The Robin Mean Value Equation I: A Random Walk Approach to the Third Boundary Value Problem

Abstract

We study the family of integral equations, called the Robin mean value equations (RMV), that are local averaged approximations to the Robin-Laplace boundary value problem (RL). When posed on $\mathcal {C}^{1,1}$-regular domains, we prove existence, uniqueness, equiboundedness and the comparison principle for solutions to (RMV). For the continuous right hand side of (RL), we show that solutions to (RMV) converge uniformly, in the limit of the vanishing radius of averaging, to the unique W^2,p solution, which coincides with the unique viscosity solution of (RL). We further prove the lower bound on solutions to (RMV), which is consistent with the optimal lower bound for solutions to (RL). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle along a suitable discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks.

Appendix: : A Proof of Uniqueness of Viscosity Solutions to ??

We first make an observation that relies on the assumed regularity of $\partial {\mathcal {D}}$.

Lemma A.1

When ${\mathcal {D}}\subset {\mathbb {R}}^{N}$ satisfies the uniform outer supporting sphere condition:

$$ \text{for every} x\in\partial{\mathcal{D}} \text{exists} B_{r}(b)\subset{\mathbb{R}}^{n}\setminus \bar{\mathcal{D}} \quad \text{such that} \quad |x-b|=r, $$

(9.1)

with some radius r > 0, then the boundary requirements in Definition 4.1 (i) and (ii) can be reduced to: $\langle p, \vec {n}(x)\rangle + \gamma u(x)\leq 0$ and $\langle p, \vec {n}(x)\rangle + \gamma u(x)\geq 0$, respectively.

Proof

Let $x\in \partial {\mathcal {D}}$ and $(p,X)\in J_{\bar {\mathcal {D}}}^{2,+}u(x)$. For each large j, consider the jet:

$$ (p_{j}, X_{j})= \left( p-\frac{1}{j}\vec{n}(x), ~ X+r Id_{N} - (r+j)\vec{n}(x)^{\otimes 2}\right). $$

We claim that $(p_{j}, X_{j})\in J_{\bar {\mathcal {D}}}^{2,+}u(x)$. In this case, we have:

$$ - \text{trace} X_{j} = -\big(\text{trace} X + (N-1)r-j\big)> f(x), $$

so by (i) there must be:$\displaystyle 0\geq \langle p_{j}, \vec {n}(x)\rangle +\gamma u(x) = \langle p, \vec {n}(x)\rangle +\gamma u(x)-\frac {1}{j}$. Passing to the limit with $j\to \infty$ we get the claimed boundary condition. To show that (p_j,X_j) is indeed a super-jet, let:

$$ \begin{array}{@{}rcl@{}} q_{j}(y-x) & = & \big\langle -\frac{1}{j}\vec{n}(x), y-x\big\rangle +\frac{1}{2}\big\langle r Id_{N} -(r+j)\vec{n}(x)^{\otimes 2} : (y-x)^{\otimes 2}\big\rangle \\ & = & -\frac{1}{j}\langle\vec{n}(x), y-x\rangle +\frac{r}{2} |(y-x)_{tan}|^{2} -\frac{j}{2}\langle \vec{n}(x), y-x\rangle^{2}. \end{array} $$

For $y\in \bar {\mathcal {D}}$ satisfying $\langle \vec {n}(x), y-x\rangle \leq 0$, we get: $\displaystyle q_{j}(y-x) \geq |\langle \vec {n}(x), y-x\rangle |\Big (\frac {1}{j}-\frac {j}{2} |\langle \vec {n}(x), y-x\rangle |\Big )\geq 0,$ if |y − x| is small enough. On the other other hand, for $y\in \bar {\mathcal {D}}$ such that $\langle \vec {n}(x), y-x\rangle \geq 0$, we get:

$$ \begin{array}{@{}rcl@{}} q_{j}(y-x) & \geq & -\frac{2}{j}\langle \vec{n}(x), y-x\rangle +\frac{r}{2} |(y-x)_{tan}|^{2}\\ & \geq & -\frac{2}{j}\big(r-\sqrt{r^{2}-|(y-x)_{tan}|^{2}}\big) + \frac{r}{2} |(y-x)_{tan}|^{2}\geq 0, \end{array} $$

as for small |y − x| and j ≫ 1 there holds: $\displaystyle r-\sqrt {r^{2}-|(y-x)_{tan}|^{2}}\leq \frac {1}{r}|(y-x)_{tan}|^{2}\leq \frac {jr}{4}|(y-x)_{tan}|^{2}.$ Thus, in both cases, the validity of Eq. ?? for (p,X), implies the same asymptotic bound for each (p_j,X_j) with sufficiently large j. □

Lemma A.2

Assume the uniform outer supporting sphere condition (9.1) with radius r > 0. Then the Robin problem ?? with $f\in \mathcal {C}(\bar {\mathcal {D}})$ has at most one viscosity solution.

Proof

1. Let u,v be two viscosity solutions to ??. We will prove that u ≤ v. In fact, the same analysis works when u is a viscosity sub-solution and v is a viscosity super-solution, in the sense of Definition 4.1 (i) and (ii), where u is assumed only to be upper-semicontinuous and v lower-semicontinuous, and where the jets sets $J_{\bar {\mathcal {D}}}^{2,+}$ and $J_{\bar {\mathcal {D}}}^{2,-}$ are replaced by their closures $\bar {J}_{\bar {\mathcal {D}}}^{2,+}$ and $\bar {J}_{\bar {\mathcal {D}}}^{2,-}$, respectively (see [9] for the details). Also, we recall that the requirements at boundary points in Definition 4.1 can be reduced as in Lemma 9.1, because of Eq. 9.1. The stated comparison principle is proved in three steps: by replacing u and v with strict sub- and super-solutions u_δ and v_δ, and by doubling the variables technique with an appropriate nonlinear corrector, separately in the cases when u_δ and v_δ achieves its maximum in ${\mathcal {D}}$ or on $\partial {\mathcal {D}}$. We now sketch these arguments.

For a sufficiently large C ≫ 1 and each δ ≪ 1, define:

$$ u_{\delta}(x) = u(x) - \frac{\delta}{2}\big(|x|^{2}-C\big), \qquad v_{\delta}(x) = v(x) + \frac{\delta}{2}\big(|x|^{2}-C\big). $$

Assume that $(p,X)\in J_{\bar {\mathcal {D}}}^{2,+}u_{\delta }(x)$, which is equivalent to: $(p-\delta x, X-\delta Id_{N})\in J_{\bar {\mathcal {D}}}^{2,+}u(x)$. Then:

$$ \begin{array}{ll} -\text{trace} X\leq f(x)-N\delta \qquad\quad\text{when} x\in{\mathcal{D}}\\ \langle p, \vec{n}(x)\rangle +\gamma u_{\delta}(x)\leq -\delta \qquad\text{when} x\in\partial{\mathcal{D}}. \end{array} $$

(9.2)

Thus, each u_δ is a strict sub-solution of ??. Similarly, each v_δ is a strict super-solution, namely $(p,X)\in J_{\bar {\mathcal {D}}}^{2,-}v_{\delta }(x)$ implies:

$$ \begin{array}{ll} -\text{trace} X\geq f(x)+N\delta \qquad\text{when} x\in{\mathcal{D}}\\ \langle p, \vec{n}(x)\rangle +\gamma v_{\delta}(x)\geq \delta \quad\quad\text{when} x\in\partial{\mathcal{D}}. \end{array} $$

(9.3)

