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We study the effects of a domain deformation to the nodal set of Laplacian eigenfunctions when the eigenvalue is degenerate. In particular, we study deformations of a rectangle that perturb one side and how they change the nodal sets…
Exploring the relationship between geometry and the resonant frequencies of a shape is of interest to pure and applied mathematicians. These resonant frequencies are related to the spectrum of the Laplacian, a partial differential operator.…
We study perturbations of the discrete magnetic Laplacian associated to discrete analogs of funnels. We perturb the metric in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away from possible…
We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and…
The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main…
We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of…
In this paper we analyse the spectrum of nonlocal Dirichlet problems with non-singular kernels in bounded open sets. The novelty is the continuity of eigenvalues with respect to domain perturbation via Lebesgue measure. Also, under…
In this work, we introduce a new difference equation which is discrete analogue of Diffusion differential equation and analyze some essential spectral properties, Diffusion difference operator is self-adjoint, eigenvalues of this problem…
We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the…
We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First,…
In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of…
A first-principle continuum-mechanics expression for the rolling friction coefficient is obtained for the rolling motion of a viscoelastic sphere on a hard plane. It relates the friction coefficient to the viscous and elastic constants of…
We consider an inverse problem for a Westervelt type nonlinear wave equation with fractional damping. This equation arises in nonlinear acoustic imaging, and we show the forward problem is locally well-posed. We prove that the smooth…
The universality of intermittency in hydrodynamic turbulence is considered based on a recent model for the velocity gradient tensor evolution. Three possible versions of the model are investigated differing in the assumed correlation…
The standard Maxwell formulation of the problem of polarized dielectrics suffers from a number of difficulties, both conceptual and practical. These difficulties are particularly significant in the case of liquid interfaces, where the…
We consider the mixed Dirichlet-conormal problem on irregular domains in $\mathbb{R}^d$. Two types of regularity results will be discussed: the $W^{1,p}$ regularity and a non-tangential maximal function estimate. The domain is assumed to be…
Fractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping…
We study an eigenvalue problem for the infinity-Laplacian on bounded domains. We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. The work also contains existence results when the parameter, in the…
This paper is devoted to a dispersion analysis of a class of perturbed p-Laplacians. Besides the p-Laplacian-like eigenvalue problems we also deal with new and non-standard eigenvalue problems, which can not be solved by the methods used in…
In the present article, we study the diffusion equations with fractional time derivatives. The aim of this paper is to investigate the best possible regularity for the initial value/boundary value problems with non-homogeneous Dirichlet…