Chapter 3

Harmonic and subharmonic functions611E147E5D10

Classical potential theory and the Dirichlet problem.

Sections and subsections

1. 3.1 Harmonic functions and the Dirichlet problem952EDF2CE146
   1. 3.1.1 The Laplacian and mean values524A2588C8CE
   2. 3.1.12 Fundamental solutions981631AD31A1
   3. 3.1.16 Poisson kernels and Harnack inequalities9C3CB4CF29A2
   4. 3.1.23 Perron envelopes and boundary barriersA3DA602C02CB
   5. 3.1.32 Harmonic measure and Green functions56F22305E141
2. 3.2 Kelvin transforms and half-space potentials9728BD930155
   1. 3.2.1 Inversion and the Kelvin transform9A02E6F8D636
   2. 3.2.4 Nonnegative harmonic functions on a half-space5C12992087C2
   3. 3.2.11 Bounded boundary dataFA92650B78EB
3. 3.3 Classical harmonic analysis4061AEE18413
   1. 3.3.1 Differentiating compact parameter integrals9767EC7AA70A
   2. 3.3.3 Regularity of ball averagesAABF9B330CC1
   3. 3.3.6 Measures on a compact metric boundaryE0A7F26D42BE
   4. 3.3.8 Radial boundary limits from symmetric densities4C55BC86FBE8
   5. 3.3.10 Fine coverings and differentiation on a sphere03E23BFDC019
   6. 3.3.13 Nontangential boundary limits of Poisson integrals3412F805AF75
   7. 3.3.15 Harmonic reflection across a sphereA9175B0B7BA5
4. 3.4 Subharmonic functions0BBC054FD2E1
   1. 3.4.1 Mean inequalities and comparisonABE9A23B5282
   2. 3.4.10 Riesz measures and pointwise representativesD48BD09AF75E
   3. 3.4.16 Compactness and Hartogs' lemmaAA374DAE79FD
   4. 3.4.21 Upper envelopes and countable selectionF1C0C53624DF
   5. 3.4.25 Polar sets and removable singularities7DF84C7DAE9E
5. 3.5 Green functions and potentials16EB8324A3E2
   1. 3.5.1 Harmonic lowering and greatest harmonic minorantsE0853C6A83C4
   2. 3.5.4 Minimal Green functions on general open sets870251679082
   3. 3.5.7 Potentials and global Riesz decomposition47255929FFC0
   4. 3.5.10 Continuity of potentials and bounded densities7AECB61AA15F
   5. 3.5.13 The generalized Laplacian589DA36A9C17
   6. 3.5.15 A global extension finite outside a prescribed ballB78F0B018AE7