Existence of a unique solution to an elliptic partial differential equation when the average value is known

1.   Introduction

In this paper, we consider the existence of a unique, classical solution $u({\bf{x}})$ to the quasilinear elliptic equation

for ${\bf{x}} \in \Omega$, which satisfies the condition that the average value

where $u_0$ is a given constant and $\frac{1}{|\Omega|}\int_{\Omega} f d{\bf{x}} = 0$. Periodic boundary conditions will be used. That is, we choose for our spatial domain the N-dimensional torus $\mathbb{T}^N$, where $N = 2$ or $N = 3$.

The purpose of this paper is to prove the existence of a unique classical solution $u$ to (1.1), (1.2). The proof of the existence theorem uses the method of successive approximations in which an iteration scheme, based on solving a linearized version of Eq (1.1), will be defined and then convergence of the sequence of approximating solutions to a unique solution satisfying the quasilinear equation will be proven. The key to the proof lies in obtaining a priori estimates for $u$. To the best of our knowledge, no other researcher has proven the existence and uniqueness of the solution to this partial differential equation when the given data is the average value of the solution.

The paper is organized as follows. The main result, Theorem 2.1, is presented and proven in the next section. The existence of a solution to the linearized equation used in the iteration scheme is proven in Appendix A. Appendix B presents lemmas supporting the proof of the theorem.

2.   Existence theorem

We will be working with the Sobolev space $H^s(\Omega)$ (where $s\geq 0$ is an integer) of real-valued functions in $L^2(\Omega)$ whose distribution derivatives up to order $s$ are in $L^2(\Omega)$. The norm is $\Vert u\Vert_{s}^2 = \sum_{0 \leq |\alpha |\leq s}\int_\Omega |D^\alpha u|^2d {\bf{x}}$. We are using the standard multi-index notation. We define $|F |_{r, \overline{G}_0} = \max\{\Big|\frac{d^{j} F}{d u^{j}} (u_{*}) \Big|: u_{*} \in \overline{G}_0, 0 \leq j \leq r \}$, where $F$ is a function of $u$ and where $\overline{G}_0\subset \mathbb{R}$ is a closed, bounded interval. Also, we let both $\nabla u$ and $Du$ denote the gradient of $u$. And $C^k(\Omega)$ is the set of real-valued functions having all derivatives of order $\leq k$ continuous in $\Omega$ (where $k =$ integer $\geq 0$ or $k = \infty$). The purpose of this paper is to prove the following theorem:

Theorem 2.1. Let $a$ be a smooth, positive function of $u$. Let $f \in H^{2}(\Omega)$ and let $\frac{1}{|\Omega|}\int_{\Omega} f d{\bf{x}} = 0$. Let the domain $\Omega = \mathbb{T}^N$, the N-dimensional torus, where $N = 2$ or $N = 3$.

There exists a constant $C_1$ which depends only on $N$, $\Omega$ such that if

and if

where $\overline{G}_0\subset \mathbb{R}$ is a closed, bounded interval, then there exists a unique solution $u \in C^2(\Omega)$ to the equation

which satisfies the condition that the average value

where $u_0$ is a given constant.

Proof.

We begin by using the following change of variables:

where the constant $a_0 = \min_{u_{*} \in \overline{G}_0} a(u_{*})$ and $\overline{G}_0\subset \mathbb{R}$ is a closed, bounded interval.

Under this change of variables the equation (2.2) becomes

And under this change of variables, (2.3) becomes

We fix closed, bounded intervals $\overline{G}_0 \subset \mathbb{R}$ and $\overline{G}_1 \subset \mathbb{R}$ by defining $\overline{G}_0 = \{u_{*} \in \mathbb{R} \quad :\quad |u_{*} - u_0|_{L^\infty} \leq \frac{R\Vert\nabla f\Vert_{0}}{a_0} \}$ and $\overline{G}_1 = \{v_{*} \in \mathbb{R} \quad :\quad |v_{*} - v_0|_{L^\infty} \leq R \}$, where $R$ is a constant to be defined later. We will prove that $v({\bf{x}})\in \overline{G}_1$ for ${\bf{x}}\in \Omega$. It follows that $u({\bf{x}})\in \overline{G}_0$ for ${\bf{x}}\in \Omega$.

