A singular energy minimizing free boundary

Journal of the European Mathematical Society, 2015

We consider the functional I (v) = [f (|Dv|) − v] dx, where is a bounded domain and f is a convex function. Under general assumptions on f , Crasta [Cr1] has shown that if I admits a minimizer in W 1,1 0 () depending only on the distance from the boundary of , then must be a ball. With some restrictions on f , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.

2014

We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin [13].

Interfaces and Free Boundaries, 2015

We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin .

Journal of Differential Geometry

In 1960s, Almgren [3, 4] initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts [34] and Schoen-Simon [37] in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves [31] shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many embedded minimal hypersurfaces with free boundary which are properly embedded. Contents 1. Introduction 1.1. The problem and main results 1.2. Main ideas of the proof 2. Definitions and preliminary results 2.1. Notations in Euclidean spaces 2.2. Stationary varifolds with free boundary 2.3. Almost proper embeddings and stability 3. Almost minimizing varifolds with free boundary 3.1. Equivalence classes of relative cycles 3.2. Two isoperimetric lemmas 3.3. Definitions of almost minimizing varifolds 4. The min-max construction 4.1. Homotopy relations 4.2. Discretization and interpolation 4.3. Tightening 4.4. Existence of almost minimizing varifold 5. Regularity of almost minimizing varifolds 5.1. Good replacement property 5.2. Tangent cones and rectifiability 5.3. Regularity of almost minimizing varifolds Appendix A. Fermi coordinates Appendix B. Proof of Theorem 3.20 References 1

Complex Variables and Elliptic Equations, 2017

Let be a bounded smooth domain in R N. We prove a general existence result of least energy solutions and least energy nodal ones for the problem − u = f (x, u) in u = 0 on ∂ (P) where f is a Carathéodory function. Our result includes some previous results related to special cases of f. Finally, we propose some open questions concerning the global minima of the restriction on the Nehari manifold of the energy functional associated with (P) when the nonlinearity is of the type f (x, u) = λ|u| s −2 u − μ|u| r−2 u, with s, r ∈ (1, 2) and λ, μ > 0.

Journal of Differential Equations, 1991

In [SWZ] we considered the constrained least gradient problem inf iI lVu[ dx: u E Cog', lVul d 1 a.e., 24 = g on asz , R 1

2013

Abstract. We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin

Transactions of the American Mathematical Society, 2006

The Steiner problem is the problem of finding the shortest network connecting a given set of points. By the singular Plateau Problem, we will mean the problem of finding an area-minimizing surface (or a set of surfaces adjoined so that it is homeomorphic to a 2-complex) spanning a graph. In this paper, we study the parametric versions of the Steiner problem and the singular Plateau problem by a variational method using a modified energy functional for maps. The main results are that the solutions of our one- and two-dimensional variational problems yield length and area minimizing maps respectively, i.e. we provide new methods to solve the Steiner and singular Plateau problems by the use of energy functionals. Furthermore, we show that these solutions satisfy a natural balancing condition along its singular sets. The key issue involved in the two-dimensional problem is the understanding of the moduli space of conformal structures on a 2-complex.

Mathematische Zeitschrift, 1988

Rendiconti Lincei - Matematica e Applicazioni, 2000

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.

Journal of Differential Equations, 2003

We establish an existence result for strongly indefinite semilinear elliptic systems with Neumann boundary condition, and we study the limiting behavior of the positive solutions of the singularly perturbed problem. r O jvj qþ1 dx: Therefore, at a critical point both functionals coincide, that is, I e ðu; vÞ ¼ J e ðw 1 ; w 2 Þ:

Journal of Differential Equations, 1999

Communications on Pure and …, 2005

Analysis & PDE

We study minimizers of a family of functionals E ε indexed by a characteristic length scale ε, whose formal limit is E ⋆ (u) =´W (∇u) for u taking values into a manifold, where W is a positive definite quadratic form: minimizers of E ⋆ are W-harmonic mapsclassical harmonic maps correspond to the isotropic case W (∇u) = |∇u| 2. We show that the convergence of minimizers of E ε to W-harmonic maps is locally uniform outside a singular set. We treat general energies, covering in particular the 3D Landau-de Gennes model for liquid crystals, with three distinct elastic constants. In the isotropic case, similar results are standard and rely on three ingredients: a monotonicity formula for the scale-invariant energy on small balls, a uniform pointwise bound, and a Bochner equation for the energy density. In the level of generality we consider in this work, all of these ingredients are absent and we have to use a very different strategy. Finding ways around the lack of monotonicity formula is particularly interesting, since this issue is why optimal estimates on the singular set of W-harmonic maps constitute an open problem. Our novel argument relies on showing appropriate decay for the energy on small balls, separately at scales smaller and larger than ε: the former is obtained from the regularity of solutions to elliptic systems while the latter is inherited from the regularity of W-harmonic maps. This also allows us to handle physically relevant boundary conditions for which, even in the isotropic case, uniform convergence up to the boundary was open.