Related papers: Carleman estimate for complex second order ellipti…

We develop a Fredholm alternative for a fractional elliptic operator~$\mathcal{L}$ of mixed order built on the notion of fractional gradient. This operator constitutes the nonlocal extension of the classical second order elliptic operators…

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If ${\mathcal L}_0=\mbox{div}…

For a second order operator on a compact manifold satisfying the strong H\"ormander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class…

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the…

In this paper, we introduce a Matlab program method to compute Carleman estimate for the fourth order partial differential operator $\gamma\partial_t+\partial_x^4\ (\gamma\in\mathbb{R})$. We obtain two kinds of Carleman estimates with…

This paper is concerned with the quantitative homogenization of $2m$-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\varepsilon)$ convergence rate in $W^{m-1, p_0}$ with…

We study resolvent approximations for elliptic differential nonselfadjoint operators with periodic coefficients in the limit of the small period. The class of operators covered by our analysis includes uniformly elliptic families with…

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\geq 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a…

In this paper we prove a H\"older propagation of smallness for solutions to second order parabolic equations whose general anisotropic leading coefficient has a jump at an interface. We assume that the leading coefficient is Lipschitz…

The computation of Kronecker coefficients is a challenging problem with a variety of applications. In this paper we present an approach based on methods from symplectic geometry and residue calculus. We outline a general algorithm for the…

We prove that the stationary magnetic potential vector and the electrostatic potential entering the dynamic magnetic Schr\"odinger equation can be Lipschitz stably retrieved through finitely many local boundary measurements of the solution.…

In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients, we are interested in a Carleman-type inequality for these solutions satisfying an additional growth condition in elliptic…

In this short note we provide a quantitative version of the classical Runge approximation property for second order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these…

In this paper, we establish optimal hypoelliptic estimates for a class of kinetic equations, which are simplified linear models for the spatially inhomogeneous Boltzmann equation without angular cutoff.

In this article, we present a novel Carleman estimate for ultrahyperbolic operators, in $ \mathbb{R}^m_t \times \mathbb{R}^n_x $. Then, we use a special case of this estimate to obtain improved observability results for wave equations with…

We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order $h^{1/2}$ for certain types of finite-difference schemes are obtained.

This paper is devoted to establishing global $W^{2, p}$ estimate for strong solutions to the Dirichlet problem of uniformly elliptic equations in the non-divergence form where the domain is a Lipschitz polyhedra.

Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…

In this paper we establish well posedness of the Neumann problem with boundary data in $L^2$ or the Sobolev space $\dot W^2_{-1}$, in the half space, for linear elliptic differential operators with coefficients that are constant in the…

We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique…