Classification of Nonnegative g −Harmonic Functions in Half-Spaces

Abstract

In this paper we present a short proof of the following classification Theorem for g −harmonic functions in half-spaces. Assume that u is a nonnegative solution to Δ_gu = 0 in {x_n > 0} that continuously vanishes on the flat boundary {x_n = 0}. Then, modulo normalization, u(x) = x_n in {x_n ≥ 0}. Our proof depends on a recent quantitative version of the Hopf-Oleı̆nik Lemma proven by the authors in Braga and Moreira (Adv. Math.334, 184–242, 2018). Moreover, in this paper, we show how to adapt the proofs in the literature to extend Carleson Estimate, Boundary Harnack Inequality and Schwartz Reflection Principle to the context of nonnegative g −harmonic functions. These results are also ingredients for the proof of the main result.

Appendix: Additional Properties of Functions in Sobolev Spaces W 1,G

Proposition 8.1 (Gluing Sobolev Functions)

Let $u\in W^{1,G}(B_{R}^{+})\cap C^{0}(\overline {B}_{R}^{+})$ and $v\in W^{1,G}(B_{R}^{-})\cap C^{0}(\overline {B}_{R}^{-})$ be such that u ≡ v on $\{x_{n}=0\}\cap \overline {B}_{R}$. Let us define

$$ w(x) = \left\{\begin{array}{ll} u(x) & \textrm{ if } \ x\in \overline{B}_{R}^{+}; \\ \\ v(x) & \textrm{ if } \ x\in \overline{B}_{R}^{-}. \end{array}\right.\\ $$

(A.27)

Then, $w\in W^{1,G}(B_{R})\cap C^{0}(\overline {B}_{R}).$

Proof

Clearly, $w\in C^{0}(\overline {B}_{R}).$ We observe that by setting

$$ X(x) = \left\{\begin{array}{ll} \nabla u(x) & \textrm{ if } \ x\in {B}_{R}^{+}; \\ \\ \nabla v(x) & \textrm{ if } \ x\in {B}_{R}^{-}, \end{array}\right. $$

(A.28)

it is immediate that $X\in L^{G}(B_{R};\mathbb {R}^{n})$ and w ∈ L^G(B_R).¹ We claim that X is the weak gradient of w in B_R. In order to show this, we need to prove that for any ${\Phi }\in C_{0}^{\infty }(B_{R}; \mathbb {R}^{n})$ a smooth vector field compactly supported in B_R, we have

$$ {\int}_{B_{R}} \langle{\Phi}(x), X(x) \rangle dx = - {\int}_{B_{R}}w(x)div({\Phi}(x))dx $$

(A.29)

In the case $supp({\Phi })\subset \subset B_{R}^{+}$ or $supp({\Phi })\subset \subset B_{R}^{-}$ it is clear that Eq. A.29 holds. We need to prove that Eq. A.29 holds when ${\Phi }\in C_{0}^{\infty }(B_{R}; \mathbb {R}^{n})$ such that supp(Φ) ∩{x_n = 0}≠∅. Let 0 < s < R so that supp(Φ) ⊂⊂ B_s. Since $u\in W^{1,G}(B_{s}^{+}), v\in W^{1,G}(B_{s}^{-})$ and $W^{1,G}(B_{s}^{\pm })\hookrightarrow W^{1, 1+\delta }(B_{s}^{\pm })$ by Theorem 2.2 in [19] we have by Green’s formula, Theorem 3.43² in [11], that

$$ {\int}_{B_{s}^{+}}\langle\nabla u, {\Phi} \rangle dx + {\int}_{B_{s}^{+}}u(x)div({\Phi}(x))dx = -{\int}_{\{x_{n}=0\}\cap\partial B_{s}^{+}} \langle u(y){\Phi}(y), e_{n}\rangle d\mathcal{H}^{n-1}(y),$$

$$ {\int}_{B_{s}^{-}}\langle\nabla v, {\Phi} \rangle dx + {\int}_{B_{s}^{-}}v(x)div({\Phi}(x))dx = {\int}_{\{x_{n}=0\}\cap\partial B_{s}^{+}} \langle v(y){\Phi}(y), e_{n}\rangle d\mathcal{H}^{n-1}(y).$$

Adding the two identities above and recalling that u ≡ v along $\{x_{n}=0\}\cap \overline {B}_{R}$ we obtain (A.29). □

Applying the Proposition above to u and its odd reflection across {x_n = 0} we immediately obtain

Corollary 8.1

Let $u\in W^{1,G}(B_{R}^{+})\cap C_{vfb}^{0}(\overline {B}_{R}^{+})$ and set

$$ u^{*}(x) = \left\{\begin{array}{ll} u(x) & \textrm{ if } \ x\in \overline{B}_{R}^{+}; \\ \\ -u(R_{n}(x)) & \textrm{ if } \ x\in \overline{B}_{R}^{-}, \end{array}\right.\\ $$

(A.30)

where $R_{n}:\mathbb {R}^{n}\to \mathbb {R}^{n}$ is the reflection across {x_n = 0} given by R_n(x₁,⋯ ,x_n) = (x₁,⋯ ,−x_n).

