Related papers: The hot spots conjecture can be false: Some numeri…

Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let $B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains of…

Existence and uniqueness of the scattering solutions is proved for a class of bounded rough obstacles which is much larger than the class of Lipschitz obstacles. Integral equations method is not used. The approach is based on the…

A simple picture for the spectrum of the one-dimensional Hubbard model is presented using a classification of the eigenstates based on an intuitive bound-state Bethe-Ansatz approach. This approach allows us to prove a "string hypothesis"…

Spectrum of the Laplacian on spherical domains is analyzed from the point of view of the heat equation on the cone. The series solution to the heat equation on the cone is known to lead to a study of the Laplacian eigenvalue problem on…

A representation formula for the solution of the $\infty$-Laplace equation is constructed in a punctured square, the prescribed boundary values being $u=0$ on the sides and $u=1$ at the centre. This so-called $\infty$-potential is obtained…

It has been empirically observed that eigenfunctions of Laplace's equation $-\Delta \phi = \lambda \phi$ with Neumann boundary conditions sometimes localize near the boundary of the domain if that boundary is rough (say, fractal). This has…

In this note, we prove the boundary H\"{o}lder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the…

In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space $\R_+^{n+1}$. We show that similar results remain valid for more general approximate…

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…

The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that…

The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about…

The Casten-Holland and Matano theorem for interior reactions states that no nonconstant stable solutions exist in convex domains $\Omega$ of $\mathbb{R}^n$ under zero Neumann boundary conditions. In this paper we establish that the…

In this paper we study a bounded domain with a small hole removed. Our main result concerns the spectrum of the Laplace operator with the Robin conditions imposed at the hole boundary. Moreover we prove that under some suitable assumptions…

In this paper, we focus primarily on the symmetry properties of the second Neumann eigenfunction $u$ with respect to the symmetry axis or symmetry center of the relevant domain $Q$, such as isosceles trapezoids, parallelograms, kite…

We consider domain walls embedded in curved backgrounds as an approximation for braneworld scenarios. We give a large class of new exact solutions, exhausting the possibilities for describing one and two walls for the cases where the…

Let $G=(V,E)$ be a locally finite graph, $\Omega\subset V$ be a bounded domain, $\Delta$ be the usual graph Laplacian, and $\lambda_1(\Omega)$ be the first eigenvalue of $-\Delta$ with respect to Dirichlet boundary condition. Using the…

A solution to the heat equation between Riemannian manifolds, where the domain is compact and possibly has boundary, will not leave a compact and locally convex set before the image of the boundary does.

We study the problem of computing the conformal modulus of rings and quadrilaterals with strong singularities and cusps on their boundary. We reduce this problem to the numerical solution of the associated Dirichlet and Dirichlet-Neumann…

We prove that a large class of parabolic final value problems is well posed.This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed…

We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space ${\bf H}^N$ we…