Related papers: Improved quantitative unique continuation for comp…

Quantitative unique continuation principles for multiscale structures are an important ingredient in a number applications, e.g. random Schr\"odinger operators and control theory. We review recent results and announce new ones regarding…

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: \begin{eqnarray}\label{eq00} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla W(t,u(t))\\…

We study positive solutions to the problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $\gamma>0$, $0<q\le p$, $\vartheta\ge0$ and…

In these lectures we present some useful techniques to study quantitative properties of solutions of elliptic PDEs. Our aim is to outline a proof of a recent result on propagation of smallness. The ideas are also useful in the study of the…

This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg -de Vries (KdV)--Benjamin-Bona-Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in…

In this paper we are interested in the blowup of a geometric constant $\mathfrak{C}(\delta)$ appearing in the optimal quantitative unique continuation property for wave operators. In a particular geometric context we prove an upper bound…

We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in addition the…

This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case. where the drift is a…

This paper is concerned with the two-species chemotaxis-competition model with degenerate diffusion, \[\begin{cases} u_t = \Delta u^{m_1} - \chi_1 \nabla\cdot(u\nabla w) + \mu_1 u (1-u-a_1v), &x\in\Omega,\ t>0,\\% v_t = \Delta v^{m_2} -…

In this paper, we study the quantitative unique continuation property of the second-order elliptic operators under the vanishing Neumann boundary condition over $C^{1,\alpha}$ or convex domains in two dimensions. We establish the optimal…

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla…

For the generalized surface quasi-geostrophic equation $$\left\{ \begin{aligned} & \partial_t \theta+u\cdot \nabla \theta=0, \quad \text{in } \mathbb{R}^2 \times (0,T), \\ & u=\nabla^\perp \psi, \quad \psi = (-\Delta)^{-s}\theta \quad…

We investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth manifolds. Quantitative uniqueness is a quantitative form of strong unique continuation property. We characterize…

In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator…

The mathematical theory of a novel variational approximation scheme for general second and fourth order partial differential equations \begin{equation}\label{eq: A} \partial_t u - \nabla\cdot\Big(u\nabla\frac{\delta\phi}{\delta…

We obtain a global unique continuation result for the differential inequality $|(i\partial_t+\Delta)u|\leq|V(x)u|$ in $\mathbb{R}^{n+1}$. This is the first result on global unique continuation for the Schr\"odinger equation with…

Using a physically motivated stress energy tensor, we prove weak and strong monotonicity formulas for solutions to the semilinear elliptic system $\Delta u=\nabla W(u)$ with $W$ nonnegative. In particular, we extend a recent two dimensional…

In this article we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the…

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \[ \left\{\begin{array} [c]{c}% u_{t}-\nu\Delta u+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. \] in $Q_{\Omega,T}=\Omega\times\left(0,T\right) ,$ where…

We present a discrete form of the Wheeler-DeWitt equation for quantum gravitation, based on the lattice formulation due to Regge. In this setup the infinite-dimensional manifold of 3-geometries is replaced by a space of three-dimensional…