Logarithmic Convexity for Supremum Norms of Harmonic Functions

On Three Balls Theorem for Discrete Harmonic Functions

“…It is well known, the standard approach is to prove logarithmic convexity for L 2 -norms and then use elliptic estimates to obtain L ∞ estimates, see [13] for the details. We give another elementary proof that will be extended to discrete situation in the next section.…”

“…The simplest quantitative unique continuation statement is the three balls theorem. For classical harmonic functions it follows from logarithmic convexity of the L 2 -norms, that in turn is obtained using the rotational symmetry and ellipticity of the Laplace operator and can be proved by expansions in eigenfunctions of the Laplace-Beltrami operator on the sphere [13]. However, the logarithmic convexity can be generalized to general elliptic equations and it has been successfully used in unique continuation problems, [1,7].…”

Propagation of Smallness and the Uniqueness of Solutions to Some Elliptic Equations in the Plane

“…Indeed, a similar argument was used already in the proof of inequality (12); see also [9]. Apply now Theorem 2.1 for the harmonic function v, with µ instead of µ, and ν = 0.…”

“…This form (1) appeared in the paper of Korevaar and Meyers [9]. Note that (1) remains true if L ∞ norm is replaced by L 2 norm.…”

Detecting nonlinear corrosion by electrostatic measurements

“…Using the notation introduced in the Proposition 3.1, we define the point P = P 0 − Now, we will use the three spheres inequality for harmonic functions (see for instance [14] or [5,Appendix E]) that is…”