Rouba Borghol, Laurent Veron
We construct N-harmonic functions in a domain with one isolated singularity on the boundary of the domain. By using solutions of the spherical p-harmonic spectral problem, we give an inductive method to produce a large variety of separable p-harmonic functions.
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In this paper, positive solutions to the Laplace equation with 1-dimensional circular singularities are investigated. First, we establish $L^p$ integrability estimates for such solutions $u$ near the singularities, in comparison with classical $L^1$ estimates. Then we characterize $-Δu$ around the circle. The method we developed may also be adapted to higher dimensional as well as more generalized cases.
In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content of the set of singular points with multiplicity $Q$.
Propagation of space-time singularities for perturbed harmonic oscillators
本文研究了时间依赖度量和势扰动下量子谐振子中时空奇点的传播，通过半经典方法重新表述波前集并比较未受扰动系统。
It is well known that sets of $p$-capacity zero are removable for bounded $p$-harmonic functions, but on metric spaces there are examples of removable sets of positive capacity. In this paper, we show that this can happen even on unweighted $\mathbf{R}^n$ when $n > p$, although only in very special cases. A complete characterization of removable singularities for bounded $\mathcal{A}$-harmonic functions on weighted $\mathbf{R}^n$, $n \ge 1$, is also given, where the weight is $p$-admissible. The same characterization is also shown to hold for bounded quasiharmonic functions on weighted $\mathb…
We study the boundary behaviour of solutions $u$ of $-Δ_{N}u+ |u|^{q-1}u=0$ in a bounded smooth domain $Ω\subset\mathbb R^{N}$ subject to the boundary condition $u=0$ except at one point, in the range $q>N-1$. We prove that if $q\geq 2N-1$ such a $u$ is identically zero, while, if $N-1<q<2N-1$, $u$ inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.
We consider harmonic maps into pseudo-Riemannian manifolds. We show the removability of isolated singularities for continuous maps, i.e. that any continuous map from an open subset of R^m into a pseudo-Riemannian manifold which is two times continuously differentiable and harmonic everywhere outside an isolated point is actually smooth harmonic everywhere.
In this article, we study the the harmonic map heat flow from a manifold with conic singularities to a closed manifold. In particular, we have proved the short time existence and uniqueness of solutions as well as the existence of global solutions into manifolds with nonpositive sectional curvature. These results are established in virtue of the maximal regularity theory on manifolds with conic singularities.
We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.
In this paper, we present a new distributional identity for the solutions of elliptic equations involving Hardy potentials with singularities located on the boundary of the domain. Then we use it to obtain the boundary isolated singular solutions of nonhomogeneous problems.
Structure of the free interfaces near triple junction singularities in harmonic maps and optimal partition problems
研究谐映射到树的能量最小化问题，证明三交点附近自由界面的正则性，展示其由三个C^{1,α}光滑的(d-1)维流形组成，以120度角交汇。
Nonunique tangent maps at isolated singularities of minimizing $p$-harmonic maps
本研究构造出首个在孤立奇点处具有非唯一切映射的极小化$p$-调和映射示例，拓展了B. White针对平稳情形$p=2$的相关结果，推进了对能量极小映射奇异集精细结构的研究。
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in domains with conic singularities and establish asymptotic expansions with respect to two normal directions simultaneously. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Key step is the study of the eigenvalues growth and eigenfunctions estimates.
Singular solutions of the harmonic Einstein evolution equation are constructed which are related to spatially global and time-local solutions for a certain class of quasilinear hyperbolic systems of second order. The constructed singularities of curvature invariants occur generically and are accessible by g.a.p. curves. The singularities are not strongly censored, and for strongly asymptotically predictable space-times, they are located in the causal past of the future null infinity, and are, hence, not shielded by a black hole. This is an alternative construction of singularities, which may b…
Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of $Ø$. In particular we provide sufficient and necessary conditions on the existence of minimizers.
Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded $H_{\infty}$-calculus in appropriate $L_{p}$-Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation.
Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequalities with weight function singular at the boundary of $Ø$. In particular we give sufficient conditions so that the best constant is achieved.
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the boundary. Similar constructions are carried out for solutions which are singular on any given embedded submanifold of the boundary.
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Navier-Stokes equations, for which the smoothness of the interface breaks down in finite time into a splash singularity.
Let $Ø$ be a bounded domain in $\R^N$ with $0\in\deØ$ and $N\ge 2$. In this paper we study the Hardy-Poincaré inequality for maps in $H^1_0(Ω)$. In particular we give sufficient and some necessary conditions so that the best constant is achieved.
Removable singularities in the boundary for quasilinear elliptic equations
该研究聚焦拟线性椭圆方程的边界奇异集可去性问题，提出利用解的无穷远渐近行为与沿奇异集方向的Sobolev范数条件两种途径开展分析。
The structure of the regular part of the free boundary close to singularities in the obstacle problem
本文证明了障碍问题自由边界正则部分在奇点附近精细渐近行为的首个结果，源于对长期猜想的部分解答和高维无界接触集全局解的部分分类。
We define renormalised energies for maps that describe the first-order asymptotics of harmonic maps outside of singularities arising due to obstructions generated by the boundary data and the mutliple connectedness of the target manifold. The constructions generalise the definition by Bethuel, Brezis and Hélein for the circle (Ginzburg-Landau vortices, 1994). In general, the singularities are geometrical objects and the dependence on homotopic singularities can be studied through a new notion of synharmony. The renormalised energies are showed to be coercive and Lipschitz-continuous. The renor…
We study properties of positive functions satisfying (E) --$Δ$u+m|$\nabla$u| q -- u p = 0 is a domain $Ω$ or in R N + when p > 1 and 1 < q < 2. We give sufficient conditions for the existence of a solution to (E) with a nonnegative measure $μ$ as boundary data, and these conditions are expressed in terms of Bessel capacities on the boundary. We also study removable boundary singularities and solutions with an isolated singularity on $\partial$$Ω$. The different results depends on two critical exponents for p = p c := N +1 N --1 and for q = q c := N +1 N and on the position of q with respect to…
On free boundary problems shaped by varying singularities
本文研究奇异幂$γ(x)$随空间变化的自由边界变分模型，在温和连续性条件下证明极小化元的最优正则性，并利用新单调性公式证明当$γ(x)$具有$W^{1,n^{+}}$正则性时自由边界局部为$C^{1,δ}$曲面。
We prove the existence of a solution of (--$Δ$) s u + f (u) = 0 in a smooth bounded domain $Ω$ with a prescribed boundary value $μ$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $μ$ is a Dirac mass, we prove the existence of several critical exponents p.
Isolated singularities of harmonic maps with generic boundary data
本文研究具有一般边界数据的调和映射孤立奇点，提出最小指标原理，证明在低维情况下奇点切映射为径向投影，并建立无穷远刚性定理。
Asymptotic behavior of minimizing $p$-harmonic maps when $p \nearrow 2$ in dimension 2
本文研究了当 $p$ 从下方趋近于 2 时，满足 Dirichlet 边界条件的极小 $p$-调和映射的渐近行为，证明其收敛至由重整化能量控制奇点位置的极小奇异可重整化调和映射。
In this contribution we are interested in the quantitative homogenization properties of linear elliptic equations with homogeneous Dirichlet boundary data in polygonal domains with corners. To begin our study of this situation, we consider the setting of an angular sector in 2 dimensions : Unlike in the whole-space, on such a sector there exist non-smooth harmonic functions (these depend on the angle of the sector). Here, we construct extended homogenization correctors corresponding to these harmonic functions and prove growth estimates for these which are quasi-optimal, namely optimal up to a…
We investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$Δu = -χ_{\{u>0\}} .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs meeting in a {\em cross}-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is non-trivial. We also show existence of a degenerate solution. This answers two of the open questions in a recent paper by R. Monneau-G.S. Weiss.
We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.
We study the boundary value problem with measures for (E1) $-\Gd u+g(|\nabla u|)=0$ in a bounded domain $\Gw$ in $\BBR^N$, satisfying (E2) $ u=\gm$ on $\prt\Gw$ and prove that if $g\in L^1(1,\infty;t^{-(2N+1)/N}dt)$ is nondecreasing (E1)-(E2) can be solved with any positive bounded measure. When $g(r)\geq r^q$ with $q>1$ we prove that any positive function satisfying (E1) admits a boundary trace which is an outer regular Borel measure, not necessarily bounded. When $g(r)=r^q$ with $1
Boundary partial regularity for a class of Dirac-harmonic maps
本文研究一类满足特定能量单调不等式的耦合 Dirac-harmonic 映射，证明其在边界附近具有部分正则性。
Our concern in this paper is to study the qualitative properties for harmonic functions related to the fractional Laplacian. Firstly we classify the polynomials in the whole space and in the half space for the fractional Laplacian defined in a principle value sense at infinity. Secondly, we study the fractional harmonic functions in half space with singularities on the boundary and the related distributional identities.