Then, $u^{*}\in W^{1,G}(B_{R})\cap C^{0}(\overline {B}_{R}).$

Next result is inspired in Lemma 8.1 of [19]. This is a delicate result, that shows the (full) integrability of the gradient up to the flat boundary for nonnegative functions in $W_{loc}^{1,G}$ that continuously vanishes on the flat boundary comes from the property of being a subsolution.

Proposition 8.2 (Integrability of the gradient up to the flat boundary)

Let $u\in C_{vfb}^{0}(\overline {B}_{R}^{+}) \cap W^{1,G}_{loc}(B_{R}^{+})$ such that Δ_gu ≥ 0 in $B_{R}^{+}.$ Then, for any 0 < r < R we have that $u^{+}\in W^{1,G}(B_{r}^{+}).$ In particular, if we set ${u}_{e}^{+}$ to be the extension of u⁺ by zero in $\overline {B}_{R}\setminus B_{R}^{+}$, then, ${u}_{e}^{+}\in W_{loc}^{1,G}(B_{R})\cap C^{0}(\overline {B}_{R})$.

Proof

It is easy to see that $u^{+}\in W_{loc}^{1,G}(B_{R}^{+})$. For the first assertion, we only need to prove that $\nabla u\in L^{G}(B_{r}^{+}).$ Let 0 < r < s < R and u_ϱ := (u − ϱ)⁺ for ϱ > 0 and $\varphi \in C_{0}^{\infty }(B_{s})$ with φ ≡ 1 in $\overline {B}_{r}^{+}$ with 0 ≤ φ ≤ 1. We have $u_{\varrho }\in W_{loc}^{1,G}(B_{R}^{+})$ and we also observe that $\varphi ^{1 + g_{0}}u_{\varrho }\in W_{0}^{1,G}(B_{R}^{+})$ and it has compact support in $B_{s}^{+}$. Hence,

$$ \begin{array}{@{}rcl@{}} 0 & \geq & {\int}_{B_{R}^{+}} H_{g}({\vert \nabla u \vert}) \nabla u \nabla (\varphi^{1 + g_{0}}u_{\varrho}) dx \\ \\ & = & {\int}_{B_{R}^{+} \cap \{ u > \varrho \}} \varphi^{1 + g_{0}} g(\vert \nabla u \vert) \vert \nabla u \vert dx + (1 + g_{0}) {\int}_{B_{R}^{+} \cap \{ u > \varrho \}} \varphi^{g_{0}} u_{\varrho} H_{g}({\vert \nabla {u} \vert}) \nabla {u} \nabla \varphi dx.\\ \end{array} $$

(A.31)

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{R}^{+} \cap \{ u > \varrho \}} \varphi^{1 + g_{0}} G(\vert \nabla u \vert) dx & \leq & {\int}_{B_{R}^{+} \cap \{ u > \varrho \}} \varphi^{1 + g_{0}} g(\vert \nabla u \vert) \vert \nabla u \vert dx \quad (\text {by} \ G_{4})\\ & \leq & (1 + g_{0}) {\int}_{B_{R}^{+} \cap \{ u > \varrho \}} \varphi^{g_{0}} u_{\varrho} g(\vert \nabla {u} \vert) \vert \nabla \varphi \vert dx \quad \text {(by (A.31))} \end{array} $$

Now, by generalized Young inequality G₅), we have

$$ \begin{array}{@{}rcl@{}} \varphi^{g_{0}} u_{\varrho} g(\vert \nabla {u} \vert) \vert \nabla \varphi \vert &\leq& \varepsilon \widetilde{G}(\varphi^{g_{0}}g(\vert \nabla {u} \vert)) + C(\varepsilon){G}(u_{\varrho}|\nabla \varphi|)\\ & \leq & C(\delta)\varepsilon \varphi^{1+g_{0}}\widetilde{G}(g(|\nabla u|) + C(\varepsilon){G}(u_{\varrho}|\nabla \varphi|) \quad (\text {by } G_{2}) \text { and since }\\ && 0\leq \varphi \leq 1)\\ & \leq & C(\delta) g_{0}\varepsilon \varphi^{1+g_{0}}G(|\nabla u|) + C(\varepsilon){G}(u_{\varrho}|\nabla \varphi|) \end{array} $$

Now, choosing ε > 0 small enough (depending only on δ,g₀) and using Eq. A.31, we arrive for a different constant C = C(ε) > 0

