Evolution of harmonic maps with Dirichlet boundary conditions

Boundary Regularity and the Dirichlet Problem for Harmonic Maps

In a previous paper [10] we developed an interior regularity theory for energy minimizing harmonic maps into Riemannian manifolds. In the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed Dirichlet boundary condition. We show that such maps are regular in a full neighborhood of the boundary, assuming appropriate regularity on the manifolds,...

S-valued harmonic maps on polyhedra with tangent boundary conditions

A unit-vector field n : P → S2 on a convex polyhedron P ⊂ R3 satisfies tangent boundary conditions if, on each face of P , n takes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices of P . We consider fields which are continuous elsewhere. We derive a lower bound E P (h) for the infimum Dirichlet energy Einf P (h) for such tangent unit-vector f...

On the Dirichlet Problem for Harmonic Maps with Prescribed Singularities

Let (M, g) be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into (M, g) with prescribed singularities along a closed submanifold of the domain. This generalizes our previous work where such maps into the hyperbolic plane were constructed. This problem, in the case wh...

A Nonlocal p-Laplacian Evolution Equation with Nonhomogeneous Dirichlet Boundary Conditions

Abstract. In this paper we study the nonlocal p-Laplacian-type diffusion equation ut(t, x) = ∫ RN J(x−y)|u(t, y)−u(t, x)|p−2(u(t, y)−u(t, x)) dy, (t, x) ∈]0, T [×Ω, with u(t, x) = ψ(x) for (t, x) ∈ ]0, T [×(RN \Ω). If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div(|∇u|p−2∇u) with Dirichlet boundary condition u(t, x) = ψ(x) on (t, x)...

Absorption semigroups and dirichlet boundary conditions