We will construct the solution of (2.5), (2.6) through an iteration scheme. To define the iteration scheme, we will let the sequence of approximate solutions be $\{v^k\}$. Set the initial iterate $v^0 = v_0$. For $k = 0, 1, 2, \dots$, construct $v^{k+1}$ from the previous iterate $v^k$ by solving the linear equation

which satisfies the condition that the average value

and using periodic boundary conditions.

The existence of a unique solution $v^{k+1} \in C^{2}(\Omega)$ to the linear equation (2.7) for fixed $k$ which satisfies (2.8) is proven in Appendix A. Lemmas supporting the proof are presented in Appendix B. We proceed now to prove convergence of the iterates as $k\to \infty$ to a unique, classical solution $v$ of (2.5), (2.6), which therefore produces a unique, classical solution $u = \frac{\Vert\nabla f\Vert_{0}}{a_0} v$ of (2.2), (2.3).

We begin by proving the following proposition:

Proposition 1. Assume that the hypotheses of Theorem 2.1 hold. Then there exist constants $C_2$, $C_3$, and $R$ such that the following inequalities hold for $k = 1, 2, 3\dots$:

where the constants $C_2$, $R$ depend on $N$ and $\Omega$, and where the constant $C_3$ depends on $R$, $u_0$, $a_0$, $\Vert \nabla f\Vert_{0}$, $\Vert \nabla f\Vert_{1}$, $\Big|\frac {da}{d u} \Big|_{2, \overline{G}_0}$, $N$, and $\Omega$. From (2.11) it follows that $v^k({\bf{x}}) \in \overline{G}_1$ for ${\bf{x}}\in \Omega$ and for $k = 1, 2, 3\dots$.

Proof. The proof is by induction on $k$. We prove in Lemma B.2 in Appendix B that if $v^k$ satisfies (2.9) and (2.11), then $v^{k+1}$ satisfies (2.9) and (2.10). See Lemma B.2 in Appendix B for the detailed proof. It only remains to prove inequalities (2.11) for $v^{k+1}-v_0$ and (2.12) for $\nabla(v^{k+1}-v^k)$.

In the estimates below, we will let $C$ denote a generic constant whose value may change from one relation to the next.

Estimate for $|v^{k+1} -v _{0}|_{L^{\infty}}$:

Lemma B.2 in Appendix B presents the proof that $\Vert \nabla v^{k+1}\Vert_{2}^2 \leq C_2$. Then by using standard Sobolev space inequalities we obtain the inequality:

where the constants $C$ and $C_2$ depend on $\Omega$, $N$. Here we used the fact that $|v^{k+1}-v _0|_{L^{\infty} } \leq C\Vert v^{k+1}-v_0 \Vert_2$ by Sobolev's Lemma. Since $\frac{1}{|\Omega|}\int_{\Omega} v^{k+1} d{\bf{x}} = v_0$ by (2.8), it follows that $v^{k+1}-v_0$ is a zero-mean function and $\Vert v^{k+1}-v_0\Vert_{0} \leq C\Vert \nabla (v^{k+1}-v_0) \Vert_{0}$ by Poincaré's inequality. Therefore $\Vert v^{k+1}-v_0 \Vert_2 \leq C\Vert \nabla (v^{k+1}-v_0) \Vert_1$. We define $R = C \sqrt{C_2}$. Then inequality (2.11) of Proposition 1 holds for $v^{k+1}-v_0$.

Estimate for $\Vert \nabla (v^{k+1}-v^k)\Vert_0^2$:

From successive iterates of Eq (2.7) we obtain the following:

In the estimates that follow, we use the notation $(h_1, h_2) = \int_{\Omega} h_1 h_2 d{\bf{x}}$ for the $L^{2}$ inner product of functions $h_1$, $h_2$. Note that $v^{k}-v^{k-1}$ is a zero-mean function because $v^{k}-v^{k-1} = (v^{k}-v_0)-(v^{k-1}-v_0)$ and $v^{k}-v_0$, $v^{k-1}-v_0$ are zero-mean functions by successive iterates of (2.8).

