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In this paper, we study certain inequalities and a related result for weighted Sobolev spaces on Hölder-α domains, where the weights are powers of the distance to the boundary. We obtain results regarding the divergence equation's solvability, and the improved Poincaré, the fractional Poincaré, and the Korn inequalities. The proofs are based on a local-to-global argument that involves a kind of atomic decomposition of functions and the validity of a weighted discrete Hardy-type inequality on trees. The novelty of our approach lies in the use of this weighted discrete Hardy inequality and a sufficient condition that allows us to study the weights of our interest. As a consequence, the assumptions on the weight exponents that appear in our results are weaker than those in the literature. [ABSTRACT FROM AUTHOR]

We consider the boundary value problem associated with the divergence operator on a bounded regular subset of R n , with homogeneous Dirichlet boundary condition. We prove the existence of a classical solution under slight assumptions on the datum. [ABSTRACT FROM AUTHOR]

We consider the equation $$\text{ div}\,\mathbb{Y }=f$$, with $$f$$ a zero average function on the torus $$\mathbb{T }^d$$. In their seminal paper, Bourgain and Brezis [J Am Math Soc 16(2):393-426, (electronic)] proved the existence of a solution $$\mathbb{Y }\in W^{1,d}\cap L^\infty $$ for a datum $$f\in L^d$$. We extend their result to the critical Sobolev spaces $$W^{s,p}$$ with $$(s+1)p=d$$ and $$p\ge 2$$. More generally, we prove a similar result in the scale of Triebel-Lizorkin spaces. We also consider the equation $$\text{ div} \,\mathbb{Y }=f$$ in a bounded domain $$\varOmega $$ subject to zero Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

We study the solvability of the divergence equation in weighted spaces and Lebesgue spaces with variable exponents, where the weights are so called Muckenhoupt weights. The question of constructing divergence free test functions, which can be used for problems arising in fluid dynamics, is also addressed. The approach is based on an explicit representation formula for solutions of the divergence equation due to Bogovskiĭ and the theory of singular integral operators. The developed methods are used to prove an existence result for fluids which satisfy a p(·)-growth condition. [ABSTRACT FROM AUTHOR]