In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second author shows that two-valued $C^{1, μ}$ harmonic functions on $n$ dimensional domains, which are typically-non-minimizing stationary points of Dirichlet energy, play an essential role in the study of multiplicity 2 branch points of stable codimension 1 rectifiable currents of dim…
We prove a quantification result for harmonic maps with free boundary from arbitrary Riemannian surfaces into the unit ball of ${\mathbb R}^{n+1}$ with bounded energy. This generalizes results obtained by Da Lio on the disc.
Asymptotic Analysis of Harmonic Maps With Prescribed Singularities
本文作为系列论文首篇，研究带规定奇点的调和映射的渐近行为，为建立多黑洞质量-角动量不等式提供正则性理论，还用于证明极端黑洞近视界极限存在并计算其几何。
Let $n\geq2$ and $ Ω\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$Δu + u^p = 0 \quad\hbox{in}\, Ω,$$ which vanish in a suitable trace sense on $\partialΩ,$ but which are singular at prescribed "edge" of $Ω$ if $p$ is equal or slightly above a certain exponent $p_0>1$ which depends on $Ω.$ Moreover, in the case which $Ω$ is unbounded, the solutions have fast decay at infinity.
In this paper we describe the propagation of smooth (C^\infty) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X=MxR, P=D_t^2-Δ_M, and u locally in H^1 solving Pu=0 with homogeneous Dirichlet or Neumann boundary conditions, we show that the wave front set of u is a union of maximally extended generalized broken bicharacteristics. This result is a smooth counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary. Our methods re…
We study the asymptotic behaviour, as a small parameter $\varepsilon$ tends to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear penalisation potential vanishing on a compact submanifold $\mathcal{N}$ and with a given $\mathcal{N}$-valued Dirichlet boundary data. We show that minimisers converge up to a subsequence to a singular $\mathcal{N}$-valued harmonic map, which is smooth outside a finite number of points around which the energy concentrates and whose singularities' location minimises a renormalised energy, generalising known results by Bethuel, Brezis and Hélein for…
In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in $L^p$ for all finite $p\geq 1$.
Let $X$ be a manifold with boundary, endowed with a metric with conic singularities at the boundary components of $X$. Let $u$ be a solution to the wave equation on $\mathbb{R} \times X$. When a singularity of $u$ strikes a cone point of $X$, it undergoes a mixture of diffractive spreading and geometric propagation.
Error estimates for approximations of harmonic functions on planar regions by subspaces spanned by the first harmonic Steklov eigenfunctions are found. They are based on the explicit representation of harmonic functions in terms of these harmonic Steklov eigenfunctions. When the region is a rectangle of aspect ratio h, some computational results regarding these approximations for problems with known explicit solutions are described.
We study the boundary behaviour of the solutions of (E) $-Δ_p u+|\nabla u|^q=0$ in a domain $Ω\subset \mathbb{R}^N$, when $N\geq p > q >p-1$. We show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_*$ there exist positive solutions of (E) with an isolated singularity on $\partialΩ$ and that these solutions belong to two different classes of singular solutions. If $q_*\leq q < p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the…
In the paper, we have introduced the notion of mild bounded ancient solutions to the Navier-Stokes equations in a half space. They play a certain role in understanding whether or not solutions to the initial boundary value problem for the Navier-Stokes system with non-slip boundary conditions have blowups of Type I.
Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary
本文证明了在域度量沿自由边界以$d^α$速率退化或奇异时，具有自由边界数据的弱平稳调和映照的部分正则性。
We study the boundary behavior of nonnegative p-harmonic functions which vanish on a portion of the boundary of a domain in the Heisenberg group H^n. Our main results are: 1) An estimate from above which shows that, under suitable geometric assumptions on the relevant domain, such a p-harmonic function vanishes at most linearly with respect to the sub-Riemannian distance to the boundary. 2) An estimate from below which shows that for a (Euclidean) C^{1,1} domain, away from the characteristic set, such a p-harmonic function vanishes exactly like the distance to the boundary. By combining 1) and…
We investigate various boundary decay estimates for $p(\cdot)$-harmonic functions. For domains in $\mathbb{R}^n, n\geq 2$ satisfying the ball condition ($C^{1,1}$-domains) we show the boundary Harnack inequality for $p(\cdot)$-harmonic functions under the assumption that the variable exponent $p$ is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson type estimate for $p(\cdot)$-harmonic functions in NTA domains in $\mathbb{R}^n$ and provide lower- and upper- growth estimates and a doubling property for a $p(\cdot)$-harmonic …
We consider the inhomogeneous Dirichlet problem on product domains. The main result is the asymptotic expansion of the solution in terms of increasing smoothness up to the boundary. In particular, we show the exact nature of the singularities of the solution at singularities of the boundary by constructing singular functions which make up an asymptotic expansion of the solution.
In this paper we are concerned with singular points of solutions to the {\it unstable} free boundary problem $$ Δu = - χ_{\{u>0\}} \qquad \hbox{in} B_1. $$ The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities - that is points at which the second derivatives of the solution are unbounded - as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the nonlinearity $χ_{\{u>0\}} $. …
Free boundary flow through cylindrical singularities
研究自由边界平均曲率流通过圆柱或半圆柱奇点的行为，证明这些奇点具有平均凸邻域，并推广Hershkovits-White结果，表明自由边界水平集流不增厚，从而确保流通过奇点的适定性。
We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with at least two boundary components. This seems to be the first result in this setting. Moreover, we allow to have conical singularities with both positive and negative orders, that is cone angles both less and grater than $2π$.
We study a 2-phase free boundary problem for harmonic measure first considered by Kenig and Toro and prove a sharp Hölder regularity result. The central difficulty is that there is no a priori non-degeneracy in the free boundary condition. Thus we must establish non-degeneracy by means of monotonicity formulae.
In this paper, we give a harmonic analysis proof of the Neumann boundary observability inequality for the wave equation in an arbitrary space dimension. Our proof is elementary in nature and gives a simple, explicit constant. We also extend the method to prove the observability inequality of a visco-elastic wave equation.
Isolated Singularities of Solutions to the Yamabe Equation with boundary in Dimension 3 and 4
研究三维和四维带边界的 Yamabe 方程正解在孤立奇点附近的渐近行为，建立了精确上界与不可去情形下的匹配下界及可去性条件，证明不可去奇点解渐近柱对称，扩展相关理论。
Given a smooth domain $Ω\subset\RR^N$ such that $0 \in \partialΩ$ and given a nonnegative smooth function $ζ$ on $\partialΩ$, we study the behavior near 0 of positive solutions of $-Δu=u^q$ in $Ω$ such that $u = ζ$ on $\partialΩ\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x \to 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.
We prove the spectral invariance of the algebra of classical pseudodifferential boundary value problems on manifolds with conical singularities in the Lp-setting. As a consequence we also obtain the spectral invariance of the classical Boutet de Monvel algebra of zero order operators with parameters. In order to establish these results, we show the equivalence of Fredholm property and ellipticity for both cases.
We study the behaviour near a boundary point a of any positive solution of a nonlinear elliptic equations with forcing term which vanishes on the boundary except at a. Our results are based upon a priori estimates for solutions and existence or non existence and uniqueness results for solutions of some nonlinear elliptic equations on the half unit sphere.
Let $M$ be a complex manifold of dimension $n$ with smooth boundary $X$. Given $q\in\{0,1,\ldots,n-1\}$, let $\Box^{(q)}$ be the $\ddbar$-Neumann Laplacian for $(0,q)$ forms. We show that the spectral kernel of $\Box^{(q)}$ admits a full asymptotic expansion near the non-degenerate part of the boundary $X$ and the Bergman projection admits an asymptotic expansion under some local closed range condition. As applications, we establish Bergman kernel asymptotic expansions for some domains with weakly pseudoconvex boundary and $S^1$-equivariant Bergman kernel asymptotic expansions and embedding th…
In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary conditions or to the nonlinearity of the equations can effect whether the equations develop finite-time singularities. In particular, we aim to underscore the idea that in analytical and computational investigations of the blow-up of three-dimensional Euler and Navier-Stokes equations, the boundary conditions may need to be taken into greater account. We als…
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