We define the constant $b_0 = \min_{v_{*} \in \overline{G}_1} b(v_{*})$, where $\overline{G}_1\subset \mathbb{R}$ is a closed, bounded interval. Note that $b_0 = 1$ by the definition of the function $b$ in (2.4). Then integration by parts and using Eq (2.13) yields

where $C$ is a constant which depends on $N$, $\Omega$. Here we used the fact that $|\nabla v^{k}|_{L^{\infty} } \leq C\Vert \nabla v^{k} \Vert_2$ by Sobolev's Lemma. And we used Poincaré's inequality to obtain $\Vert v^{k}-v^{k-1}\Vert_{0} \leq C\Vert \nabla(v^{k}-v^{k-1})\Vert_{0}$, since $v^{k}-v^{k-1}$ is a zero-mean function.

Using the facts that $\frac{1}{b_0} = 1$ and that $\Big|\frac{d b}{d v} \Big|_{0, \overline{G}_1}^2 = \frac{1}{a_0^4} \Vert\nabla f\Vert_{0}^2\Big|\frac{d a}{d u} \Big|_{0, \overline{G}_0}^2 \leq C_1$ by the definition of $b(v)$ in (2.4) and by the statement of the theorem, and using the fact that $\Vert\nabla v^{k}\Vert_2^2 \leq C_2$ by the induction hypothesis, we obtain from re-arranging terms in (2.14) the inequality

where we define the constant $C_1$ to be sufficiently small so that $C C_1 C_2 \leq \frac 12$. And the constants $C$, $C_1$, $C_2$ depend on $N$, $\Omega$.

By repeatedly applying inequality (2.15) it follows that

where the initial iterate $v^{0} = v_0$, which is a constant, and where $\Vert \nabla v^{1} \Vert_{0}^2 \leq \Vert\nabla v^{1}\Vert_2^2 \leq C_2$ by Lemma B.2 in Appendix B. Therefore inequality (2.12) of Proposition 1 holds for $\nabla(v^{k+1}-v^k)$.

This completes the proof of Proposition 1.

We now complete the proof of Theorem 2.1. By (2.16), $\Vert \nabla(v^{k+1}-v^k) \Vert _{0}\rightarrow 0$ as $k\to \infty$. By Poincaré's inequality, $\Vert v^{k+1}-v^{k}\Vert_{0}^2 \leq C\Vert \nabla(v^{k+1}-v^{k})\Vert_{0} ^2$. It follows that $\Vert v^{k+1}-v^k \Vert _{0}\rightarrow 0$ as $k\to \infty$. We next use the standard interpolation inequality $\Vert v^{k+1} -v ^{k} \Vert _{r} \leq C\Vert v^{k+1} -v^{k}\Vert _{0}^\beta \Vert v^{k+1} -v^{k} \Vert _{4}^{1-\beta }$, where $\beta = \frac{4-r}{4}$, and $0 < r < 4$. Then since $\Vert v^{k+1} -v^{k} \Vert _{4}^2 \leq C\big(\Vert v^{k+1} \Vert_{4}^2+ \Vert v^{k} \Vert _{4}^2\big) \leq C C_3$ by (2.10) in Proposition 1, it follows that $\Vert v^{k+1}-v^k \Vert _{r}\rightarrow 0$ as $k\to \infty$ for $0 < r < 4$.

Therefore there exists $v \in H^{r}(\Omega)$, where $r < 4$, such that $\Vert v^{k}-v \Vert_{r}\rightarrow 0$ as $k\to \infty$. The fact that $v \in H^{4}(\Omega)$ can be deduced using boundedness in high norm and a standard compactness argument (see, for example, Embid ^[2], Majda ^[6]). Sobolev's Lemma implies that $v \in C^2(\Omega)$.

