A type of homogenization problem

Rendiconti Lincei - Matematica e Applicazioni, 2000

Calculus of Variations and Partial Differential Equations, 2004

We extend to the degenerate case p = 2, Simon's approach to the classical regularity theory of harmonic maps of Schoen & Uhlenbeck, by proving a "p-Harmonic Approximation Lemma". This allows to approximate functions with p-harmonic functions in the same way as the classical harmonic approximation lemma (going back to De Giorgi) does via harmonic functions. Finally, we show how to combine this tool with suitable regularity estimates for solutions to degenerate elliptic systems with a critical growth right hand side, in order to obtain partial C 1,α -regularity of p-harmonic maps.

Communications in Mathematical Physics, 1992

In this paper, we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds. This is motivated from one of Hawking's conjectures on the uniqueness of Kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum Einstein equation in general relativity.

Journal of the American Mathematical Society, 2013

Journal of Differential Geometry

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial. Contents 132 J. JOST, Y. L. XIN & L. YANG 4.3. Mollified Green function 155 4.4. Telescoping lemma 157 5. Regularity of weakly harmonic maps and Liouville type theorems 159 5.1. Pointwise estimates 159 5.2. Image shrinking property 162 5.3. Estimating the oscillation 164 5.4. Holder estimates 166 5.5. A Liouville type theorem 6. Analytic and geometric conclusions 168 6.1. Simple manifolds 168 6.2. Manifolds with nonnegative Ricci curvature 171 6.3. Bernstein type theorems 171 References 173

Indiana University Mathematics Journal, 2018

The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the geometric-driven profile of ground states. In this work we study, under very general assumptions, the convergence of minimizers towards harmonic maps. We show that the convergence is locally uniform up to the boundary, away from the lower dimensional singular set. Our results generalize related findings, most notably in the theory of liquid-crystals, to all dimensions n ≥ 3, and to general nonlinearities. Our proof follows a well-known scheme, relying on small energy estimate and monotonicity formula. It departs substantially from previous studies in the treatment of the small energy estimate at the boundary, since we do not rely on the specific form of the potential. In particular this extends existing results in 3-dimensional settings. In higher dimensions we also deal with additional difficulties concerning the boundary monotonicity formula.

2010

We study the regularity of solutions to degenerate A-harmonic equations under suitable integrability assumptions on the ellipticity and growth coefficients. In particular we show a self-improving property of the gradient of the solutions, extending previous results by Gehring and Iwaniec-Sbordone valid in the uniformly elliptic setting.

Analysis & PDE, 2012

We continue the development, by reduction to a first-order system for the conormal gradient, of L 2 a priori estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence-form second-order complex elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning a priori almost everywhere nontangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying a posteriori a separate work on bounded domains. Andreas Rosén was formerly called Andreas Axelsson.

arXiv (Cornell University), 2009

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.

arXiv (Cornell University), 2022

We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their C 1,α or C ∞-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the L p-theory for strongly elliptic linear systems and Sobolev's embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required.

arXiv (Cornell University), 2018

Let {u n } be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold K ⊂ N satisfying sup n ∇u n L 2 (M) + τ(u n) L 2 (M) ≤ Λ, where τ(u n) is the tension field of the map u n. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time. 1. introduction Let (M, g) be a compact Riemannian manifold with smooth boundary and (N, h) be a compact Riemannian manifold of dimension n. Let K ⊂ N be a k−dimensional closed submanifold where 1 ≤ k ≤ n. For a mapping u ∈ C 2 (M, N), the energy density of u is defined by e(u) = 1 2 |∇u| 2 = Trace g u * h, where u * h is the pull-back of the metric tensor h. The energy of the mapping u is defined as E(u) = M e(u)dvol g. Define C(K) = u ∈ C 2 (M, N); u(∂M) ⊂ K. A critical point of the energy E over C(K) is a harmonic map with free boundary u(∂M) on K. The problem of the existence, uniqueness and regularity of such harmonic maps with a free boundary was first systematically investigated in [8]. By Nash's embedding theorem, (N, h) can be isometrically embedded into some Euclidean space R N. Then we can get the Euler-Lagrange equation ∆ g u = A(u)(∇u, ∇u),

Transactions of the American Mathematical Society, 1981

Let u u be a bounded harmonic function on a noncompact rank one symmetric space M = G / K ≈ N − A , N − A K M = G/K \approx {N^ - }A,{N^ - }AK being a fixed Iwasawa decomposition of G G . We prove that if for an a 0 ∈ A {a_0} \in A there exists a limit u ( n a 0 ) ≡ c 0 u(n{a_0}) \equiv {c_0} , as n ∈ N − n \in {N^ - } goes to infinity, then for any a ∈ A a \in A , u ( n a ) = c 0 u(na) = {c_0} . For M = S U ( n , 1 ) / S ( U ( n ) × U ( 1 ) ) = B n M = SU(n,1)/S(U(n) \times U(1)) = {B^n} , the unit ball in C n {{\mathbf {C}}^n} with the Bergman metric, this is a result of Hulanicki and Ricci, and in this case it reads (via the Cayley transformation) as a theorem on convergence of a bounded harmonic function to a boundary value at a fixed boundary point, along appropriate, tangent to ∂ B n \partial {B^n} , surfaces.

Communications in Analysis and Geometry, 1993

Comptes Rendus Mathematique, 2015

In this note we study the boundary regularity of minimizers of a family of weak anchoring energies that model the states of liquid crystals. We establish optimal boundary regularity in all dimensions n ≥ 3. In dimension n = 3, this yields full regularity at the boundary which stands in sharp contrast with the observation of boundary defects in physics works. We also show that, in the cases of weak and strong anchoring, regularity of minimizers is inherited from that of their corresponding limit problems.The analysis rests in a crucial manner on the fact that the surface and Dirichlet energies scale differently; we take advantage of this fact to reduce the problem to the known regularity of tangent maps with zero Neumann conditions.