Related papers: Complex interpolation with Dirichlet boundary cond…
In this paper we will discuss the Dirichlet problem of nonlinear second order partial differential equations resolved with any derivatives. First, we transform it into generalized integral equations. Next, we discuss the existence of the…
We give analytic description for the completion of $C_0^\infty ( \mathbf{R}_+)$ in Dirichlet space $D^{1,p}(\mathbf{R}_+, \omega):= \{ u:\mathbf{R}_+\rightarrow \mathbf{R}: u\ \hbox{ is locally absolutely continuous on} \ \mathbf{R}_+ \…
We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of…
Defect of compactness for non-compact imbeddings of Banach spaces can be expressed in the form of a profile decomposition. This paper extends the profile decomposition for Sobolev spaces proved by Solimini (AIHP 1995) to the non-reflexive…
Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except $a^{11}$) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient…
We characterize preduals and K\"othe duals to a class of Sobolev multiplier type spaces. Our results fit in well with the modern theory of function spaces of harmonic analysis and are also applicable to nonlinear partial differential…
We study the interpolation sets for the Hardy-Sobolev spaces defined on the unit ball of ${\bf C}^n$. We begin by giving a natural extension to ${\bf C}^n$ of the condition that is known to be necessay and sufficient for interpolation sets…
We prove uniqueness and stability for the inverse boundary value problem of the two dimensional Schr\"odinger equation. We do not assume the potentials to be continuous or even bounded. Instead, we assume that some of their positive…
We investigate stability properties of indirectly damped systems of evolution equations in Hilbert spaces, under new compatibility assumptions. We prove polynomial decay for the energy of solutions and optimize our results by interpolation…
We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the…
We generalize the Abstract Interpolation Lemma proved by the authors in [2]. Using this extension, we show in a more general context, the persistence property for the generalized Korteweg-de Vries equation, see (1.2), in the weighted…
We study an infinite system of ordinary differential equations that models the evolution of coagulating and fragmenting clusters, which we assume to be composed of identical units. Under very mild assumptions on the coefficients we prove…
We compute the Borel complexity of some classes of Banach spaces such as different versions of diameter two properties, spaces satisfying the Daugavet equation or spaces with an octahedral norm. In most of the above cases our computation is…
A mixed boundary value problem for the Stokes system in a polyhedral domain is considered. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the sides of the polyhedron. The…
A large class of initial-boundary value problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit formulae are presented for the generalized…
We establish necessary and sufficient conditions for the boundedness and compactness of weighted composition operators acting on weighted Dirichlet spaces and determine the spectrum of a certain class of such operators. Our results extend…
The author proves the existence of strong solutions of the Dirichlet problem for the nonstationary Stokes system in polygonal domain. Here, the solutions are elements of weighted Sobolev spaces, where the weight function is a power of the…
We identify explicitly the fractional power spaces for the $L^2$ Dirichlet Laplacian and Dirichlet Stokes operators using the theory of real interpolation. The results are not new, but we hope that our arguments are relatively accessible.
In this paper n-dimensional Sobolev type spaces $ E_{\alpha}^{s,p}(\R^n_+)$ $(\alpha\in \R^n,\;\;\alpha_1> -\frac{1}{2},...,\alpha_n>-\frac{1}{2}, s\in \R, p\in [1,+\infty])$ are defined on $\R^n_+$ by using Fourier-Bessel transform. Some…
The orthogonality of Hilbert spaces whose elements can be represented as simple and double layer potentials is determined. Conditions of well-posed solvability of integral equations for the sum of simple and double layer potentials…