Poisson Equation and Volume Potentials

This chapter analyzes volume potentials and the solvability of the Poisson equation by means of such volume potentials. To go into detail, we first consider the case of weakly singular kernels, and then the case in which the first order partial derivatives of the kernel are also weakly singular. That been done, we turn to singular kernels and, after that, to weakly singular kernels with singular first order partial derivatives. Finally, we prove the basic properties of the corresponding volume potentials in spaces of Hölder continuous functions, in Schauder spaces, and in the Roumieu classes. Although similar properties in Schauder spaces can be found in the classical monographs of Günter (Potential theory and its applications to basic problems of mathematical physics. Translated from the Russian by John R. Schulenberger. Frederick Ungar Publishing Co., New York, 1967) and of Kupradze, Gegelia, Basheleishvili, and Burchuladze (Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North-Holland series in applied mathematics and mechanics, vol 25, Russian edition. North-Holland Publishing Co., Amsterdam, New York, 1979. Edited by V. D. Kupradze), here we prove the corresponding statements with optimal Hölder exponents. To do so, we follow a proof of Miranda (Atti Accad Naz Lincei Mem Cl Sci Fis Mat Natur Sez I (8), 7, 303–336, 1965) and we exploit a known lemma (cf. Majda and Bertozzi, Vorticity and incompressible flow. Cambridge texts in applied mathematics, vol 27, Prop. 8.12, pp 348–350. Cambridge University Press, Cambridge, 2002) for which we provide a proof of Mateu, Orobitg, and Verdera (J Math Pures Appl (9), 91(4), 402–431, 2009). To conclude the chapter, we use the properties proven for the volume potentials to study the solvability of the basic boundary value problems for the Poisson equation.