By Tarantello G.

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**Additional info for An Harnak inequality for Liouville-type equations with singular sources**

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3 Boundary Problem problem, we show how these unknowns are calculated. To make things simple, we assume that no charges are distributed on the plane. The problem under consideration examines in a section of a disk in which boundaries have fixed potential values FHr, j = 0L = 0, FHr, j = aL = 0, and FHr = R, jL = F0 Hj L. 5. 5. R FHr,j=0L=0 Boundary conditions on a disk segment. The domain G is free of charges. The domain G is free of any charges and the potential FHr, jL is regular and finite for r Ø 0.

5 Exercises 1. Create some pictures for a quadrupole arrangement of charges using the package PointCharge'. Choose the location of the charges in the representation plane of the potential section. What changes are required if your choice of coordinates for the charges is outside the representation plane? Perform some experiments with a larger number of charges. 2. , F0 = sin(j) or F0 = j). What changes occur in the potential if we change the angle a? Examine the influence of the upper summation index N on the accuracy of the solution.

As a consequence, we get n = n p ê a with n = 0, 1, 2, 3, ... The angular part of the solution thus reduces to np hn HjL = An sin H ÅÅÅÅaÅÅÅÅ jL. 41) that bn = 0. 42) where dn = an An . 3 Boundary Problem representations. Values for dn are determined by applying the boundary condition on the circle FHr = R, jL = F0 HjL. 44) or in explicit form, 2 a np Å F HjL sinH ÅÅÅÅaÅÅÅÅ jL d j. 45) includes the boundary condition and only contains known parameters. Thus, we can determine dn 's numerical value if we know the boundary condition and if we specify the index m of the expansion in Eq.

### An Harnak inequality for Liouville-type equations with singular sources by Tarantello G.

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