BOUNDARY REGULARITY FOR QUASILINEAR ELLIPTIC SYSTEMS

ABSTRACT

We consider boundary regularity for solutions of certain systems of second-order nonlinear elliptic equations, and obtain a general criterion for a weak solution to be regular in the neighbourhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.     

Uniform partial regularity of quasi minimizers for the perimeter

Quasi minimizers for the perimeter are measurable subsets G of such that for all variations of G with and for a given increasing function such that . We prove here that, given , G a reduced quasi minimizer, and , there are , with , and , homeomorphic to a closed ball with radius t in , such that for some absolute constant . The constant above depends only on n, and . If moreover for some , we prove that we can find such a ball such that is a dimensional graph of class . This will be obtained proving that a quasi minimizer is equivalent to some set which satisfies the condition B. This condition gives some kind of uniform control on the flatness of the boundary and then criterions proven by Ambrosio-Paolini and Tamanini can be applied to get the required regularity properties. Received: July 12, 1999 / Accepted: October 1, 1999     

Boundary regularity for nonlinear elliptic systems

We consider questions of boundary regularity for solutions of certain systems of second-order nonlinear elliptic equations. We obtain a general criterion for a weak solution to be regular in the neighbourhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighbourhood. This result is new for the situation under consideration (general nonlinear second order systems in divergence form, with inhomogeneity obeying the natural growth conditions). Received: 6 July 2001 / Accepted: 27 September 2001 / Published online: 28 February 2002     

Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth

We prove a small excess regularity theorem for almost minimizers of a quasi-convex variational integral of subquadratic growth. The proof is direct, and it yields an optimal modulus of continuity for the derivative of the almost minimizer. The result is new for general almost minimizers, and in the case of absolute minimizers it considerably simplifies the existing proof. Mathematics Subject Classification (2000) 49N60, 26B25     

Higher Critical Points in an Elliptic Free Boundary Problem

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied extensively. But because the functionals are not smooth, standard variational methods cannot be used directly to prove the existence of higher critical points. Here we find a nontrivial critical point of mountain pass type and prove many of the same estimates known for minimizers, including Lipschitz continuity and nondegeneracy. We then show that the free boundary is smooth in dimension 2 and prove partial regularity in higher dimensions.     

Regularity of Dirichlet Nearly Minimizing Multiple-Valued Functions

In this article, we extend the related notions of Dirichlet quasiminimizer, ω-minimizer and almost minimizer to the framework of multiple-valued functions in the sense of Almgren and prove Hölder regularity results. Concerning the regularity of the graph, we show that, in contrast to absolute minimizers, there are examples with various large complicated branching sets.     

Rotationally symmetric 1-harmonic maps from D 2 to S 2

We consider rotationally symmetric 1-harmonic maps from D ² to S ² subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate non-convex functional with linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique (up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution. R. Dal Passo passed away on 8th August 2007. Endowed with great strength, creativity and humanity, Roberta has been an outstanding mathematician, an extraordinary teacher and a wonderful friend. Farewell, Roberta.     

A class of stable energy minimisers for the rotating drop problem

We implement variational techniques and an implicit function theorem to derive constraints on angular velocity under which we may verify the existence, boundary regularity, and stability of an energy-minimising family of rotating liquid drops in a neighbourhood of the closed unit ball in Rn+1.     

Existence of solutions of a thermoviscoplastic model and associated optimal control problems

A quasistatic, thermoviscoplastic model at small strains with linear kinematic hardening, von Mises yield condition and mixed boundary conditions is considered. The existence of a unique weak solution is proved by means of a fixed-point argument, and by employing maximal parabolic regularity theory. The weak continuity of the solution operator is also shown. As an application, the existence of a global minimizer of a class of optimal control problems is proved.     

Analytic regularity of a free boundary problem

In this paper, we consider a free boundary problem with volume constraint. We show that positive minimizer is locally Lipschitz and the free boundary is analytic away from a singular set with Hausdorff dimension at most n − 8.     