From Lemma A.1 in Appendix A, $v^{k+1}\in C^2(\Omega)$ is a solution of the linear equation $\nabla \cdot (b(v^k) \nabla v^{k+1}) = g$ for each $k\geq 0$, and $v^{k+1}$ satisfies the condition that $\frac{1}{|\Omega|}\int_{\Omega} v^{k+1} d{\bf{x}} = v_0$. It follows that $v$ is a classical solution of the equation $\nabla \cdot (b(v) \nabla v) = g$, and $v$ satisfies the condition that $\frac{1}{|\Omega|}\int_{\Omega} v d{\bf{x}} = v_0$. The uniqueness of the solution follows by a standard proof using estimates similar to the estimates used in the proof of inequality (2.12). Therefore, there exists a unique classical solution $u = \Big(\frac{\Vert \nabla f\Vert_{0}}{a_0}\Big) v$ of $\nabla \cdot (a(u) \nabla u) = f$ which satisfies the condition that $\frac{1}{|\Omega|}\int_{\Omega} u d{\bf{x}} = u_0$. This completes the proof of the theorem.

3.   Conclusion

We have proven that if

and if

where $\overline{G}_0\subset \mathbb{R}$ is a closed, bounded interval and where the constant $C_1$ depends on $N$, $\Omega$, then there exists a unique solution $u \in C^2(\Omega)$ to the equation

which satisfies the condition that the average value

where $u_0$ is a given constant, under periodic boundary conditions. We remark that in the trivial case in which $\nabla f = 0$ (and therefore $f = 0$), it follows that $u = u_0$ is the unique solution.

Conflict of interest

The author confirms that there is no conflict of interest.

A.   Existence for the linear equation

In this appendix, we present the proof of the existence of a unique, classical solution to the linear problem (2.7), (2.8).

Lemma A.1. Let $b$ be a smooth positive function of $w$. Let $w \in C^2(\Omega)$, let $g \in H^{2}(\Omega)$, and let $\frac{1}{|\Omega|} \int_{\Omega} g d{\bf{x}} = 0$. Let the domain $\Omega = \mathbb{T}^N$, the N-dimensional torus, where $N = 2$ or $N = 3$. Then there exists a unique solution $v \in C^2(\Omega)$ of the equation

which satisfies the condition

where $v_0$ is a given constant.

Proof.

We define the zero-mean function

The existence of a unique zero-mean solution $\overline{v} \in C^2 (\Omega)$ to equation (A.1) under periodic boundary conditions is a well-known result from the standard theory of elliptic equations (see, e.g., Embid ^[2], Evans ^[3], Gilbarg and Trudinger ^[4]).

It follows that the function $v$ defined by

is the unique solution to equation (A.1) which satisfies the condition (A.2) that $\frac{1}{|\Omega|}\int_{\Omega} v d{\bf{x}} = v_0$.

This completes the proof of the lemma.

B.   A priori estimates

In this appendix, we present lemmas supporting the proof of the theorem.

We begin by listing several standard Sobolev space inequalities.

Lemma B.1. (Standard Sobolev Space Inequalities)

(a) Let $b$ be a smooth function of $w$, and let $w({\bf{x}})$ be a continuous function such that $w({\bf{x}}) \in \overline{G}_1$ for ${\bf{x}} \in \Omega$ where $\overline{G}_1 \subset \mathbb{R}$ is a closed, bounded interval. And let $w \in H^{r+1}(\Omega)$ where $r\geq 0$.

Then

where $|\frac{db}{dw} |_{r, \overline{G}_1} = \max\{\Big|\frac{d^{j+1} b}{d w^{j+1}} (w_{*}) \Big|: w_{*} \in \overline{G}_1, 0 \leq j \leq r \}$. And the constant $C$ depends on $r$, $N$, $\Omega$.

(b) If $f\in H^{n}(\Omega)$, where $\Omega \subset \mathbb{R}^{N}$, and $r = \beta m+(1-\beta)n$, with $0\leq\beta \leq 1$ and $m < n$, then

Here $C$ is a constant which depends on $m$, $n$, $N$, $\Omega$.

