We investigate the relation between Besov spaces generated by the Dirichlet Laplacian and the Neumann Laplacian in one space dimension from the view point of the boundary value of functions. Derivatives on spaces with such boundary conditions are defined, and it is proved that the derivative operator is isomorphic from one to the other.

The purpose of this paper is to study functions in the unit disk $\mathbb D$ through the family of Toeplitz operators $\{T_{φdσ_{t}}\}_{t∈[0,1)}$, where $T_{φdσ_{t}}$ is the Toeplitz operator acting the Bergman space of $\mathbb D$ and where $dσ_t$ is the Lebesgue measure in the circle $tS^1$. In particular for $1\le p \lt \infty$ we characterize the harmonic functions $φ$ in the Hardy space $h^{p}(\mathbb D)$ by the growth in $t$ of the $p$-Schatten norms of $T_{φdσ_{t}}$. We also study the dependence in $t$ of the norm operator of $T_{adσ_{t}}$ when $a∈H^p_{at}$, the atomic Hardy space in the unit circle with $1/2 \lt p \le 1$.

We give blow-up behavior for a Brezis and Merle's problem with Dirichlet and Hölderian conditions. Also we derive a compactness criterion as in the work of Brezis and Merle.

Our aim is to give a simple estimate of the Bloch constant applying some fundamental facts on complex analysis. Our method is based on the Cauchy estimate, the maximum modulus principle, the Schwarz lemma and the Rouche theorem.

This paper deals with the global existence and stability of solutions of a new class of partial integral equations of Hadamard fractional order.

In this paper, we consider the Neumann problem for the Laplace operator with a given data containing a divergence of a vector field. We demonstrate the existence and regularity of a weak solution. As an application, we consider the existence and regularity of a weak solution in regard to the Maxwell-Stokes type equation.

We propose here a mathematical study of the small oscillations of a heavy viscous liquid in an arbitrary open rigid container supported by an elastic structure, i.e a spring-mass-damper system. From the equations of motion of the system, we deduce a variational formulation of the problem and after, an operatorial equation in a suitable Hilbert space. Then, we can study the spectrum of the problem. At first, we prove that it is formed by eigenvalues that are located in the right half-plane, so that the equilibrium position is stable. Besides, we show that the operator pencil of the problem is a well-known pencil, whose we prove by a simple method that it has two branches of real eigenvalues having as points of accumulation zero and the infinity and a number at most finite of complex eigenvalues. Finally, we prove the existence and the unicity of the solution of the associated evolution problem by means of Lions method. Afterwards, we consider the case where the damper is removed, that is very different. We prove in this case that the equilibrium position is stable, but the problem is reduced to the study of a Krein-Langer pencil, so that in particular, there exist always non oscillatory eigenmotions.