Related papers: The Dirichlet Principle for Inner Variations

When a birational surface map is expanding on cohomology there is a canonical way to associate positive closed currents to the map and its inverse. In this paper we use a version of Dirichlet energy to construct the wedge product of these…

The purpose of the present paper is to establish appropriate cut-off resolvent estimates for the Dirichlet Laplacian on exterior domains. The geometrical assumptions on domains are rather general, for example, non-trapping condition is not…

We consider maps into Riemannian manifolds of non-positive curvature and start developing a systematic PDE theory. We control the Sobolev $H^{2,2}$-norm of such a map in terms of its energy, the $L^2$-norm of its tension field and a…

In this paper we study 1-equivariant wave maps of finite energy from 1+3-dimensional Minkowski space exterior to the unit ball at the origin into the 3-sphere. We impose a Dirichlet boundary condition at r=1, meaning that the unit sphere in…

We introduce pointwise map smoothness via the Dirichlet energy into the functional map pipeline, and propose an algorithm for optimizing it efficiently, which leads to high-quality results in challenging settings. Specifically, we first…

The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy. Specifically, among all homeomorphisms f : R -> R* between bounded doubly connected domains such that Mod (R) < Mod (R*)…

The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to…

We extend the celebrated theorem of Kellogg for conformal mappings to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimiser of Dirichlet energy of Sobolev mappings between double connected domains $D$ and…

This note is to concern a generalization to the case of twisted coefficients of the classical theory of Abelian differentials on a compact Riemann surface. We apply the Dirichlet's principle to a modified energy functional to show the…

Domain Incremental Learning is a critical scenario that requires models to continuously adapt to new data domains without retraining. However, domain shifts often cause severe performance degradation. To address this, we propose Hybrid…

We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green…

Let O be a closed geodesic polygon in S^2. Maps from O into S^2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S^2, we compute the infimum…

A standard Hilbert-space proof of Dirichlet's principle is simplified, using an observation that a certain form of min-problem has unique solution, at a specified point. This solves Dirichlet's problem, after it is recast in the required…

For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary…

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this…

We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = \psi(x)$ where $f$ is a "natural" differential operator, with a restricted domain $F$, on a manifold $X$. By "natural" we mean operators that arise intrinsically…

The temperature in natural convection problems is, under mild data assumptions, uniformly bounded in time. This property has not yet been proven for the standard finite element method (FEM) approximation of natural convection problems with…

An existence result is shown for the asymptotic Dirichlet problem for harmonic maps from the product of the hyperbolic planes to the hyperbolic space, where the Dirichlet data is given on the distinguished boundary (the product of the…

J. Kigami has laid the foundations of what is now known as analysis on fractals, by allowing the construction of an operator of the same nature of the Laplacian, defined locally, on graphs having a fractal character. The Sierpinski gasket…

In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of…