Operators In Sobolev Spaces Research Articles

Continuity of families of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces, elliptic regularity, Green's formula and trace theorems for Sobolev spaces, well-posed boundary conditions, duality of spaces and operators in Hilbert space, and the interpolation theorem for operators in Sobolev spaces.

Existence Results for Nonlinear Elliptic Equations with Leray–Lions Operators in Sobolev Spaces with Variable Exponents

Existence Results for Nonlinear Elliptic Equations with Leray–Lions Operators in Sobolev Spaces with Variable Exponents

Existence of Solutions for a Class of Fractional Kirchhoff-type Systems in $\mathbb{R}^N$ with Non-standard Growth

This paper is concerned with the existence and multiplicity of nontrivial solutions for a class of Kirchhoff-type systems in $\mathbb{R}^N$ involving the fractional pseudo-differential operators defined as the generalizations of the $p(x)$-Laplace operator. Our main tools come from a direct variational methods, the Mountain Pass Theorem, the symmetric Mountain Pass Theorem and the Fountain Theorem in critical point theory. The obtained results of this note significantly contribute to the study of Kirchhoff-type systems in the sense that our situation covers not only differential operators of fractional order but also nonhomogeneous differential operators in Sobolev spaces with variable exponent.

On Variations of the Neumann Eigenvalues of p-Laplacian Generated by Measure Preserving Quasiconformal Mappings

We study variations of the first nontrivial eigenvalue of the two-dimensional p-Laplace operator, p > 2, generated by measure preserving quasiconformal mappings. The study is based on the geometric theory of composition operators in Sobolev spaces and sharp embedding theorems. Using a sharp version of the reverse Holder inequality, we obtain a lower estimate for the first nontrivial eigenvalue in the case of Ahlfors type domains.

Sobolev regularity for commutators of the fractional maximal functions

In this paper the Sobolev regularity properties are investigated of the commutators of fractional maximal functions, both in the global and local case. Some new bounds for the derivatives of the above commutators will be established. As several applications, the boundedness for these operators in Sobolev spaces as well as the bounds of these operators on the Sobolev spaces with zero boundary values in the local setting are obtained.

Minimality of eigenfunctions and associated functions of ordinary differential operators

Let E, F, H be Banach spaces, $$A,B\in L(E,F)$$ such that $$A+\lambda B$$ and its adjoint $$A^*+\lambda B^*$$ have a biorthogonal system of eigenvectors and associated vectors $$(y_i)_{i=1}^\infty$$ and $$(v_i)_{i=1}^\infty$$ . It is assumed that a closed finite codimensional subspace $$E_0$$ of E exists such that $$E_0$$ is dense in H and such that the restriction of B to $$E_0$$ has a continuous extension to H. It is shown that a subsystem $$(y_i)_{i=k+1}^\infty$$ is minimal in H if $$(v_i)_{i=1}^k$$ has a certain basis property. This abstract result can be applied to regular ordinary linear differential operators in Sobolev spaces whose boundary conditions may depend linearly on the eigenvalue parameter for proving minimality in $$L_p$$ . Explicit examples for Sturm–Liouville problems with one or both boundary conditions depending on $$\lambda$$ are considered.

Regularity and Continuity of Local Multilinear Maximal Type Operators

This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators, $$\begin{aligned} {\mathfrak {M}}_{\alpha ,\Omega }(\vec {f})(x)=\sup \limits _{0<r<\mathrm{dist}(x,\Omega ^c)}\frac{r^\alpha }{|B(x,r)|^m}\prod \limits _{i=1}^m\int _{B(x,r)}|f_i(y)|\mathrm{{d}}y,\quad \hbox {for \ }0\le \alpha <mn, \end{aligned}$$ where $$\Omega $$ is a subdomain in $${\mathbb {R}}^n$$ , $$\Omega ^c={\mathbb {R}}^n\setminus \Omega $$ and B(x, r) is the ball in $${\mathbb {R}}^n$$ centered at x with radius r. Several new pointwise estimates for the derivative of the local multilinear maximal function $${\mathfrak {M}}_{0,\Omega }$$ and the fractional maximal functions $${\mathfrak {M}}_{\alpha ,\Omega }$$ $$(0<\alpha < mn)$$ will be presented. These estimates will not only enable us to establish certain norm inequalities for these operators in Sobolev spaces, but also give us the opportunity to obtain the bounds of these operators on the Sobolev space with zero boundary values.

Continuity of composition operators in Sobolev spaces

We prove that all the composition operators Tf(g):=f∘g, which take the Adams-Frazier space Wpm∩W˙mp1(Rn) to itself, are continuous mappings from Wpm∩W˙mp1(Rn) to itself, for every integer m≥2 and every real number 1≤p<∞. The same automatic continuity property holds for Sobolev spaces Wpm(Rn) for m≥2 and 1≤p<∞.

Nonhomogeneous p(x)-Laplacian Steklov problem with weights

ABSTRACTThis paper is concerned with a weighted Steklov problem involving the -Laplacian operator in Sobolev spaces with variable exponents Our approach is based on variational method and Ekeland's principle, we establish that the above problem admits a nontrivial weak solution under appropriate conditions.

Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation

We model the electrical behavior of several biological cells under external stimuli by extending and computationally improving the multiple traces formulation introduced in Henríquezet al.[Numer. Math.136(2016) 101–145]. Therein, the electric potential and current for a single cell are retrieved through the coupling of boundary integral operators and non-linear ordinary differential systems of equations. Yet, the low-order discretization scheme presented becomes impractical when accounting for interactions among multiple cells. In this note, we consider multi-cellular systems and show existence and uniqueness of the resulting non-linear evolution problem in finite time. Our main tools are analytic semigroup theory along with mapping properties of boundary integral operators in Sobolev spaces. Thanks to the smoothness of cellular shapes, solutions are highly regular at a given time. Hence, spectral spatial discretization can be employed, thereby largely reducing the number of unknowns. Time-space coupling is achievedviaa semi-implicit time-stepping scheme shown to be stable and second order convergent. Numerical results in two dimensions validate our claims and match observed biological behavior for the Hodgkin–Huxley dynamical model.

Fundamental solutions of nonlocal Hörmander’s operators II

Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$: \[\mathcal{L}^{(\alpha)}_{\sigma,b}f(x):=\mbox{p.v.}\int_{|z|<\delta}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}\,\mathrm{d}z+b(x)\cdot\nabla f(x)+{\mathscr{L}}f(x),\] where $\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ and $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are smooth functions and have bounded partial derivatives of all orders greater than $1$, $\delta$ is a small positive number, p.v. stands for the Cauchy principal value and ${\mathscr{L}}$ is a bounded linear operator in Sobolev spaces. Let $B_{1}(x):=\sigma(x)$ and $B_{j+1}(x):=b(x)\cdot\nabla{B}_{j}(x)-\nabla{b(x)}\cdot B_{j}(x)$ for $j\in\mathbb{N}$. Suppose $B_{j}\in C_{b}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ for each $j\in\mathbb{N}$. Under the following uniform Hörmander’s type condition: for some $j_{0}\in\mathbb{N}$, \[\inf_{x\in\mathbb{R}^{d}}\inf_{|u|=1}\sum_{j=1}^{j_{0}}|uB_{j}(x)|^{2}>0,\] by using Bismut’s approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator $\mathcal{L}^{(\alpha)}_{\sigma,b}$. In particular, we answer a question proposed by Nualart [Sankhyā A 73 (2011) 46–49] and Varadhan [Sankhyā A 73 (2011) 50–51].

On the equivalence of the electromagnetic problem of diffraction by an inhomogeneous bounded dielectric body to a volume singular integro-differential equation

The paper is concerned with the smoothness of the solutions to the volume singular integrodifferential equations for the electric field to which the problem of electromagnetic-wave diffraction by a local inhomogeneous bounded dielectric body is reduced. The basic tool of the study is the method of pseudo-differential operators in Sobolev spaces. The theory of elliptic boundary problems and field-matching problems is also applied. It is proven that, for smooth data of the problem, the solution from the space of square-summable functions is continuous up to the boundaries and smooth inside and outside of the body. The results on the smoothness of the solutions to the volume singular integro-differential equation for the electric field make it possible to resolve the issues on the equivalence of the boundary value problem and the equation.

Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell’s equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.

Integrodifferential Equations of the Vector Problem of Electromagnetic Wave Diffraction by a System of Nonintersecting Screens and Inhomogeneous Bodies

The vector problem of electromagnetic wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire spaceR3but rather everywhere except for the screen edges. The original boundary value problem for Maxwell’s equations system is reduced to a system of integrodifferential equations in the regions occupied by the bodies and on the screen surfaces. The integrodifferential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator.

Maximal potentials, maximal singular integrals, and the spherical maximal function

We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.

On the continuity of discrete maximal operators in Sobolev spaces

We investigate the continuity of discrete maximal operators in Sobolev space W 1;p (R n ). A counterexample is given as well as it is shown that the continuity follows under certain sucient assumptions. Especially, our research verifies that for the continuity in Sobolev spaces the role of the partition of the unity used in the construction of the maximal operator is very delicate.

Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation

We exhibit the optimal constant for Sobolev inequalities in Lorentz spaces for a mean oscillation, and its relation with a boundedness of the Hardy–Littlewood maximal operator in Sobolev spaces. Some applications to a scale invariant form of Hardyʼs inequality in a limiting case are also considered.

Sobolev spaces, Lebesgue points and maximal functions

In this note, we study boundedness of a large class of maximal operators in Sobolev spaces that includes the spherical maximal operator. We also study the size of the set of Lebesgue points with respect to convergence associated with such maximal operators.

Normally resolvable operators and direct decompositions of Sobolev spaces

We present an approach to the study of some normally resolvable differential operators in Sobolev spaces. Within the framework of this approach, we obtain decompositions of some function spaces and, in particular, generalize the Weyl–Helmholtz decomposition. The key role is played by the assertion about the complementability of subspaces of a normed space. Bibliography: 15 titles.

A note on immersions of domains of fractional powers of certain sectorial operators in Sobolev spaces

In this note we give a proof of a result on immersions of domains of fractional powers of certain sectorial operators associated to strongly elliptic operators in Sobolev spaces; such immersions preserve information on fractional derivatives. We also briefly comment on the application of this result to a problem of optimal control of mosquito populations.