Related papers: Lipschitz property of minimisers between double co…

The question of unique continuation of harmonic functions in a domain $\Omega$ $\subset$ R d with boundary $\partial$$\Omega$, satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset $\omega$ of the…

In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on…

In this paper, we give a uniqueness theorem for the Dirichlet problem of minimal maps into general Riemannian manifolds with non-positive sectional curvature, improving Theorem 5.2 of Lee-Ooi-Tsui's paper published in J. Geom. Anal.. The…

We study the existence and regularity of minimizers of the neo-Hookean energy in the closure of classes of deformations without cavitation. The exclusion of cavitation is imposed in the form of the divergence identities, which is equivalent…

For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties. On the one hand, it is Lipschitz with…

We present a new method to compare the shapes of genus-zero surfaces. We introduce a measure of mutual stretching, the symmetric distortion energy, and establish the existence of a conformal diffeomorphism between any two genus-zero…

Recent work of the authors and their collaborators has uncovered fundamental connections between the Dirichlet-to-Neumann map, the spectral flow of a certain family of self-adjoint operators, and the nodal deficiency of a Laplacian…

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of…

Let $\Omega=\widetilde{\Omega}\setminus \overline{D}$ where $\widetilde{\Omega}$ is a bounded domain with connected complement in $\mathbb C^n$ (or more generally in a Stein manifold) and $D$ is relatively compact open subset of…

We consider an energy functional combining the square of the local oscillation of a one--dimensional function with a double well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the…

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…

We prove local Lipschitz property of the map which puts in correspondence to each $N$--net different from $(N-1)$--net its Chebyshev center. If dimension of Eucledean or Lobachevskii space is greater than 1 and net consists of more than 2…

We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in Besov…

The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this…

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…

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics…

We extend recent work of Gurel-Gurevich--Jerison--Nachmias (2020) and Bou-Rabee--Gwynne (2024) by showing that as the mesh of our lattice tends to $0$, we have a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps…

We study the budget map and the indirect utility function of a parametric consumer problem in a Banach space setting by some advanced tools from set-valued and variational analysis. The Lipschitz-likeness and differentiability properties of…

That is, given a compact set $B \subset \mathbb{R}^n$ (the boundary) and a subgroup $L$ of the \v{C}ech homology group $\check{H}_{d-1}(B;G)$ of dimension $d$ over some commutative group $G$, we find a compact set $E \supset B$ such that…

An approximation theorem of Youngs (1948) asserts that a continuous map between compact oriented topological 2-manifolds (surfaces) is monotone if and only if it is a uniform limit of homeomorphisms. Analogous approximation of Sobolev…