A twice continuously differentiable \(u\colon\Omega\to\RR\) is harmonic if \(\Delta u=u_{xx}+u_{yy}=0\). The real part of a holomorphic function is harmonic. On a simply connected domain every harmonic \(u\) is the real part of a holomorphic function, unique up to an imaginary constant: the form \(-u_y\,dx+u_x\,dy\) is closed, so \(v(z)=\int_{z_0}^{z}(-u_y\,dx+u_x\,dy)\) is independent of the path and \(u+iv\) is holomorphic. Harmonic functions satisfy the mean value property \(u(a)=\frac{1}{2\pi}\int_0^{2\pi}u(a+re^{i\theta})\,d\theta\) for every closed disc \(\abs{z-a}\le r\) in \(\Omega\), and hence the maximum and minimum principles. For \(\varphi\) continuous on the unit circle, the Poisson integral \(u(re^{i\theta})=\frac{1}{2\pi}\int_0^{2\pi}\frac{1-r^2}{1-2r\cos(\theta-t)+r^2}\,\varphi(e^{it})\,dt\) solves the Dirichlet problem on the unit disc: \(u\) is harmonic on \(\DD\) and extends continuously to the closed disc with boundary values \(\varphi\).
Laplacian and harmonic functions (Tag D-CFBSA)
Let \(\Omega\subseteq\RR^2\) be open and let \(u\colon\Omega\to\RR\) be a \(C^2\) function of \((x,y)\). The Laplacian of \(u\) is \[ \Delta u\coloneqq\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}, \] and \(u\) is harmonic on \(\Omega\) if \(\Delta u=0\) on \(\Omega\).
Mean value property for harmonic functions (Tag PR-UWGI6)
Let \(\Omega\subseteq\CC\) be open, let \(u\) be harmonic on \(\Omega\), and let \(z_0=x_0+iy_0\in\Omega\) and \(r>0\) satisfy \(\overline{D_r(z_0)}\subseteq\Omega\). Then \[ u(z_0)=\frac{1}{2\pi r}\oint_{\bd D_r(z_0)}u\,ds=\frac{1}{\pi r^2}\iint_{D_r(z_0)}u(x,y)\dx\dy, \] where \(ds\) is arc length on the circle \(\bd D_r(z_0)\).
Strict maximum principle, real version (Tag PR-6WOTK)
Let \(D\subseteq\CC\) be a connected open set, let \(u\colon D\to\RR\) be harmonic, and let \(M\in\RR\) satisfy \(u(z) \leq M\) for all \(z \in D\). If \(u(z_0)=M\) for some \(z_0 \in D\), then \(u(z)=M\) for all \(z \in D\).
Strict maximum principle for complex-valued harmonic functions (Tag PR-QW3ZK)
Let \(D\subseteq\CC\) be a connected open set and let \(h=u+iv\) be a complex-valued function on \(D\) whose real and imaginary parts are harmonic. If \(\abs{h(z)}\le M\) for all \(z\in D\) and \(\abs{h(z_0)}=M\) for some \(z_0\in D\), then \(h\) is constant on \(D\).
Write \(h(z_0)=Me^{i\varphi}\) and put \(U\coloneqq\Re\big(e^{-i\varphi}h\big)=u\cos\varphi+v\sin\varphi\), a real harmonic function on \(D\). Then \(U\le\abs{h}\le M\) on \(D\) and \(U(z_0)=M\), so by the strict maximum principle for real harmonic functions on a connected open set, \(U\equiv M\). Since \(M=U\le\abs{e^{-i\varphi}h}\le M\), the imaginary part of \(e^{-i\varphi}h\) vanishes, so \(e^{-i\varphi}h\equiv M\) and \(h\equiv Me^{i\varphi}\).
Maximum principle for complex-valued harmonic functions (Tag C-KOFDQ)
Let \(D\subseteq\CC\) be a bounded domain, and let \(h\colon D\to\CC\) be harmonic and extend continuously to \(\overline D\). If \(\abs{h(z)}\le M\) for all \(z\in\bd D\), then \(\abs{h(z)}\le M\) for all \(z\in D\).
Dirichlet problem on the unit disc (Tag D-XR64P)
Let \(u\colon S^1\to\RR\) be bounded and piecewise continuous. The Dirichlet problem on \(\DD\) with boundary data \(u\) asks for a bounded harmonic function \(\tilde u\colon\DD\to\RR\) such that \[ \lim_{\substack{z\to\zeta\\ z\in\DD}}\tilde u(z)=u(\zeta) \] at every point \(\zeta\in S^1\) where \(u\) is continuous. A solution of the Dirichlet problem is such a function \(\tilde u\).
When \(u\) is continuous on \(S^1\), the boundary condition says that \(\tilde u\) extends to a continuous function on \(\overline{\DD}\) that equals \(u\) on \(S^1\).