Related papers: A remark on unique continuation for the Cauchy-Rie…
We obtain an intertwining relation between some Riemann-Liouville operators of order a in (1,2) connecting through a certain multiplicative identity in law the one-dimensional marginals of reflected completely asymmetric a-stable L\'evy…
In this article, we obtain existence and uniqueness results to some problems involving complex nonlinear fractional differential equations (FDEs) in the closed unit disc of C. By help of these results, we prove that some IVPs for some…
In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of $\omega$-ultradifferentiable functions in the sense of Braun, Meise and Taylor, for…
We develop an operator-theoretical method for the analysis on well posedness of partial differential equations that can be modeled in the form \begin{equation*} \left\{ \begin{array}{rll} \Delta^{\alpha} u(n) &= Au(n+2) + f(n,u(n)), \quad n…
We study the existence and uniqueness of solution of a nonlinear Cauchy problem involving the $\psi$-Hilfer fractional derivative. In addition, we discuss the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of its solutions. A few examples…
We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and…
An abstract version of the celebrated inequality is described by means of the spectral bound of an operator defined on a Banach lattice. As a consequence, uniqueness and continuous dependence results for the general semilinear problem…
We study the following quasilinear partial differential equation with two subdifferential operators: $${\frac{\partial u}{\partial s}(s,x)} + (\mathcal{L}u)(s,x,u(s,x),(\nabla u(s,x))^\ast\sigma(s,x,u(s,x))) + f(s,x,u(s,x),(\nabla…
We prove weighted $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators in higher dimensions. The vector-valued extensions of these inequalities are also given.
In this article we prove quantitative unique continuation results for wave operators of the form $\partial$ 2 t -- div(c(x)$\nabla$$\bullet$) where the scalar coefficient c is discontinuous across an interface of codimension one in a…
This paper provides a connection between two distinct branches of research in CR geometry -- namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR…
We obtain a vanishing result for solutions of the inequality $|\Delta u|\le q_1|u|+q_2|\nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is…
In this work, a boundary value problem for Sturm-Liouville operator with discontinuous coefficient is examined. The main equation is obtained which has an important role in solution of inverse problem for boundary value problem and…
Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D…
We quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product we obtain a new stability estimate in the linear case. We also show the…
We investigate differentiability of functions defined on regions of the real quaternion field and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as…
The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}} {U}(t,x) + \mathbb{A}(D) {U} (t,x)=H(t,x) $. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of…
We prove an analog of the quaternionic Borel-Pompieu formula in the sense of proportional fractional $\psi$-Cauchy-Riemann operators via Riemann-Liouville derivative with respect to another function.
Based on the three-ball inequality and the doubling inequality established in [23], we quantify the strong unique continuation established by Koch and Tataru [21] for elliptic operators with unbounded lower-order coefficients. We also…
The main objective of this article is to discuss the local existence of the solution to an initial value problem involving a non-linear differential equation in the sense of Riemann-Liouville fractional derivative of order $\sigma\in(1,2),$…