV\text{mbox}asi\to \infty .$$

$$\Vert \nabla r_m-\mathbf{r}{\Vert }_{L_p(M)}\to 0\text{mbox}asm\to \infty .$$

$$\phi \mapsto {\int }_M|\nabla (u-\phi ){|}^{p}dV,\phi \in \stackrel{\mathrm{o}}{W}{}_p^1(M),$$

$${\int }_M|\nabla (u-r_m){|}^{p}dV\le \underset{j\ge m}{\sup }{\int }_M|\nabla (u-{\phi }_{i_j}){|}^{p}dV;$$

$${\int }_M|\nabla (u-r_m){|}^{p}dV\to \underset{\phi \in C_0^{\infty }(M)}{\inf }{\int }_M|\nabla (u-\phi ){|}^{p}dV\text{mbox}asm\to \infty .$$

$${\alpha }_m=\frac{1}{\mathrm{mes}\omega }{\int }_{\omega }{r}_{m}dV,$$

$$\Vert r_{m_j}-v{\Vert }_{W_p^1(G)}\to 0\text{mbox}asj\to \infty$$

$${\int }_M|\nabla (u-v){|}^{p}dV=\underset{\phi \in C_0^{\infty }(M)}{\inf }{\int }_M|\nabla (u-\phi ){|}^{p}dV.$$

$${v|}_K=h-w.$$

$${{\psi }_j|}_K=h-w,j=1,2,\dots .$$

Full text

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with the bounded Dirichlet integral, where $M$ is an oriented complete Riemannian manifold with boundary and $h\in W_{p,loc}^{1}(M)$, $p>1$.

1. Introduction

Let $M$ be an oriented complete Riemannian manifold with boundary. We consider solutions of the problem

[TABLE]

[TABLE]

where $\Delta_{p}u=\nabla_{i}(g^{ij}|\nabla u|^{p-2}\nabla_{j}u)$ is the $p$-Laplacian and $h\in W_{p,loc}^{1}(M)$, $p>1$.

As a condition at infinity, we assume that the Dirichlet integral is bounded, i.e.

[TABLE]

As is customary, by $g_{ij}$ we denote the metric tensor consistent with the Riemanian connection and by $g^{ij}$ we denote the tensor dual to the metric one. In so doing, $|\nabla u|=(g^{ij}\nabla_{i}u\nabla_{j}u)^{1/2}$. As in [10], by $W_{p,loc}^{1}(\omega)$, where $\omega\subset M$ is an open set, we mean the space of measurable functions belonging to $W_{p}^{1}(\omega^{\prime}\cap\omega)$ for any open set $\omega^{\prime}\subset M$ with compact closure. The space $L_{p,loc}(\omega)$ is defined analogously.

A function $u\in W_{p,loc}^{1}(M)$ is called a solution of (1.1) if

[TABLE]

for all $\varphi\in C_{0}^{\infty}(M\setminus\partial M)$, where $dV$ is the volume element of the manifold $M$. In its turn, condition (1.2) means that $(u-h)\psi\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(M\setminus\partial M)}$ for all $\psi\in C_{0}^{\infty}(M)$.

Boundary value problems for differential equations in unbounded domains and on smooth manifolds have been studied by a number of authors [1]–[8], [12]. In the case where $M$ is a domain in ${\mathbb{R}}^{n}$ bounded by a surface of revolution, a criterion for the existence of solutions of (1.1)–(1.3) was obtained in [12]. However, the method used in [12] cannot be generalized to the case of an arbitrary Riemannian manifold. Theorem 2.1 proved in our article does not have this shortcoming.

Let $K\subset M$ be a compact set. We denote by $C_{0}^{\infty}(M,K)$ the set of functions from $C^{\infty}(M)$ that are equal to zero in a neighborhood of $K$. In its turn, by ${\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(\omega,K)}$, where $\omega$ is an open subset of $M$, we denote the closure of $C_{0}^{\infty}(M,K)\cap W_{p}^{1}(\omega)$ in $W_{p}^{1}(\omega)$. By definition, a function $\varphi\in W_{p,loc}^{1}(M)$ satisfies the condition

[TABLE]

where $\psi\in W_{p,loc}^{1}(M)$, if $\varphi-\psi\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(\omega,K)}$ for some open set $\omega$ containing $K$.

