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For a bounded domain$$\Omega \subset {\mathbb {R}}^n$$Ω⊂Rnwith smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion for the integral of the trace of the Stokes semigroup$$e^{-t S}$$e-tSas$$t\rightarrow 0^+$$t→0+. These coefficients (i.e., spectral invariants) provide precise information for the volume of the domain$$\Omega $$Ωand the surface area of the boundary$$\partial \Omega $$∂Ωby the spectrum of the Stokes problem. As an application, we show that ann-dimensional ball is uniquely determined by its Stokes spectrum among all Euclidean bounded domains with smooth boundary.

For a compact connected Riemannian n-manifold $$(\Omega ,g)$$ with smooth boundary, we explicitly calculate the first two coefficients $$a_0$$ and $$a_1$$ of the asymptotic expansion of $$\sum _{k=1}^\infty \mathrm{{e}}^{-t \tau _k^{\mp }}= a_0t^{-n/2} {\mp } a_1 t^{-(n-1)/2} +O(t^{1-n/2})$$ as $$t\rightarrow 0^+$$ , where $$\tau ^-_k$$ (respectively, $$\tau ^+_k$$ ) is the k-th Navier–Lame eigenvalue on $$\Omega $$ with Dirichlet (respectively, Neumann) boundary condition. These two coefficients provide precise information for the volume of the elastic body $$\Omega $$ and the surface area of the boundary $$\partial \Omega $$ in terms of the spectrum of the Navier–Lame operator. This gives an answer to an interesting and open problem mentioned by Avramidi in (Non-Laplace type operators on manifolds with boundary, analysis, geometry and topology of elliptic operators. World Sci. Publ., Hackensack, pp. 107–140, 2006). As an application, we show that an n-dimensional ball is uniquely determined by its Navier–Lame spectrum among all bounded elastic bodies with smooth boundary.

This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat trace asymptotic expansion. In particular, we explicitly give the expressions for the first four coefficients. These coefficients are spectral invariants which provide precise information concerning the volume and curvatures of the boundary of the manifold and some physical quantities by the magnetic Steklov eigenvalues., 26 pages. This is a new work for eigenvalues of the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary

In this paper, by explicitly calculating the principal symbols of pseudodifferential operators, we establish two Weyl-type asymptotic formulas with sharp remainder estimates for the counting functions of the two classes of biharmonic Steklov eigenvalue problems of smooth bounded domains in a Riemannian manifold.

For a given bounded domain Ω with smooth boundary in a smooth Riemannian manifold ( M , g ) , by decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferential operator, and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as t → 0 + . In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients provide precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem.

In this paper, we obtain the sharp k-th order Sobolev inequalities in the hyperbolic space $${\mathbb{H}^n}$$ for all k = 1, 2, 3, . . . . This gives an answer to an open question raised by Aubin in [Aubin, Princeton University Press, Princeton (1982), pp. 176–177] for $${W^{k,2}(\mathbb{H}^n)}$$ with k > 1. In addition, we prove that the associated Sobolev constants are optimal.

We prove the radial symmetry of the solutions of second-order nonlinear elliptic equations for overdetermined Dirichlet and Neumann boundary value problems. In addition, a global uniqueness theorem of Holmgren type is given for nonlinear elliptic equations.

The well-known Schiffer conjecture saying that for a smooth bounded domain Ω ⊂ R n , if there exists a positive Neumann eigenvalue such that the corresponding Neumann eigenfunction u is constant on the boundary of Ω , then Ω is a ball. In this paper, we shall prove that the Schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding Neumann eigenfunction is constant on the boundary. We also prove a similar result to the Berenstein conjecture for the overdetermined Dirichlet eigenvalue problem.

Second-order elliptic operators realized under Dirichlet boundary conditions in bounded domains are shown to generate strongly continuous semigroups in the topology of uniform convergence. The stochastic representation and integral representation with kernel for the strongly continuous semigroups are given. Furthermore, several useful formulae for the principal eigenvalue of second-order elliptic operators are obtained.

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues in bounded domains of Riemannian manifolds. We also obtain the Weyl-type asymptotic formulas for the eigenvalues of the buckling and clamped plate problems in bounded domains of Riemannian manifolds. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case.

For classical Neumann eigenvalue, buckling eigenvalue and clamped plate eigenvalue, we give the corresponding Rellich type identities. As an application of these results, then, we obtain a new necessary and sufficient condition for a domain without the Pompeiu property.

Let Ω be a bounded domain with C 2 -smooth boundary in an n-dimensional oriented Riemannian manifold. It is well known that for the biharmonic equation Δ 2 u = 0 in Ω with the condition u = 0 on ∂Ω, there exists an infinite set { u k } of biharmonic functions in Ω with positive eigenvalues { λ k } satisfying Δ u k + λ k ϱ ∂ u k ∂ ν = 0 on ∂Ω. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues λ k .