Related papers: Singularities, initial and boundary problems for t…
Consider a bounded domain with the Dirichlet condition on a part of the boundary and the Neumann condition on its complement. Does the spectrum of the Laplacian determine uniquely which condition is imposed on which part? We present some…
Here a mixed problem for a nonlinear hyperbolic equation with Neumann boundary value condition is investigated, and a priori estimations for the possible solutions of the considered problem are obtained. These results demonstrate that any…
Consider a planar, bounded, $m$-connected region $\Omega$, and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\bord\Omega$, where each 2-cell is either a triangle or a quadrilateral. From…
Linearisation is often used as a first step in the analysis of nonlinear initial boundary value problems. The linearisation procedure frequently results in a confusing contradiction where the nonlinear problem conserves energy and has an…
Mather and Yau showed that an isolated complex hypersurface singularity is completely determined by its moduli algebra. It is shown, for the simple elliptic singularities, how to construct continuous invariants from the moduli algebras and,…
In this paper we work out explicit lagrangians describing superpotential coupling to the boundary of a 5D orientifold, as relevant to a number of quasi-realistic models of nature. We also make a number of general comments on orientifold…
The relativistic membrane equation can be rewritten as a first order hyperbolic system. Making use of the characteristic decomposition method, a new blow-up theorem is established. As an application, it demonstrates the formation of…
We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic…
We discuss first order optimality conditions for geometric optimization problems with Neumann boundary conditions and boundary observation. The methods we develop here are applicable to large classes of state systems or cost functionals.…
We study boundary conditions of topological sigma models with the goal of generalizing the concepts of anomalous symmetry and symmetry protected topological order. We find a version of 't Hooft's anomaly matching conditions on the…
A boundary control problem for the pure Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first-order necessary conditions for optimality are proved. Key words: Cahn-Hilliard equation,…
The null-timelike initial-boundary value problem for a hyperbolic system of equations consists of the evolution of data given on an initial characteristic surface and on a timelike worldtube to produce a solution in the exterior of the…
We introduce boundary special generic maps, a class of submersions from manifolds with boundary to Euclidean spaces whose restriction to the boundary has only boundary definite fold points as its singular points. We derive the…
After quick survey of some key results and open questions about the structure of singularities of minimal surfaces, we discuss recent work~\cite{Sim23} on singularities of stable minimal hypersurfaces, including some simplifications of the…
We establish well-posedness of initial-boundary value problems for continuity equations with BV (bounded total variation) coefficients. We do not prescribe any condition on the orientation of the coefficients at the boundary of the domain.…
We consider the inhomogeneous Dirichlet problem on product domains. The main result is the asymptotic expansion of the solution in terms of increasing smoothness up to the boundary. In particular, we show the exact nature of the…
The initial-boundary value problems for linear non-autonomous first order evolution equations are examined. Our assumptions provide a unified treatment which is applicable to many situations, where the domains of the operators may change…
We consider the N=1 supersymmetric two-dimensional non-linear sigma model with boundaries and nonzero B-field. By analysing the appropriate currents we describe the full set of boundary conditions compatible with N=1 superconformal…
We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power $\gamma(x)$ and how it changes. Under a mild continuity assumption on…
We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and…