Proposition 1.1**.**

A function $u\in W_{p,loc}^{1}(\Omega)$ satisfies (1.2) if and only if

[TABLE]

for any compact set $K\subset\partial M$.

Proof.

At first, let (1.2) hold and $K$ be a compact subset of $\partial M$. Take an open pre-compact set $\omega$ containing $K$ and a function $\psi\in C_{0}^{\infty}(M)$ such that

[TABLE]

By (1.2), the function $(u-h)\psi$ belongs to the closure of $C_{0}^{\infty}(M\setminus\partial M)$ in the space $W_{p}^{1}(M\setminus\partial M)$. Assuming that functions from ${C_{0}^{\infty}(M\setminus\partial M)}$ are extended by zero to $\partial M$, we obtain $u-h\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(\omega,K)}.$

Now, assume that condition (1.6) is valid and let $\psi\in C_{0}^{\infty}(M)$. We consider the compact set $K=\operatorname{supp}\psi\cap\partial M$. In view of (1.6), there exists an open set $\omega$ such that $K\subset\omega$ and, moreover, $u-h\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(\omega,K)}$ or, in other words,

[TABLE]

for some sequence of functions $\varphi_{i}\in C_{0}^{\infty}(M,K)\cap W_{p}^{1}(\omega)$, $i=1,2,\ldots$. We denote $\tilde{K}=\operatorname{supp}\psi\setminus\omega$. Since $\tilde{K}$ is a compact set belonging to $M\setminus\partial M$, there is a function $\tau\in C_{0}^{\infty}(M\setminus\partial M)$ equal to one in a neighborhood of $\tilde{K}$. It is easy to see that $(1-\tau)\psi\varphi_{i}\in C_{0}^{\infty}(\omega\setminus\partial M),$ $i=1,2,\ldots$. At the same time, by (1.7), we have

[TABLE]

therefore, one can assert that $(1-\tau)\psi(u-h)\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(M\setminus\partial M)}.$ It is also obvious that $\tau\psi(u-h)\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(M\setminus\partial M)}.$ Thus, we obtain $\psi(u-h)=(1-\tau)\psi(u-h)+\tau\psi(u-h)\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(M\setminus\partial M)}.$ ∎

Let $\Omega$ be an open subset of $M$. The capacity of a compact set $K\subset M$ associated with a function $\psi\in W_{p,loc}^{1}(M)$ is defined as

[TABLE]

where the infimum is taken over all functions $\varphi\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(\Omega)}$ for which (1.5) is valid. In so doing, we assume that the functions from ${\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(\Omega)}$ are extended by zero beyond $\Omega$. For an arbitrary closed set $E\subset M$, we put

[TABLE]

where the supremum is taken over all compact sets $K\subset E$. If $\Omega=M$, we write $\operatorname{cap}_{\psi}(K)$ instead of $\operatorname{cap}_{\psi}(K,M)$. In the case of $\psi=1$ and $p=2$, the capacity $\operatorname{cap}_{\psi}(K)$ coincides with the well-known Wiener capacity [9].

It is not difficult to verify that the capacity introduced above has the following natural properties.

1. (a)

Let $K_{1}\subset K_{2}$ and $\Omega_{2}\subset\Omega_{1}$, then

[TABLE] 2. (b)

Suppose that $\lambda$ is a real number, then

[TABLE] 3. (c)

Let $\psi_{1},\psi_{2}\in W_{p,loc}^{1}(M)$, then

[TABLE]

We say that $u\in W_{p,loc}^{1}(M)$ is a solution of (1.1) under the condition

[TABLE]

if the integral identity (1.4) holds for all $\varphi\in C_{0}^{\infty}(M)$. The set of solutions of problem (1.1), (1.8) with bounded Dirichlet integral (1.3) is denoted by $\mathfrak{H}$.

2. Main result

Theorem 2.1**.**

Problem (1.1)–(1.3) has a solution if and only if

[TABLE]

for some $w\in\mathfrak{H}$.

The proof of Theorem 2.1 is based on the following two lemmas known as Poincare’s inequalities.