2. We will show that u_δ ≤ v_δ in $\bar {\mathcal {D}}$, for all δ ≪ 1. By contradiction, fix δ > 0 and assume that:

$$ \max_{\bar{\mathcal{D}}}\big(u_{\delta}-v_{\delta}\big)>0. $$

We first treat the case of $\max \limits _{\bar {\mathcal {D}}}\big (u_{\delta }-v_{\delta }\big ) > \max \limits _{\partial \bar {\mathcal {D}}}\big (u_{\delta }-v_{\delta }\big )$. We apply [9, Proposition 3.7] to: Φ(x,y) = u_δ(x) − v_δ(y) and ${\Psi }(x,y) = \frac {1}{2}|x-y|^{2}$ and obtain a sequence $\{(x_{\alpha }, y_{\alpha })\}_{\alpha \to \infty }$ of maximizers to Φ − αΨ on $\bar {\mathcal {D}}\times \bar {\mathcal {D}}$ that converges to some diagonal element (z₀,z₀) such that z₀ is a maximizer of u_δ − v_δ and thus $z_{0}\in {\mathcal {D}}$. Applying [9, Theorem 3.2] to: w(x,y) = u_δ(x) − v_δ(y) and $\phi (x,y) = \frac {\alpha }{2}|x-y|^{2}$, we further obtain sequences of matrices $X_{\alpha }, Y_{\alpha }\in {\mathbb {R}}^{N\times N}_{\text {sym}}$ satisfying:

$$ \begin{array}{ll} \big(\alpha (x_{\alpha} - y_{\alpha}), X_{\alpha}\big) \in J_{{\mathcal{D}}}^{2,+}u_{\delta}(x_{\alpha}),\qquad \big(\alpha (x_{\alpha} - y_{\alpha}), Y_{\alpha}\big) \in J_{\mathcal{D}}^{2,-}v_{\delta}(y_{\alpha})\\ \left[\begin{array}{cc} X_{\alpha}&0\\ 0& -Y_{\alpha} \end{array}\right] \leq 3\alpha \left[\begin{array}{cc} Id_{N}& - Id_{N}\\ - Id_{N}& Id_{N} \end{array}\right]. \end{array} $$

(9.4)

The first two assertions above, together with (9.2), (9.3) yield:

$$ -\text{trace} (X_{\alpha} - Y_{\alpha})\leq f(x_{\alpha}) - f(y_{\alpha})-2N\delta\to -2\delta \quad \text{as} \alpha\to\infty, $$

which contradicts the last assertion in Eq. 9.4 that implies: trace(X_α − Y_α) ≤ 0.

3. We now treat the remaining case, namely that of:

$$ \max_{\bar{\mathcal{D}}}\big(u_{\delta}-v_{\delta}\big) = \big(u_{\delta}-v_{\delta}\big)(z_{0})>0 \quad \text{for some} z_{0}\in\partial{\mathcal{D}}. $$

(9.5)

Applying [9, Proposition 3.7] to:

$$ {\Phi}(x,y) = u_{\delta}(x) - v_{\delta}(y) - \gamma u_{\delta}(z_{0})\langle y-x, \vec{n}(z_{0})\rangle - |x-z_{0}|^{2},\qquad {\Psi}(x,y) = \frac{1}{2}|x-y|^{2}, $$

we obtain a sequence $\{(x_{\alpha }, y_{\alpha })\}_{\alpha \to \infty }$ of maximizers to Φ − αΨ that converges to some (z,z), where z is a maximizer of Φ(z,z) = u_δ(z) − v_δ(z) −|z − z₀|⁴. Hence there must be z = z₀. Also:

$$ \alpha|x_{\alpha}-y_{\alpha}|^{2}\to 0\quad \text{as}\alpha\to\infty. $$

(9.6)

We now apply [9, Theorem 3.2] to:

$$ w(x,y) = u_{\delta}(x) - v_{\delta}(y), \qquad \phi(x,y) = \frac{\alpha}{2}|x-y|^{2} +\gamma u_{\delta}(z_{0})\langle y-x, \vec{n}(z_{0})\rangle, $$

which yields existence of sequences of matrices $X_{\alpha }, Y_{\alpha }\in {\mathbb {R}}^{N\times N}_{\text {sym}}$ satisfying:

$$ \begin{array}{ll} \big(\alpha (x_{\alpha} - y_{\alpha}) - \gamma u_{\delta}(z_{0})\vec{n}(z_{0})+4|x_{\alpha}-z_{0}|^{2}(x_{\alpha}-z_{0}), X_{\alpha}\big) \in J_{\bar{\mathcal{D}}}^{2,+}u_{\delta}(x_{\alpha}),\\ \big(\alpha (x_{\alpha}- y_{\alpha}) - \gamma u_{\delta}(z_{0})\vec{n}(z_{0}), Y_{\alpha}\big) \in J_{\bar{\mathcal{D}}}^{2,-}v_{\delta}(y_{\alpha}), \\ \left[\begin{array}{cc} X_{\alpha}&0 \\ 0& -Y_{\alpha} \end{array}\right] \leq \nabla^{2}\phi(x_{\alpha}, y_{\alpha}) +\frac{1}{\alpha}\nabla^{2}\phi(x_{\alpha}, y_{\alpha})^{2}. \end{array} $$

