Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property: for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that $$ u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y)\, dy,\hskip .1in\text{(Discrete Mean-value property).}\tag{1}$$ what can we say about the harmonicity of u? which kind of harmonicity do we have? $$ $$ Recall that if $(1)$ holds independently on $r_x$ (that is for every $r>0$ such that $B(x,r)\subset \Omega$), then $u$ is harmonic.

Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property: for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that $$ u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y)\, dy,\hskip .1in\text{(Discrete Mean-value property).}\tag{1}\label{1}$$ What can we say about the harmonicity of $u$? which kind of harmonicity do we have?

Recall that if \eqref{1} holds independently of $r_x$ (that is for every $r>0$ such that $B(x,r)\subset \Omega$), then $u$ is harmonic.

Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property: for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that $$ u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y)dy,\hskip .1in\text{(Discrete Mean-value property).}\tag{1}$$ what can we say about the harmonicity of u? which kind of harmonicity do we have? $$ $$ Recall that if (1) holds independently on $r_x$ (that is for every $r>0$ such that $B(x,r)\subset \Omega$), then $u$ is harmonic.

Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property: for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that $$ u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y)\, dy,\hskip .1in\text{(Discrete Mean-value property).}\tag{1}$$ what can we say about the harmonicity of u? which kind of harmonicity do we have? $$ $$ Recall that if $(1)$ holds independently on $r_x$ (that is for every $r>0$ such that $B(x,r)\subset \Omega$), then $u$ is harmonic.