SPDEs driven by space-time white noise in high dimensions: absolute continuity of the law and convergence of solutions

In this paper, we consider stochastic partial differential equations driven by space-time white noise in high dimensions. We prove, under reasonable conditions, that the law of the solution admits a density with respect to Lebesgue measure. The stability of the equation, as the higher order differential operator tends to zero, is also studied in the paper.     

On the discrete spectrum of non-selfadjoint operators

We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of eigenvalues of Schrödinger operators with complex potentials.     

Nontrivial solutions of superlinear p-Laplacian equations

We consider p-Laplacian equations on a bounded domain, where the nonlinearity is superlinear but dose not satisfy the usual Ambrosetti-Rabinowitz condition near infinity, or its dual version near zero. Nontrivial solutions are obtained by computing the critical groups and Morse theory.     

Universal bounds for eigenvalues of the biharmonic operator

In this paper, we study the eigenvalues of the clamped plate problem:

     

Refined asymptotics for eigenvalues on domains of infinite measure

In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure. Also, we derive some estimates for the spectral counting function of the Laplace operator on unbounded two-dimensional domains.     

Completion problems for real matrices, II

Thirty years ago, G. N. de Oliveira has proposed the following completion problems: Describe the possible characteristic polynomials of , where C_1,1 and C_2,2 are square submatrices, when some of the blocks C_i,j are fixed and the others vary. Several of these problems remain unsolved. This paper gives the solution, over the field of real numbers, of Oliveira’s problem where the blocks C_1,1,C_2,2 are fixed and the others vary.     

Eigenvalues and edge-connectivity of regular graphs

In this paper, we show that if the second largest eigenvalue of a d-regular graph is less than , then the graph is k-edge-connected. When k is 2 or 3, we prove stronger results. Let ρ(d) denote the largest root of x³-(d-3)x²-(3d-2)x-2=0. We show that if the second largest eigenvalue of a d-regular graph G is less than ρ(d), then G is 2-edge-connected and we prove that if the second largest eigenvalue of G is less than , then G is 3-edge-connected.     

Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erd?s proved that if a graph $G$ with $n$ vertices satisfies

$e\left(G\right)>max\left\{\left(\genfrac{}{}{0ex}{}{n?k}{2}\right)+k^2,\left(\genfrac{}{}{0ex}{}{?(n+1)∕2?}{2}\right)+{?\frac{n?1}{2}?}^2\right\},$

where the minimum degree $\delta \left(G\right)\ge k$ and $1\le k\le (n?1)∕2$, then it is Hamiltonian. For $n\ge 2k+1$, let $E_n^k=K_k\vee (k{K}_1+K_{n?2k})$, where “$\vee$” is the “join” operation. One can observe $e\left(E_n^k\right)=\left(\genfrac{}{}{0ex}{}{n?k}{2}\right)+k^2$ and $E_n^k$ is not Hamiltonian. As $E_n^k$ contains induced claws for $k\ge 2$, a natural question is to characterize all 2-connected claw-free non-Hamiltonian graphs with the largest possible number of edges. We answer this question completely by proving a claw-free analog of Erd?s’ theorem. Moreover, as byproducts, we establish several tight spectral conditions for a 2-connected claw-free graph to be Hamiltonian. Similar results for the traceability of connected claw-free graphs are also obtained. Our tools include Ryjá?ek’s claw-free closure theory and Brousek’s characterization of minimal 2-connected claw-free non-Hamiltonian graphs.     

Eigenvalues of the Wentzell–Laplace operator and of the fourth order Steklov problems

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell–Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Steklov problems on a Euclidean ball.