(c) If $f\in H^{s_0}(\Omega)$ where $\Omega \subset \mathbb{R}^{N}$, $N = 2$ or $N = 3$, and $s_0 = [\frac N2]+1$, then

Here $C$ is a constant which depends on $N$, $\Omega$.

(d) If $Df\in H^{r_1}(\Omega)$, $g\in H^{r-1}(\Omega)$, where $r\geq 1$ and where $r_1 = \max\{r-1, s_0\}$ and $s_0 = [\frac N2]+1$, then:

where $|\alpha| = r$ and where the constant $C$ depends on $r$, $N$, $\Omega$.

These inequalities are well-known. Proofs may be found, for example, in ^[5], ^[7]. These inequalities also appear in ^[1], ^[2].

Lemma B.2. Let the function $w\in C^2(\Omega)$ satisfy (2.9), (2.11) in Proposition 1 and let the hypotheses in the statement of Theorem 2.1 hold. Let $b$ be a smooth, positive function of $w$. Let $g \in H^{2}(\Omega)$ and let $\frac{1}{|\Omega|}\int_{\Omega} g d{\bf{x}} = 0$. Let (2.4) define the functions $b$, $g$. Let the domain $\Omega = \mathbb{T}^N$, the N-dimensional torus, where $N = 2$ or $N = 3$.

Let $v$ be the solution from Lemma A.1 in Appendix A of

which satisfies the condition

where $v_0$ is a given constant.

Then $\nabla v$ and $v$ satisfy the following inequalities:

where the constant $C_2$ depends on $N$ and $\Omega$ and where the constant $C_3$ depends on $R$, $u_0$, $a_0$, $\Vert \nabla f\Vert_{0}$, $\Vert \nabla f\Vert_{1}$, $\Big|\frac {da}{d u} \Big|_{2, \overline{G}_0}$, $N$, and $\Omega$.

Proof.

In the estimates below, we will let $C$ denote a generic constant whose value may change from one relation to the next. We use the notation $(h_1, h_2) = \int_{\Omega} h_1 h_2 d{\bf{x}}$ for the $L^{2}$ inner product of two functions $h_1$, $h_2$. And we use the notation $h_{\alpha} = D^{\alpha} h$ for differentiation with a multi-index $\alpha$.

Estimate for $\Vert \nabla v \Vert_{0}^2$:

Using integration by parts and then substituting equation (B.5) yields

where $b_0 = \min_{w_{*} \in \overline{G}_1} b(w_{*}) = 1$ by definition of the function $b$, and $\Vert \nabla g\Vert_{0} = 1$ by definition of the function $g$. Here we used the fact that $g$ and $v-\frac{1}{|\Omega|} \int_{\Omega}v d{\bf{x}}$ are zero-mean functions and we used Poincaré's inequality for a zero-mean function $h$, namely $\Vert h\Vert_{0} \leq C\Vert \nabla h\Vert_{0}$. The constant $C$ depends on $N$, $\Omega$.

It follows that

where the generic constant $\tilde{C}$ depends on $N$, $\Omega$.

Estimate for $\Vert \nabla v \Vert_{1}^2$: To begin, let $|\alpha| \geq 1$. Using integration by parts and then substituting equation (B.5) yields

where $b_0 = \min_{w_{*} \in \overline{G}_1} b(w_{*})$. If $|\alpha| = 1$ in (B.9) then

Re-arranging the terms in (B.10) and adding the resulting inequality over $|\alpha| = 1$ yields

where the generic constants $C$, $\tilde{C}$ depend on $N$, $\Omega$. Here we used the facts that $\frac{1}{b_0} = 1$, $\Vert \nabla g \Vert_{0} = 1$, and $\Vert \nabla w\Vert_{2}^2 \leq C_2$. And $\Vert \nabla v\Vert_{0}^2 \leq \tilde{C}$ from (B.8). And we used the fact that $\Big|\frac{d b}{d w} \Big|_{0, \overline{G}_1}^2 \; = \Big(\frac{\Vert\nabla f\Vert_{0}^2 }{a_0^4}\Big)\Big|\frac{d a }{d u} \Big|_{0, \overline{G}_0}^2 \leq C_1$ by definition of the function $b$ in (2.4) and by the statement of Theorem 2.1.

