Inégalité de Sobolev fractionnaire

Details

Supervisors

Ponce, Augusto

Faculty

Faculté des sciences

Degree label

Master [120] en sciences mathématiques, à finalité approfondie

Abstract

enThe objective of this masters’ thesis is to explore an analogue result to the Sobolev inequality in the context of fractional Sobolev spaces. We will define these spaces using the Gagliardo semi-norm and introducing an object playing the role of a fractional gradient in this environment. The primary result of this thesis is to reconstruct the Sobolev inequality for this new object, the properties of which we will discuss in detail. We will prove this inequality in two disjoint cases. The first case, if p>1, relies on an estimation of the Riesz kernel. As this estimation is not valid for p=1, we study a counterpart of the vector-valued Riesz kernel estimation which we will adapt to the fractional gradient. We finish by studying a counter-example proving that this result is not true in dimension 1 for p=1.