We start this chapter by introducing the embedded property of normed spaces. Next the Sobolev-Gagliardo-Nirenberg inequality and its proof are given. We then define the cone, Lipschitz and Cm class conditions and relation between them is analyzed by an example. The remaining part is dedicated to Sobolev embedding theorems with further inequalities such as Sobolev Poincar´e theorem, Poincar´e inequality and logarithmic Sobolev inequalities. Compact embedding and embedding in Hs(Rn) Sobolev spaces are investigated in the final part of the chapter.
Keywords: Embedding property of normed spaces, Sobolev embedding theorems, Sobolev-Gagliardo-Nirenberg inequality, Poincar´e inequality, Cm condition, Sobolev Poincar´e theorem, compact embedding, cone condition, logarithmic Sobolev inequality, interpolation inequality.