相关论文: On the nodal set of solutions to Dirac equations

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…

In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers…

We study boundary value problems for bounded uniform domains in $\mathbb{R}^n$, $n\geq 2$, with non-Lipschitz (and possibly fractal) boundaries. We prove Poincar\'e inequalities with trace terms and uniform constants for uniform…

The combined work of Guaraco, Hutchinson, Tonegawa and Wickramasekera has recently produced a new proof of the classical theorem that any closed Riemannian manifold of dimension $n + 1 \geq 3$ contains a minimal hypersurface with a singular…

We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.

We study the singular set of free interface in an optimal partition problem for the Dirichlet eigenvalues. We prove that its upper $(n-2)$-dimensional Minkowski content, and consequently, its $(n-2)$-dimensional Hausdorff measure are…

In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff…

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an…

The article [HPS] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering…

We show that on a compact Riemmanian manifold $(M,g)$, nodal sets of linear combinations of any $p+1$ smooth functions form an admissible $p-$sweepout provided these linear combinations have uniformly bounded vanishing order. This applies…

We analyze the existence and stability of localized solutions in the one-dimensional discrete nonlinear Schr\"{o}dinger (DNLS) equation with a combination of competing self-focusing cubic and defocusing quintic onsite nonlinearities. We…

In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries for operators whose coefficients exhibit small oscillations analogous to the…

We show that Bowen's equation, which characterises the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In…

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable…

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in H^s for s > 1/2. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the…

Let $u$ be a solution to an elliptic equation $\text{div}(A\nabla u)=0$ with Lipschitz coefficients in $\mathbb{R}^n$. Assume $|u|$ is bounded by $1$ in the ball $B=\{|x|\leq 1\}$. We show that if $|u| < \varepsilon$ on a set $ E \subset…

We establish the dimension version of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions $d=2$ or $3$, we obtain the first…

In this paper, we generalize the main results of [1] and [31] to Lorentz spaces, using a simple procedure. The main results are the following. Let $n\geq 3$ and let $u$ be a Leray-Hopf solution to the $n$-dimensional Navier-Stokes equations…

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [21, p. 505, Comm. Math. Phys., 2010]{[RS3]} for the Navier-Stokes equations. Firstly, we prove that the upper box dimension of the potential…

In this paper, we study the singular set of 3-dimensional Navier-Stokes equations. Under the condition$\frac{1}{R^{\frac{3s}{q}+2-s}}\int^{R^{2}}_{0}(\int_{B_{R}}|u|^{q}dx)^{\frac{s}{q}}ds <C,$ for $(q,s)\in\{(2,5),(5,2)\},$ we use the…