Generalizations On Several Inequalities In The Sobolev Spaces With Singularities

The thesis mainly investigate H?lder's and Minkoswki's inequalities as well as in-terpolation inequalities of sum form in the Sobolev spaces with conical singularities.In addition to this,we get the generalizations of Troisi's inequality in the Sobolev spaces with singularities of cone,wedge and corner type.To be specific,the first two inequali-ties are generalized via the motivations from their proofs in the classical cases.For the interpolation inequalities,however,it will be a little more complex,in which we estab-lish in the first place the one of dimension 1 in the sense of general derivatives with high orders,then another one in the form of Fuchs derivatives of dimension 1 and high or-ders,and also a relation between Fuchs derivatives and general ones.Based upon these,the desired interpolation inequalities is obtained on cubes in R_+×R~n.Eventually,the well-known Besicovitch cover lemma allows us to derive the result on a general domain included in R_+×R~n.As for the Troisi's inequality which has a close connection with Sobolev embedding inequalities,there is a method depending on properties of calculus to acquire its generalizations in the spaces of conical singularities,wedge singularities,and corner singularities.