Separable anisotropic differential operators in weighted abstract spaces and applications

In this paper operator-valued multiplier theorems in Banach-valued weighted Lp spaces are studied. Also weighted Sobolev-Lions type spaces are discussed when E₀, E are two Banach spaces and E₀ is continuously and densely embedded on E. Several conditions are found that ensure the continuity of the embedding operators that are optimally regular in these spaces in terms of interpolations of E₀. These results permit us to show the separability of the anisotropic differential operators in an E-valued weighted Lp space. By using these results the maximal regularity of infinite systems of quasi elliptic partial differential equations are established.     

Maximal B-regular boundary value problems with parameters

This study focuses on non-local boundary value problems (BVP) for elliptic differential-operator equations (DOE) defined in Banach-valued Besov (B) spaces. Here equations and boundary conditions contain certain parameters. This study found some conditions that guarantee the maximal regularity and fredholmness in Banach-valued B-spaces uniformly with respect to these parameters. These results are applied to non-local boundary value problems for a regular elliptic partial differential equation with parameters on a cylindrical domain to obtain algebraic conditions that guarantee the same properties.     

On strong solutions of a Robin problem modelling heat conduction in materials with corroded boundary

Motivated by a practical problem on a corrosion process, we shall study a third kind of BVP for a large class of elliptic equations in vector-valued L_p spaces. Particularly we will determine optimal spaces for boundary data and get maximal regularity for inhomogeneous equations. Then based on these results we shall treat some nonlinear problems. Our approach will be based on the semigroup theory, the interpolation theory of Banach spaces, fractional powers of positive operators, operator-valued Fourier multiplier theorems and the Banach fixed point theorem.     

Maximal B-Regular Integro-Differential Equation

By using Fourier multiplier theorems, the maximal B-regularity of ordinary integro-differential operator equations is investigated. It is shown that the corresponding differential operator is positive and satisfies coercive estimate. Moreover, these results are used to establish maximal regularity for infinite systems of integro-differential equations.     

Coercive boundary value problems for regular degenerate differential-operator equations

This study focuses on nonlocal boundary value problems (BVP) for degenerate elliptic differential-operator equations (DOE), that are defined in Banach-valued function spaces, where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients. Several conditions obtained, that guarantee the maximal L_p regularity and Fredholmness. These results are also applied to nonlocal BVP for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties.     

Estimates of approximation numbers and applications

In this study, the estimates of approximation numbers of embedding operators in weighted spaces have been analyzed. These estimates depend on orders of differential operators, dimensions of function spaces and weighted functions. This fact implies that the associated embedding operators belong to Schatten class of compact operators. By using these estimates, the discreetness of spectrum and completion of root elements relating to principal nonselfedjoint degenerate differential operators is obtained.     

Embedding Theorems in B-Spaces and Applications

This study focuses on the anisotropic Besov-Lions type spaces B^lp,θ（Ω;E0,E） associated with Banach spaces E0 and E. Under certain conditions, depending on l =（l1,l2,…,ln）and α=（α1,α2,…,αn）,the most regular class of interpolation space Eα between E0 and E are found so that the mixed differential operators D^α are bounded and compact, from B^l＋s p,θ（Ω;E0,E） to B^s p,θ（Ω;Eα）.These results are applied to concrete vector-valued function spaces and to anisotropic differential-operator equations with parameters to obtain conditions that guarantee the uniform B separability with respect to these parameters. By these results the maximal B-regularity for parabolic Cauchy problem is obtained. These results are also applied to infinite systems of the quasi-elliptic partial differential equations and parabolic Cauchy problems with parameters to obtain sufficient conditions that ensure the same properties.     

Linear and nonlinear nonlocal boundary value problems for differential-operator equations

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal L_p -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.     

Singular perturbation degenerate problems occurring in atmospheric dispersion of pollutants

The boundary value problems for the degenerate differential-operator equations with small parameters generated on all boundary are studied. Several conditions for the separability and the fredholmness in Banach-valued L_p-spaces of are given. In applications, maximal regularity of degenerate Cauchy problem for parabolic equation arising in atmospheric dispersion of pollutants studied.     

Compact embedding in Besov spaces and B-separable elleptic operators

Necessary and sufficient conditions for compactness of sets in Banach space valued Besov class Bp,q s(Ω;E) is derived.The embedding theorems in Besov-Lions type spaces B l,s p,q(Ω;E0,E) are studied,where E0,E are two Banach spaces and E 0  E.The most regular class of interpolation space E α,between E 0 and E are found such that the mixed differential operator D α is bounded and compact from B p,q l,s (Ω;E 0,E) to B p,q s (Ω;E α) and Ehrling-Nirenberg-Gagliardo type sharp estimates established.By using these results the separability of differential operators with variable coefficients and the maximal B-regularity of parabolic Cauchy problem are obtained.In applications,the infinite systems of the elliptic partial differential equations and parabolic Cauchy problems are studied.     

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coeffcients in Banach-valued Lp-spaces are studied.To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations,the suffcient conditions are founded.By using these facts,the unique continuation properties are established.In the application part,the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.     

Free Boundary Value Problems for Abstract Elliptic Equations and Applications

The free boundary value problems for elliptic differential-operator equations are studied. Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L _p -spaces are given. In application, the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.     

Abstract Elliptic Equations with Integral Boundary Conditons

This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in $L_{p}$ spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed $L_{\mathbf{p}}$ norm.     

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

The weighted Sobolev-Lions type spaces W pl,γ(Ω; E0, E) = W pl,γ(Ω; E) ∩ Lp,γ (Ω; E0) are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω∈ Rn in W pl,γ(; E0, E) is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ(Ω; E0, E) to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

Singular functional differential equations of neutral type in Banach spaces

The well-posedness of a large class of singular partial differential equations of neutral type is discussed. Here the term singularity means that the difference operator of such equations is nonatomic at zero. This fact offers many difficulties in applying the usual methods of perturbation theory and Laplace transform technique and thus makes the study interesting. Our approach is new and it is based on functional analysis of semigroup of operators in an essential way, and allows us to introduce a new concept of solutions for such equations. Finally, we study the well-posedness of a singular reaction-diffusion equation of neutral type in weighted Lebesgue's spaces.     

On quasispectral maximal subspaces of a class of Volterra-type operators

The concept of quasispectral maximal subspaces for quasinilpotent (but not nilpotent) operators was introduced by M. Omladi\v{c} in 1984. As an application a class of quasinilpotent operators on -spaces, close to the Volterra kernel operator, was studied. In the present Banach function space setting we determine all quasispectral maximal subspaces of analogues of such operators and prove that these subspaces are all the invariant bands. An example is given showing that (in general) they are not all the closed, invariant ideals of the operator.