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In this thesis, it has been introduced Hardy inequality and it’s extends. The fractional Hardy inequality for Ω ⊆ Rn and f ∈ C ∞ (Ω) is:1f (x) − f ( y)2∫α +n2dx dy ≥ kn,α ∫f (x)dx( )αΩ×Ωx − yΩ M α x In this thesis we are going to introduce the fractional and derivative forms of Hardy inequality then the Hardy inequality will be proved to fractional form on Euclidean and Hyperbolic domains, and finally we will get right into Hardy inequality with remainder.
 

The controllability problem may be formulated roughly as follows. Consider an evolution system (either described in terms of partial or ordinary differential equations) on which we are allowed to act by means of a suitable choice of the control (the right-hand side of the system, the boundary conditions, etc.). Given a time interval 0 < t < T, and initial and final states, the goal is to determine whether there exists a control driving the given initial data to the given final ones in time T. Now, consider the simplest parabolic equation, namely heat equation and suppose that one could act by appropraite controls on this system. The null controllability problem which is one of the very... 

The aim of this work is to study the optimal exponent p to have solvability of semilinear biharmonic problem with a singular term in a smooth and bounded domain such that contains origin in Euclidean space with dimension greater than 4. The singular term is related to the Hardy inequality. First of all, it is not difficult to show that any positive supersolution of problem is unbounded near the origin and then additional hypotheses on p are needed to ensure existence of solutions. We will say that problem blows up completely if the solutions to the truncated problems (with a bounded weight instead of the Hardy singularity) tend to infinity for every x in domain as n goes infinity. The main... 

Singular differential equations have a wide range of applications. Hardy singularities, which are connected to inequalities of the same name and have various extensions, are the most well-known singularities. The application of Hardy inequalities in quantum physics and also in the linearization of reaction-diffusion equations in thermodynamics and combustion theory motivates researchers to examine them. Singularities on a domain's boundary are another well-known type of singularity. In the study of fluid mechanics and pseudoplastic flows, these singularities emerge.Differential equations with coefficients or functions that are simply functions belonging to $ L^1 $, or bounded Radon measures,...