(9.7)

Since $\nabla ^{2}\phi (x_{\alpha }, y_{\alpha }) = \alpha \left [\begin {array}{cc} Id_{N}& - Id_{N}\\ - Id_{N}& Id_{N} \end {array}\right ] + \mathcal {O}\big (|x_{\alpha }-z_{0}|^{2}\big )$, the last assertion above implies:

$$ \text{trace} (X_{\alpha}-Y_{\alpha})\leq \mathcal{O}\big(|x_{\alpha}-z_{0}|^{2} + \frac{1}{\alpha}|x_{\alpha}-z_{0}|^{4}\big) \quad \to 0\quad \text{as }\alpha\to\infty. $$

(9.8)

Note that for large α ≫ 1 there must be $x_{\alpha }, y_{\alpha }\in {\mathcal {D}}$. Indeed, if $x_{\alpha }\in \partial {\mathcal {D}}$ then Eq. 9.2 and the first assertion in Eq. 9.7 yield the contradiction in:

$$ \begin{array}{@{}rcl@{}} -\delta & \geq & \alpha \langle x_{\alpha} - y_{\alpha}, \vec{n}(x_{\alpha})\rangle - \gamma u_{\delta}(z_{0})\langle \vec{n}(z_{0}), \vec{n}(x_{\alpha})\rangle\\ &&+ 4|x_{\alpha} - z_{0}|^{2}\langle x_{\alpha}-z_{0}, \vec{n}(x_{\alpha})\rangle + \gamma u_{\delta}(x_{\alpha}) \geq -\frac{\alpha}{2r}|x_{\alpha} - y_{\alpha}|^{2} \\&&+ \gamma u_{\delta}(z_{0})\langle \vec{n}(z_{0}), \vec{n}(x_{\alpha})\rangle + \mathcal{O}\big(|x_{\alpha} - z_{0}|^{3}\big) + \gamma u_{\delta}(x_{\alpha}) \\ & \to& 0\quad \text{as }\alpha\to\infty, \end{array} $$

where we used (9.1) for the bound $\langle y_{\alpha } - x_{\alpha }, \vec {n}(x_{\alpha })\rangle \leq \frac {1}{2r}|x_{\alpha } - y_{\alpha }|^{2}$, followed by Eq. 9.8. Similarly, if $y_{\alpha }\in \partial {\mathcal {D}}$ then Eq. 9.3 and the second assertion in Eq. 9.7 brings the contradiction with Eq. 9.5, as:

$$ \begin{array}{@{}rcl@{}} \delta & \leq & \alpha \langle x_{\alpha} - y_{\alpha}, \vec{n}(y_{\alpha})\rangle - \gamma u_{\delta}(z_{0})\langle \vec{n}(z_{0}), \vec{n}(y_{\alpha})\rangle + \gamma v_{\delta}(y_{\alpha})\\ & \leq & \frac{\alpha}{2r}|x_{\alpha} - y_{\alpha}|^{2} - \gamma u_{\delta}(z_{0})\langle \vec{n}(z_{0}), \vec{n}(y_{\alpha})\rangle +\gamma v_{\delta}(y_{\alpha})\\ &\to& \gamma\big(v_{\delta}(z_{0}) - u_{\delta}(z_{0})\big)\quad \text{as} \alpha\to\infty. \end{array} $$

The fact of $x_{\alpha }, y_{\alpha }\in {\mathcal {D}}$ established, we use Eq. 9.7 together with Eqs. 9.2, 9.3, to obtain:

$$ -\text{trace} \big(X_{\alpha} - Y_{\alpha}\big)\leq f(x_{\alpha})-f(y_{\alpha}) - 2N\delta \to -2N\delta\quad \text{as} \alpha\to\infty, $$

contradicting (9.8). This ends the proof of the Lemma. □