Lemma 2.1**.**

Let $G\subset M$ be a pre-compact Lipschitz domain and $\omega$ be a subset of $G$ of non-zero measure. Then

[TABLE]

for all $u\in W_{p}^{1}(G)$, where the constant $C>0$ does not depend on $u$.

Lemma 2.2**.**

Let $\omega\subset M$ be a pre-compact Lipschitz domain. Then

[TABLE]

for all $\varphi\in W_{p}^{1}(\omega)$, where

[TABLE]

and the constant $C>0$ does not depend on $\varphi$.

Proof of Theorem 2.1.

We show that the existence of a solution of (1.1)–(1.3) implies the validity of (2.1). Consider a sequence of functions $\varphi_{i}\in C_{0}^{\infty}(M)$, $i=1,2,\ldots$, such that

[TABLE]

Since the sequence $\nabla\varphi_{i}$, $i=1.2,\ldots$, is bounded in $L_{p}(M)$, there is a subsequence $\nabla\varphi_{i_{j}}$, $j=1,2,\ldots$, that converges weakly in $L_{p}(M)$ to some vector-function ${\mathbf{r}}\in L_{p}(M)$. Let $R_{m}$ be the convex hull of the set $\{\varphi_{i_{j}}\}_{j\geq m}$. By Mazur’s theorem, there exists a sequence $r_{m}\in R_{m}$, $m=1,2,\ldots$, such that

[TABLE]

In view of the convexity of the functional

[TABLE]

we have

[TABLE]

therefore,

[TABLE]

Let $\omega\subset M$ be a pre-compact Lipschitz domain. Denoting

[TABLE]

we obtain in accordance with Lemma 2.2 that the sequence $r_{m}-\alpha_{m}$, $m=1,2,\ldots$, is fundamental in $W_{p}^{1}(\omega)$. By Lemma 2.1, this sequence is also fundamental in $W_{p}^{1}(G)$ for any pre-compact Lipschitz domain $G\subset M$.

At first, we assume that the sequence $\alpha_{m}$, $m=1,2,\ldots$, is bounded. Extracting from it a convergent subsequence $\alpha_{i_{j}}$, $j=1,2,\ldots$, we have that the sequence of the functions $r_{m_{j}}$, $j=1,2,\ldots$, is fundamental in $W_{p}^{1}(G)$ for any pre-compact Lipschitz domain $G\subset M$. Hence, there exists $v\in W_{p,loc}^{1}(M)$ such that

[TABLE]

for any pre-compact Lipschitz domain $G\subset M$. In view of (2.2), we have $\nabla v={\mathbf{r}};$ therefore,

[TABLE]

Thus, by the variational principle, the function $w=u-v$ belongs to $\mathfrak{H}$.

Let us show the validity of inequality (2.1). Let $K\subset\partial\Omega$ be some compact set. It is easy to see that

[TABLE]

Take a function $\tau\in C_{0}^{\infty}(M)$ equal to one in a neighborhood of $K$. Putting $\psi_{j}=\tau v+(1-\tau)r_{m_{j}},$ $j=1,2,\ldots,$ we obtain a sequence of functions from ${\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(M)}$ satisfying the condition

[TABLE]

In so doing, we obviously have

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whence it follows immediately that

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In view of the arbitrariness of the compact set $K\subset\partial\Omega$, the last formula implies the estimate

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Now, assume that the sequence $\alpha_{m}$, $m=1,2,\ldots$, is not bounded. Without loss of generality, we can also assume that $|\alpha_{m}|\to\infty$ as $m\to\infty$. If this is not the case, then we replace $\alpha_{m}$, $m=1,2,\ldots$, with a suitable subsequence. Applying Lemma 2.2, we arrive at the inequality

[TABLE]

for all $m=1,2,\ldots$, where the constant $C>0$ does not depend on $m$, whence we have

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For any positive integer $m$ we take a positive integer $s_{m}\geq m$ such that

[TABLE]

and

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Putting further

[TABLE]

we obtain

[TABLE]

By Lemma 2.2, the estimate

[TABLE]

is valid, where the constant $C>0$ does not depend on $m$ and $l$. At the same time, condition (2.7) allows us to assert that

[TABLE]