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{R}^{+} \cap \{ u > \varrho \}} \varphi^{1 + g_{0}} G(\vert \nabla u \vert) dx & \leq & C(\varepsilon) {\int}_{B_{R}^{+} \cap \{ u > \varrho \}} {G}(u_{\varrho}|\nabla \varphi|) dx \\ & \leq & C(\varepsilon) {\int}_{B_{R}^{+} \cap \{ u > \varrho \}\cap (supp(\varphi))} {G}(|u| |\nabla \varphi|)dx \\ & \leq & C(\varepsilon) {\int}_{B_{R}^{+} \cap \{ u > \varrho \}\cap (supp(\varphi))} {G}\Big(||u||_{L^{\infty}(B_{R}^{+})} \!\cdot \!||\nabla \varphi||_{L^{\infty}(B_{R}^{+})}\Big)dx\\ && \quad (\text {by $G_{1}$}) \\ & \leq & C(\varepsilon)(1+g_{0})\Big(||u||_{L^{\infty}(B_{R}^{+})}+1\Big)^{1+g_{0}} \\ &&\Big(||\nabla \varphi||_{L^{\infty}(B_{R}^{+})}+1\Big)^{1+g_{0}} G(1)\cdot |supp(\varphi)| < \infty \end{array} $$

Now, since G(0) = 0, by letting ϱ → 0⁺ we conclude that

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{r}^{+}} G(\vert \nabla u^{+} \vert) dx &=& {\int}_{B_{r}^{+}} G(\vert \nabla u \chi_{\{ u > 0 \}} \vert) dx \\ & = & {\int}_{B_{r}^{+}} G(\vert \nabla u \vert\chi_{\{ u > 0 \}}) dx \\ & = & {\int}_{B_{r}^{+}} G(\vert \nabla u \vert)\chi_{\{ u > 0 \}} dx \\ &= & {\int}_{B_{r}^{+} \cap \{ u > 0 \}} G(\vert \nabla u \vert) dx \\ & = & \lim\limits_{\rho\to 0^{+}} {\int}_{B_{r}^{+} \cap \{ u > \varrho \}} G(\vert \nabla u \vert) dx\\ & \leq & \lim\limits_{\rho\to 0^{+}} {\int}_{B_{R}^{+} \cap \{ u > \varrho \}} \varphi^{1 + g_{0}} G(\vert \nabla u \vert) dx \quad (\text {once}\ \varphi \equiv 1 \ \text { in } B_{r}^{+}) \\ & \leq & C(\varepsilon)\Big(||u||_{L^{\infty}(B_{R}^{+})}+1\Big)^{1+g_{0}} \Big(||\nabla \varphi||_{L^{\infty}(B_{R}^{+})}+1\Big)^{1+g_{0}} G(1)\\ &&\cdot |supp(\varphi)| <\infty \quad(\text {by $G_{1}$}), \end{array} $$

and this finishes the proof of the first part. For the claims about $u_{e}^{+}$, we observe that clearly $u_{e}^{+}\in C^{0}(\overline {B}_{R}).$ The fact that $u_{e}^{+}\in W_{loc}^{1,G}(B_{R})$ follows from Proposition 8.1. □

We can now obtain an improvement of Corollary 8.1 by dismissing the full integrability of the gradient up to the flat boundary.

Corollary 8.2 (Sobolev regularity of the odd-reflection)

Let $u\in W_{loc}^{1,G}(B_{R}^{+})\cap C_{vfb}^{0}(\overline {B}_{R}^{+})$ such that Δ_gu = 0 in $\mathbb {R}_{+}^{n}$ and set

$$ u^{*}(x) = \left\{\begin{array}{ll} u(x) & \textrm{ if } \ x\in \overline{B}_{R}^{+}; \\ \\ -u(R_{n}(x)) & \textrm{ if } \ x\in \overline{B}_{R}^{-}, \end{array}\right.\\ $$

(A.32)

where $R_{n}:\mathbb {R}^{n}\to \mathbb {R}^{n}$ is the reflection across {x_n = 0} given by R_n(x₁,⋯ ,x_n) = (x₁,⋯ ,−x_n). Then, $u^{*}\in W_{loc}^{1,G}(B_{R})\cap C^{0}(\overline {B}_{R}).$

Proof

Indeed, since $\pm u\in W_{loc}^{1,G}(B_{R}^{+})\cap C_{vfb}^{0}(\overline {B}_{R}^{+})$ and Δ_gu = Δ_g(−u) ≥ 0 in $B_{R}^{+}$, Proposition 8.2 gives that u⁺ and (−u)⁺ = u⁻ are both in $W^{1,G}(B_{r}^{+})$ for any 0 < r < R. In particular, $u=u^{+}-u^{-}\in W^{1,G}(B_{r}^{+})$ for any 0 < r < R. The result now follows from Proposition 8.1. □