A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory

We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in supercritical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.     

Boundary regularity of flows under perfect slip boundary conditions

We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions and we assume that the nonlinear elliptic operator satisfies non-standard ?-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity.     

Boundary continuity of solutions to elliptic equations with nonstandard growth

We study regularity properties of weak solutions to elliptic equations involving variable growth exponents. We prove the sufficiency of a Wiener type criterion for the regularity of boundary points. This criterion is formulated in terms of the natural capacity involving the variable growth exponent. We also prove the Hölder continuity of weak solutions up to the boundary in domains with uniformly fat complements, provided that the boundary values are Hölder continuous.     

Boundary Properties of Solutions to Second-Order Parabolic Equations in Domains with Special Symmetry

We consider the first boundary-value problem for second-order nondivergent parabolic equations with, in general, discontinuous coefficients. We study the regularity of a boundary point assuming that in a neighborhood of this point the boundary of the domain is a surface of revolution. We prove a necessary and sufficient regularity condition in terms of parabolic capacities; for the heat equation this condition coincides with Wiener's criterion.     

A Free Boundary Problem of Fourth Order: Classical Solutions in Weighted Hölder Spaces

We prove short-time existence and uniqueness of classical solutions in weighted Hölder spaces for the thin-film equation with linear mobility, zero contact angle, and compactly supported initial data. We furthermore show regularity of the free boundary and optimal regularity of the solution in terms of the regularity of the initial data. Our approach relies on Schauder estimates for the operator linearized at the free boundary, obtained through a variant of Safonov's method that is solely based on energy estimates.     

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem.     

Boundary regularity for almost minimizers of quasiconvex variational problems

We consider boundary regularity for almost minimizers of quasiconvex variational integrals with polynomial growth of order p ≥ 2, and obtain a general criterion for an almost minimizer to be regular in the neighbourhood of a given boundary point. Combined with existing results on interior partial regularity, the proof yields directly the optimal regularity for an almost minimizer in this neighbourhood.     

Regularity of boundary wavelets

The conventional way of constructing boundary functions for wavelets on a finite interval is by forming linear combinations of boundary-crossing scaling functions. Desirable properties such as regularity (i.e. continuity and approximation order) are easy to derive from corresponding properties of the interior scaling functions. In this article we focus instead on boundary functions defined by recursion relations. We show that the number of boundary functions is uniquely determined, and derive conditions for determining regularity from the recursion coefficients. We show that there are regular boundary functions which are not linear combinations of shifts of the underlying scaling functions.     

A trace regularity result for thermoelastic equations with application to optimal boundary control

We consider a mixed problem for a Kirchoff thermoelastic plate model with clamped boundary conditions. We establish a sharp regularity result for the outer normal derivative of the thermal velocity on the boundary. The proof, based upon interpolation techniques, benefits from the exceptional regularity of traces of solutions to the elastic Kirchoff equation. This result, which complements recent results obtained by the second and third authors, is critical in the study of optimal control problems associated with the thermoelastic system when subject to thermal boundary control. Indeed, the present regularity estimate can be interpreted as a suitable control-theoretic property of the corresponding abstract dynamics, which is crucial to guarantee well-posedness for the associated differential Riccati equations.     

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.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part I: Shape Differentiability of Pseudo-homogeneous Boundary Integral Operators

In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators.     

Evolution temporelle d'une poche de tourbillon singuliere

We study here regularity of solutions of the two dirnensional incompressible Euler system whose initial data are vortez patches with singular boundary. We show the persislance of conormal regularity apart from a closed subset. As a consequence, we obtain the following result for vories patches : if the initial boundary is regular apart from a closed subset, it remairw regular for all time apart from the closed subset transported by the pow.     

Regularity of the Solutions for Parabolic Two-Phase Free Boundary Problems

In this paper we start to develop the regularity theory of general two-phase free boundary problems for parabolic equations. In particular we consider uniformly parabolic operators in nondivergence form and we are mainly concerned with the optimal regularity of the viscosity solutions. We prove that under suitable nondegenerate conditions the solution is Lipschitz across the free boundary.     

A Discrete Phenomenon in the C and Analytic Regularity of the Blow-Up Curve of Solutions to the Liouville Equation in One Space Dimension

We give a complete discussion of the C^∞ or analytic regularity of blow-up curves for Cauchy problems or some mixed problems for the Liouville equation in one space dimension. In the case of mixed problems, the regularity results depend on the boundary condition: actually, we show the existence of a sequence of boundary conditions for which the regularity of the blow-up curve is better than in the general case.     

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.     

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.     

Boundary regularity ofp-harmonic maps with free and partially constrained boundary conditions

We study the boundary regularity ofp-harmonic maps with free and partially constrained boundary conditions and give estimates on the size of the singular subset of the boundary.     

On the sufficient condition for regularity of a boundary point for singular parabolic equations with non‐standard growth

We investigate the continuity of solutions for general nonlinear parabolic equations with non‐standard growth near a nonsmooth boundary of a cylindrical domain. We prove a sufficient condition for regularity of a boundary point.     

在边界附近蜕化的椭圆型Monge-Ampère方程解的正则性(英文)

我们证明了在边界附近蜕化的椭圆型Ｍｏｎｇｅ－Ａｍｐèｒｅ方程Ｄｉｒｉｃｈｌｅｔ问题解的整体Ｃ１,１正则性,并举例说明了我们的结果是最佳的。     

The obstacle problem for a fractional Monge–Ampère equation

Abstract

We study the obstacle problem for a nonlocal, degenerate elliptic Monge–Ampère equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.