Related papers: Counter example to Strichartz estimates for the wa…

Consider the metric cone $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$ where the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. Let $\Delta_g$ be the Friedrich extension positive…

Consider the following equation $$\partial_t u_t(x)=\frac{1}{2}\partial _{xx}u_t(x)+\lambda \sigma(u_t(x))\dot{W}(t,\,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution…

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\geq 3$. In particular the so called the interior determination problem. This non-linear wave…

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $\Omega\subset {\bf R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric…

We consider the time-harmonic scalar wave scattering problems with Dirichlet, Neumann, impedance and transmission boundary conditions. Under this setting, we analyze how sensitive diffracted fields and Cauchy data are to small perturbations…

Using coordinates $(x,y)\in \mathbb R\times \mathbb R^{d-1}$, we introduce the notion that an unbounded domain in $\mathbb R^d$ is star shaped with respect to $x=\pm \infty$. For such domains, we prove estimates on the resolvent of the…

We consider the wave equation with a distributional Dirac damping and Dirichlet boundary conditions on a compact interval. It is shown that the spectrum of the corresponding wave operator is fully determined by zeroes of an entire function.…

We consider a second order self-adjoint operator in a domain which can be bounded or unbounded. The boundary is partitioned into two parts with Dirichlet boundary condition on one of them, and Neumann condition on the other. We assume that…

We firstly prove Strichartz estimates for the fractional Schr\"odinger equations on $\mathbb{R}^d$ endowed with a smooth bounded metric $g$. We then prove Strichartz estimates for the fractional Schr\"odinger and wave equations on compact…

We study the defocusing semilinear wave equation in ${\mathbb{R}}\times{\mathbb{R}}^2\backslash{\mathcal K}$ with the Dirichlet boundary condition, where ${\mathcal K}$ is a star-shaped obstacle with smooth boundary. We first show that the…

The classical Rellich inequalities imply that the $L^2$-norms of the normal and tangential derivatives of a harmonic function are equivalent. In this note, we prove several refined inequalities, which make sense even if the domain is not…

We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to…

In this paper, we show that if the bounded solutions to the parabolic Dirichlet problem on a Lipshitz-$\left[1,\frac{1}{2}\right]$ domain obey a Carleson measure estimate, then the corresponding parabolic measure on the boundary will belong…

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for…

We prove scattering for the defocusing energy-critical non-linear wave equation with Dirichlet boundary conditions outside two strictly convex obstacles in dimension three. This is the first large data scattering result for such an equation…

In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\gg\phi=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$…

We study bounded pseudoconvex domains in complex Euclidean spaces. We find analytical necessary conditions and geometric sufficient conditions for a domain being of trivial Diederich--Forn\ae ss index (i.e. the index equals to 1). We also…

Consider three-dimensional fractional MHD equations in an exterior domain with the Dirichlet boundary condition assumed. Asymptotic behaviours of weak solutions to the three-dimensional exterior fractional MHD equations are studied. $L^2$…

We consider the wave equation in a bounded domain (eventually convex). Two kinds of inequality are described when occurs trapped ray. Applications to control theory are given. First, we link such kind of estimate with the damped wave…

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…