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Potential theory studies harmonic functions, subharmonic functions, and the Laplace operator, including harmonic measure, regular boundary points, and the Dirichlet problem. The Brownian representation of the Dirichlet problem connects these analytic objects with exit probability distributions.

A harmonic function on a punctured ball that is bounded near the missing point extends uniquely to a harmonic function on the whole ball. The extension can be defined by the mean value property for harmonic functions; interior estimates then give smoothness across the point.

In the unit half-cylinder $z\ge 0$, specifying only the curved-side boundary values does not determine a bounded harmonic function. The displayed nonzero function, where $j_{0,1}$ is a positive zero of the Bessel function of the first kind $J_0$, is regular at the axis, bounded, harmonic, and zero on $r=1$. It may be added to any particular solution without changing the side data, and even decays at infinity. Boundary information at the base $z=0$ is still required for uniqueness.

For a Gaussian free field with boundary heights $-\lambda$ and $+\lambda$ on the two sides of the starting point, its distinguished zero-height interface has chordal ${\mathrm{SLE}}_4$ law. Conditional on an initial curve segment, the field is a zero-boundary field in the slit domain plus $m_t(z)=\lambda -(2\lambda /\pi )\arg (g_t(z)-W_t)$. Its mean bracket equals minus the variation of its conditional Green covariance. Subtracting the initial harmonic function gives the corresponding coupling to a zero-boundary field.

For a spatial initial slice with vanishing extrinsic curvature, the vacuum momentum constraint is automatic and the Hamiltonian constraint requires zero Ricci scalar. In three spatial dimensions, the indicated conformally flat metric has $\mathcal{R}=-8{\psi }^{-5}\Delta \psi$, with a nonzero conformal factor. Thus a harmonic function supplies the data on its domain. A positive example is $\psi =1+a/r$ on a punctured or exterior region. There is no nonconstant globally smooth example on all of Euclidean three-space tending to one at infinity, by the maximum principle for harmonic functions. Time symmetry of an initial slice does not imply that its entire evolved spacetime is static.

A spherical squirmer with tangential surface slip velocity $(I-nn)(A+Bn)$, where $B$ is a symmetric second-rank tensor and a traceless second-rank tensor, translates with $U=-2A/3$ and does not rotate in an unbounded quiescent fluid. With $S=x\cdot Bx$ its exterior Stokes flow isThis follows from the Unscaled Papkovich–Neuber representation using harmonic functions as potentials $\Phi =-a^{2}Bx/(2r^3)$ and $\chi =a^{3}A\cdot x/(3r^3)+a^{4}S/(2r^5)$. Matching the surface coefficients proves the formula. The two modes generate a potential dipole and a stresslet, respectively.

For planar Brownian motion started at $(x,y)$ with $y>0$, the probability that its first hit of the real axis lies in $(0,\infty )$ isThis is the bounded harmonic function in the upper half-plane with boundary values zero on the negative half-axis and one on the positive half-axis.

For a smooth harmonic function on a manifold without boundary, assume its nonzero superlevel sets are compact. On a regular positive superlevel set $\{U>\epsilon \}$, integration by parts gives $\int |\nabla U{|}^2=-\epsilon {\int }_{\partial \{U>\epsilon \}}|\nabla U|\le 0$. Applying the same argument to $-U$ proves $U=0$. This avoids assuming an unstated decay rate for the boundary flux at infinity.

An element $h\in H_{\mathrm{loc}}^1(U)$ is weakly harmonic on an open set $U$ when ${\int }_U\nabla h\cdot \nabla \varphi =0$ for every test function $\varphi \in C_c^{\infty }(U)$. This is the statement $\Delta h=0$ in distributions. It requires no regularity of the domain boundary. The Weyl lemma, a form of elliptic regularity, identifies such an element with a smooth harmonic function.

The harmonic function $v(x,y)=xy$ vanishes on both axes but is nonzero in the quadrant, and $v_z=-iz/2$. More generally $\mathrm{Im}{z}^{2m}$, for a positive integer $m$, vanishes on both axes and is harmonic. Adding these unbounded functions leaves all edge Dirichlet boundary conditions unchanged. Thus decay of the boundary data does not imply decay of every solution, and an edge-data-only formula cannot determine an arbitrary solution's Wirtinger derivative. A boundedness or suitable growth condition must select the solution.