Quantitative uniqueness of solutions to parabolic equations

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 quantitative uniqueness by the rate of vanishing. We can obtain the vanishing order of solutions by $C^{1, 1}$ norm of the potential functions, as well as the $L^\infty$ norm of the coefficient functions. Some quantitative Carleman estimates and three cylinder inequalities are established.Comment: 23 page

Quantitative uniqueness of elliptic equations

Based on a variant of frequency function, we improve the vanishing order of solutions for Schr\"{o}dinger equations which describes quantitative behavior of strong uniqueness continuation property. For the first time, we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions. Furthermore, strong unique continuation is established for higher order elliptic equations using this variant of frequency function.Comment: Add more references and polish the paper! To appear in AJ

Nodal sets of Robin and Neumann eigenfunctions

We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the interior nodal sets is obtained for Robin eigenfunctions in the smooth domain. For the analytic domain, the sharp upper bounds of the interior nodal sets was shown for Robin eigenfunctions. More importantly, we obtain the sharp upper bounds for the boundary nodal sets of Neumann eigenfunctions with new quantitative global Carleman estimates. Furthermore, the sharp doubling inequality and vanishing order of Robin eigenfunctions on the boundary of the domain are obtained.Comment: 37 page

Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients

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 continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established.Comment: 33 page

Interior nodal sets of Steklov eigenfunctions on surfaces

We investigate the interior nodal sets $\mathcal{N}_\lambda$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $C\lambda$. The singular sets $\mathcal{S}_\lambda$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_\lambda)\leq C\lambda^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_\lambda)\leq C\lambda^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.Comment: Correct some typo

A lower bound for the nodal sets of Steklov eigenfunctions

We consider the lower bound of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds with boundary--the eigenfunctions of the Dirichlet-to-Neumann map. Let $N_\lambda$ be its nodal set. Assume that zero is a regular value of Steklov eigenfunctions. We show that $$H^{n-1}(N_\lambda)\geq C\lambda^{\frac{3-n}{2}}$$ for some positive constant $C$ depending only on the manifold

Upper bounds of nodal sets for eigenfunctions of eigenvalue problems

The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. The examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions. It is done through analytic estimates of Morrey-Nirenberg and Carleman estimates.Comment: Update the wording and reference

The Maximum Principles and Symmetry results for Viscosity Solutions of Fully Nonlinear Equations

This paper is concerned about maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We obtain the radial symmetry and monotonicity properties for nonnegative viscosity solutions of begin{equation}F (D^2 u)+u^p=0 \quad \quad \mbox{in}\ \mathbb R^n \label{abs}\end{equation} under the asymptotic decay rate $u=o(|x|^{-\frac{2}{p-1}})$ at infinity, where $p>1$ (Theorem 1, Corollary 1). As a consequence of our symmetry results, we obtain the nonexistence of any nontrivial and nonnegative solution when $F$ is the Pucci extremal operators (Corollary 2). Our symmetry and monotonicity results also apply to Hamilton-Jacobi-Bellman or Isaccs equations. A new maximum principle for viscosity solutions to fully nonlinear elliptic equations is established (Theorem 2). As a result, different forms of maximum principles on bounded and unbounded domains are obtained. Radial symmetry, monotonicity and the corresponding maximum principle for fully nonlinear elliptic equations in a punctured ball are shown (Theorem 3). We also investigate the radial symmetry for viscosity solutions of fully nonlinear parabolic partial differential equations (Theorem 4).Comment: Comments are welcom

Quantitative uniqueness of solutions to second order elliptic equations with singular potentials in two dimensions

In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the Lebesgue norms of the lower order terms for all admissible exponents. Then we show that a scaling argument allows us to pass from these vanishing order estimates to estimates for the rate of decay of solutions at infinity. Our proofs rely on a new $L^p - L^q$ Carleman estimate for the Laplacian in $\mathbb{R}^2$.Comment: 24 page

Doubling inequality and nodal sets for solutions of bi-Laplace equations

We investigate the doubling inequality and nodal sets for the solutions of bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for Laplace eigenfunctions by Logunov. In addition, we derive an implicit upper bound for the nodal sets of solutions. We show two types of doubling inequalities for the solutions of bi-Laplace equations. As a consequence, the rate of vanishing is given for the solutions.Comment: 40 page