From (B.8), (B.11) it follows that

Estimate for $\Vert \nabla v\Vert_{2}^2$:

Letting $|\alpha| = 2$ in inequality (B.9) and then using integration by parts with $|\gamma| = 1$ produces

where we used Cauchy's inequality with $\epsilon$ and we define $\epsilon = \frac{1}{10}$. We also used Sobolev's Lemma, i.e., $|h|_{L^{\infty}}\leq C\Vert h\Vert_{2}$.

Re-arranging terms in (B.13), and then adding the resulting inequality over $|\alpha| = 2$ and $|\gamma| = 1$, produces

where we can assume that $C_1 < 1$ and that $C_2 > 1$. Here we used the fact that $\Vert \nabla w\Vert_{2}^2 \leq C_2$. And we used the fact that $\Big|\frac{d b}{d w} \Big|_{0, \overline{G}_1}^2$ = $\Big(\frac{\Vert\nabla f\Vert_{0}^2 }{a_0^4}\Big)\Big|\frac{d a }{d u} \Big|_{0, \overline{G}_0}^2 \leq C_1$. And we used the fact that $\Big|\frac{d^2 b}{d w^2} \Big|_{0, \overline{G}_1}^2 = \; \Big(\frac{\Vert\nabla f\Vert_{0}^4 }{a_0^6}\Big)\Big|\frac{d^2 a }{d u^2} \Big|_{0, \overline{G}_0}^2 \; \leq \Big(\frac{\Vert\nabla f\Vert_{0}^4 }{a_0^8}\Big)\Big|\frac{d a }{d u} \Big|_{0, \overline{G}_0}^4 \leq C_1^2$ by the definition of $b(v)$ in (2.4) and by the statement of the theorem. And we used the facts that $\Vert \nabla g\Vert_{0} = 1$ and that $\frac{1}{b_0} = 1$.

From (B.14) and from inequality (B.12) for $\Vert \nabla v\Vert_{1}^2$, it follows that

where the generic constants $C$, $\tilde{C}$ depend on $N$, $\Omega$, and where $C_1$ is sufficiently small so that $C C_1 C_2 \tilde{C} \leq \frac 12$ and so that $C C_1 C_2^2 \leq \frac 12$.

Re-arranging terms in (B.15) yields

where we define $C_2 = 4\tilde{C}+C$, and where the constant $C_2$ depends on $N$, $\Omega$.

Estimate for $\Vert \nabla v\Vert_{3}^2$:

Letting $|\alpha| = 3$ in inequality (B.9) and then using integration by parts with $|\gamma| = 1$ produces

where $\epsilon = \frac 14$ and where we used the Sobolev space inequality (B.4) from Lemma B.1 with $r = |\alpha| = 3$ and $r_1 = 2$.

Re-arranging the terms in (B.17) and then adding the resulting inequality over $|\alpha| = 3$ and $|\gamma| = 1$ yields

where we used the Sobolev space inequality (B.1) from Lemma B.1 for $\Vert Db \Vert_{2}^2$. We also used the facts that $|w-v_{0}|_{L^{\infty}}\leq R$, $\Vert \nabla w\Vert_{2}^2 \leq C_2$, and $b_0 = 1$. And we used the fact that $g$ is a zero-mean function, so that $\Vert g\Vert_{0} \leq C \Vert \nabla g\Vert_{0}$ by Poincaré's inequality.

From (B.18) and from inequality (B.16) for $\Vert \nabla v\Vert_{2}^2$, it follows that

Estimate for $\Vert v\Vert_{4}^2$:

From (B.19) it follows that

Here we used Poincaré's inequality for the zero-mean function $v-v_0$, where the constant $v_0 = \frac{1}{|\Omega|}\int_{\Omega} v d{\bf{x}}$. And we used the definition of the functions $v$, $b$, $g$ from (2.4). This completes the proof of the lemma.