The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study of $L^p$-theory of the homotopy operator applied to differential forms. Specifically, we will discuss various local and global norm estimates for the homotopy operator T and its compositions with other operators, such as Green’s operator and potential operator.

We establish the Poincaré-type inequalities for the composition of the homotopy operator, exterior derivative operator, and the projection operator with $L^{\phi }\text{-}$norm applied to the nonhomogeneous $A$-harmonic equation in $L^{\phi }(\mathrm{\Omega })$-averaging domains.

The inverse eigenvalue problem is a classical and difficult problem in matrix theory. In the case of real spectrum, we first present some sufficient conditions of a real r-tuple (for $r=2$; 3; 4; 5) to be realized by a symmetric stochastic matrix. Part of these conditions is also extended to the complex case in the case of complex spectrum where the realization matrix may not necessarily be symmetry. The main approach throughout the paper in our discussion is the specific construction of realization matrices and the recursion when the targeted r-tuple is updated to a $(r+1)$-tuple.

We establish the $A_r(D)$-weighted integral inequality for the composition of the Homotopy $T$ and Green’s operator $G$ on a bounded convex domain and also motivated it to the global domain by the Whitney cover. At the same time, we also obtain some $(p,q)$-type norm inequalities. Finally, as applications of above results, we obtain the upper bound for the $L^p$ norms of $T(G(u))$ or $(T(G(u)){)}_B$ in terms of $L^q$ norms of $u$ or $du$.

We introduce a class of variational integrals whose Euler equations are nonhomogeneous $𝒜$-harmonic equations. We investigate the relationship between the minimization problem and the Euler equation and give a simple proof of the existence of some nonhomogeneous $𝒜$-harmonic equations by applying direct methods of the calculus of variations. Besides, we establish some interesting results on variational integrals.

We first establish the local Poincaré inequality with $L^{𝒜}$-averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenous $A$-harmonic equation. Then, according to the definition of $L^{𝒜}$-averaging domains and relative properties, we demonstrate the global Poincaré inequality with $L^{𝒜}$-averaging domains. Finally, we give some illustrations for these theorems.

We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneous $A$-harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equation $d^{\mathrm{\star }}A(x,d\omega )=B(x,d\omega )$ in the weighted variable exponent Sobolev space $W_d^{p(x)}(\mathrm{\Omega },{\mathrm{\Lambda }}^l,\mu ).$

This work is concerned with the abstract Cauchy problems that depend on parameters. The goal is to study continuity in the parameters of the classical solutions of the Cauchy problems. The situation considered in this work is when the operator of the Cauchy problem is not densely defined. By applying integrated semigroup theory and the results on continuity in the parameters of C ₀-semigroup and integrated semigroup, we obtain the results on the existence and continuity in parameters of the classical solutions of the Cauchy problems. The application of the obtained abstract results in a parabolic partial differential equation is discussed in the last section of the paper.