Hence, the sequence $v_{m}$, $m=1,2,\ldots$, is fundamental in $L_{p}(\omega)$. According to Lemma 2.1, this sequence is also fundamental in $W_{p}^{1}(G)$ for any pre-compact Lipschitz domain $G\subset M$. Let us denote by $v$ the limit of this sequence. In view of (2.2) and (2.8), we have

[TABLE]

therefore, $v$ satisfies relation (2.3) and in accordance with the variational principle the function $w=u-v$ belongs to $\mathfrak{H}$. In so doing, for any compact set $K\subset\partial M$ condition (2.4) is obviously valid. Thus, putting $\psi_{j}=\tau v+(1-\tau)v_{j},$ $j=1,2,\ldots,$ where $\tau\in C_{0}^{\infty}(M)$ is some function equal to one in a neighborhood of $K$, we obtain

[TABLE]

whence we again arrive at relation (2.5) from which (2.6) follows.

It remains to show that condition (2.1) implies the existence of a solution of problem (1.1)–(1.3). Let (2.1) be valid for some $w\in\mathfrak{H}$. We take pre-compact Lipschitz domains $\Omega_{i}\subset\Omega_{i+1}$, $i=1,2,\ldots$, whose union coincides with the entire manifold $M$. Consider the functions $\varphi_{i}\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(M)}$ such that

[TABLE]

In view of (2.1), the sequence $\nabla\varphi_{i}$, $i=1,2,\ldots$, is bounded in the space $L_{p}(M)$. Hence, there exists a subsequence $\nabla\varphi_{i_{j}}$, $j=1,2,\ldots$, of this sequence that weakly converges in $L_{p}(M)$ to some vector-function ${\mathbf{r}}\in L_{p}(M)$. As above, we denote by $R_{m}$ the convex hull of the set $\{\varphi_{i_{j}}\}_{j\geq m}$. By Mazur’s theorem, there exists a sequence $r_{m}\in R_{m}$, $m=1,2,\ldots$, such that (2.2) holds. Since the functional

[TABLE]

is convex, we obtain

[TABLE]

Also, it can be seen that

[TABLE]

One can assume without loss of generality that $\Omega_{1}\cap\partial M\neq\emptyset$. Thus, we have

[TABLE]

for all $\varphi\in{\stackrel{{\scriptstyle\rm\scriptscriptstyle o}}{{W}}\!\!{}_{p}^{1}(\Omega_{1},\overline{\Omega}_{1}\cap\partial M)},$ where the constant $C>0$ does not depend on $\varphi$. In particular,

[TABLE]

for all $i,j=1,2,\ldots$, whence it follows that the sequence $r_{i}$, $i=1,2,\ldots$, is fundamental in $L_{p}(\Omega_{1})$. Lemma 2.1 implies that this sequence is also fundamental in $W_{p}^{1}(G)$ for any pre-compact Lipschitz domain $G\subset M$. Let us denote by $u_{1}$ the limit of this sequence. In view of (2.9) and (2.10), we obtain

[TABLE]

and

[TABLE]

Let us construct a solution of problem (1.1)–(1.3). This time we take a sequence of functions $\varphi_{i}\in{C_{0}^{\infty}(M\setminus\partial M)},$ $i=1,2,\ldots$, such that

[TABLE]

By (2.11), the sequence $\nabla\varphi_{i}$, $i=1,2,\ldots$, is bounded in $L_{p}(\Omega)$. Thus, it has a subsequence $\nabla\varphi_{i_{j}}$, $j=1,2,\ldots$, that weakly converges in $L_{p}(M)$ to some vector-function ${\mathbf{r}}\in L_{p}(M)$. According to Mazur’s theorem, there exists a sequence $r_{m}\in R_{m}$, $m=1,2,\ldots$, satisfying relation (2.2). Since $r_{m}\in C_{0}^{\infty}(M\setminus\partial M)$, $m=1,2,\ldots$, this sequence is fundamental in $W_{p}^{1}(G)$ for any pre-compact domain $G\subset M$. Denoting by $u_{0}$ the limit of this sequence, we have

[TABLE]

To complete the proof, it remains to note that, in view of (2.12) and the variational principle, the function $u=u_{1}+w-u_{0}$ is a solution of (1.1)–(1.3). ∎