Related papers: A note on the shift theorem for the Laplacian in p…
Levinson's theorem for Dirac particles constraints the sum of the phase shifts at threshold by the total number of bound states of the Dirac equation. Recently, a stronger version of Levinson's theorem has been proven in which the value of…
We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary…
We prove the Pleijel theorem in non-collapsed RCD spaces, providing an asymptotic upper bound on the number of nodal domains of Laplacian eigenfunctions. As a consequence, we obtain that the Courant nodal domain theorem holds except at most…
We present an extension to the Poisson-Boltzmann model where the dipolar features of solvent molecules are taken explicitly into account. The formulation is derived at mean-field level and can be extended to any order in a systematic…
I propound a non-linear generalization of the Poisson equation describing a "medium" in D dimensions with a "dielectric constant" proportional to the field strength to the power D-2. It is the only conformally invariant scalar theory that…
We study the effect of domain perturbation on invariant manifolds for semilinear parabolic equations subject to Dirichlet boundary condition. Under Mosco convergence assumption on the domains, we prove the upper and lower semicontinuity of…
Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in special bounded domains of Euclidean $d$-space. The boundary of such a domain is an embedded simplicial complex…
We study positive solutions of semilinear elliptic equations in a planar triangular domain under mixed boundary conditions, consisting of homogeneous Dirichlet boundary conditions on one side and homogeneous Neumann boundary conditions on…
Reciprocal space methods for solving Poisson's equation for finite charge distributions are investigated. Improvements to previous proposals are presented, and their performance is compared in the context of a real-space density functional…
We provide some versions of the Zaremba-Hopf-Oleinik boundary point lemma for general elliptic and parabolic equations in divergence form under the sharp requirements on the coefficients of equations and on the boundaries of domains.
We provide an alternative unified approach for proving the Pythagorean theorem (in dimension $2$ and higher), the law of sines and the law of cosines, based on the concept of shape derivative. The idea behind the proofs is very simple: we…
We obtain a multidimensional Tauberian theorem for Laplace transforms of Gelfand-Shilov ultradistributions. The result is derived from a Laplace transform characterization of bounded sets in spaces of ultradistributions with supports in a…
We consider the non-relativistic limit of general relativity coupled to a $(p+1)$-form gauge field and a scalar field in arbitrary dimensions and investigate under which conditions this gives rise to a Poisson equation for a Newton…
This short note aims to give an insight to Arveson's boundary theorem by means of non-commutative Poisson boundaries and its applications.
In this report it is proposed to generalize the definition of Poisson brackets in order to treat spatial integrals of divergences as Hamiltonians which generate a kind of Hamiltonian equations on the boundary. Nonlinear Schrodinger equation…
We use an iteration procedure propped up by a a classical form of the maximum principle to show the existence of solutions to a nonlinear Poisson equation with Dirichlet boundary conditions. These methods can be applied to the case of…
We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality…
We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence…
Let Omega be a bounded, simply connected domain with boundary of class C^{1,1} except at finitely many points S_j where the boundary is locally a corner of aperture alpha_j<=pi/2. Improving on results of Grisvard, we show that the solution…
We prove the existence of at least three solutions for a weighted $p$-Laplacian operator involving Dirichlet boundary condition in a weighted Sobolev space. The main tool we use here is a three solution theorem in reflexive Banach spaces…