General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Basic Interface Problems)

Here we discuss three-dimensional so-called basic and mixed boundary value problems (BVP) for steady state oscillations of piecewise homogeneous anisotropic bodies imbedded into an infinite elastic continuum. Uniqueness is shown with the help of generalized Sommerfeld–Kupradze radiation conditions, while existence follows for arbitrary values of the oscillation parameter by the reduction of the original interface transmission BVPs to equivalent uniquely solvable boundary integral or pseudodifferential equations on the interfaces. For the basic BVPs, we show classical regularity and, in addition for the mixed BVPs that the solutions are Hölder continuous with exponent α ∈ (0, 1/2) in the neighbourhood of the curves of discontinuity of the boundary and transmission conditions.     

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C^∞, and let g_ij = δ_ij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W^2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C³ away from a certain singular set Σ and C^∞ away from a larger singular set Σ ∪ Σ₀. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ₀. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc.     

The smoothness of the free boundary for a class of vector-valued problems

We consider a minimizer u: Rⁿ Ω→ⁿ of Dirichlet's integral under the additional side condition In(.) ? M for a smooth domain M?R^N. Imposing certain geometric conditions on M we show that the coincidence set {x?Ω: u(x) ? ?M) is of locally finite perimeter in Ω. Moreover, we formulate a condition implying the regularity of the free boundary near a given point.     

Regularity Properties of <Emphasis Type="Italic">H</Emphasis>-Convex Sets

We study the first- and second-order regularity properties of the boundary of H-convex sets in the setting of a real vector space endowed with a suitable group structure: our starting point is indeed a step two Carnot group. We prove that, locally, the noncharacteristic part of the boundary has the intrinsic cone property and that it is foliated by intrinsic Lipschitz continuous curves that are twice differentiable almost everywhere.     

Boundary characteristic point regularity for semilinear reaction-diffusion equations: towards an ode criterion

The classical problem of regularity of boundary characteristic points for semilinear heat equations with homogeneous Dirichlet conditions is considered. The Petrovskii ( 2?{loglog} ) \left( {2\sqrt {{\log \log }} } \right) criterion (1934) of the boundary regularity for the heat equation can be adapted to classes of semilinear parabolic equations of reaction–diffusion type and takes the form of an ordinary differential equation (ODE) regularity criterion. Namely, after a special matching with a boundary layer, the regularity problem reduces to a onedimensional perturbed nonlinear dynamical system for the first Fourier-like coefficient of the solution in an inner region. A similar ODE criterion, with an analogous matching procedures, is shown formally to exist for semilinear fourth order biharmonic equations of reaction-diffusion type. Extensions to regularity problems of backward paraboloid vertices in \mathbbR^N {\mathbb{R}^N} are discussed. Bibliography: 54 titles. Illustrations: 1 figure.     

Regularity for Shape Optimizers: The Nondegenerate Case

We consider minimizers of (1) where F is a function strictly increasing in each parameter, and is the k^th Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C^1,α graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n – 3. We also obtain a new regularity result for vector‐valued Bernoulli‐type free boundary problems.© 2018 Wiley Periodicals, Inc.     

Strong well‐posedness of a three phase problem with nonlinear transmission condition

We prove existence and uniqueness of strong solutions to a quasilinear parabolic‐elliptic system modelling an ionic exchanger. This chemical system consists of three phases connected with nonlinear boundary conditions. The most interesting difficulty of our problem manifests in the nonlinear transmission condition, as almost all quantities are non‐linearly involved in this boundary equation. Our approach is based on the contraction mapping principle, where maximal L_p‐regularity of the associated linear problem is used to obtain a fixed point equation of the starting problem. Copyright © 2012 John Wiley & Sons, Ltd.     

Regularity and estimates for J-holomorphic discs attached to a maximal totally real submanifold

We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity to the boundary from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yields explicit estimates of Hölderian norms of such discs.     

On the nonexistence of boundary branch points for minimal surfaces spanning smooth contours II

This is the second in a series of two papers discussing the elementary but beautiful and fundamental question (open for some eighty years) of whether or not a minimal surface spanning a sufficiently smooth curve, which is a local minimizer, is immersed up to and including the boundary. We show that C ^k minimizers of energy or area cannot have nonexceptional boundary branch points.