Numerous elliptic and parabolic variational problems arising in physics and geometry (Ginzburg-Landau equations, harmonic maps, Yang-Mills fields, Omega-instantons, Yamabe equations, geometric flows in general...) possess a critical dimension in which an invariance group (similitudes, conformal groups) acts. This common feature generates, in all these different situations, the same non-linear effect. One observes a strict splitting in space between an almost linear regime and a dominantly non-linear regime which has two major characteristics : it requires a quantized amount of energy and arises along rectifiable objects of special geometric interest (geodesics, minimal surfaces, J-holomorphic curves, special Lagrangian manifolds, mean-curvature flows...).
We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examine the possible lack of C1 convergence for sequences of surfaces with bounded Willmore energy and find that the mechanism responsible for this is a bubbling phenomenon, where energy escapes to infinity.
In this paper we prove a convergence result for sequences of Willmore immersions with simple minimal bubbles. To this end we replace the total curvature control in T. Rivière's proof of the $\varepsilon$-regularity for Willmore immersions by a control of the local Willmore energy.
We study a Sobolev critical fast diffusion equation in bounded domains with the Brezis-Nirenberg effect. We obtain extinction profiles of its positive solutions, and show that the convergence rates of the relative error in regular norms are at least polynomial. Exponential decay rates are proved for generic domains. Our proof makes use of its regularity estimates, a curvature type evolution equation, as well as blow up analysis. Results for Sobolev subcritical fast diffusion equations are also obtained.
Let $E = \Bbb C/Λ$ be a flat torus and $G$ be its Green function with singularity at $0$. Consider the multiple Green function $G_n$ on $E^n$: $$G_{n}(z_1,\cdots,z_n) := \sum_{i < j} G(z_{i} - z_{j}) - n \sum_{i = 1} ^{n} G(z_{i}).$$ A critical point $a = (a_1, \cdots, a_n)$ of $G_n$ is called trivial if $\{a_1, \cdots, a_n\} = \{-a_1, \cdots, -a_n\}$. For such a point $a$, two geometric quantities $D(a)$ and $H(a)$ arising from bubbling analysis of mean field equations are introduced. $D(a)$ is a global quantity measuring asymptotic expansion and $H(a)$ is the Hessian of $G_n$ at $a$. By way …
In this paper, we study the fractional harmonic gradient flow on $S^1$ taking values in $S^{n-1} \subset \mathbb{R}^n$ for every $n \geq 2$, in particular addressing uniqueness and regularity of solutions in the so-called energy class with sufficiently small energy, adding to the existing body of knowledge which includes existence of solutions and certain bubbling phenomena.
We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their …
The goal of this paper is to discuss some of the results in [31] and [32] and expand upon the work there by proving a global weak existence result as well as a first bubbling analysis in finite time. In addition, an alternative local existence proof is presented based on a fixed-point argument. This leads to two possible outlooks until a conjecture by Sire, Wei and Zheng is settled, see [27]: Either there always exists a global smooth solution (thus the solution constructed here is actually as regular as desired) or finite-time bubbling may occur in a similar way as for the harmonic gradient f…
In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equation with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced by the exponential nonlinearity. In this paper, we consider the following semilinear Neumann elliptic problem with the Trudinger-Moser exponential growth: \begin{equation*}\begin{cases} -dΔu_d+u_d=u_d(e^{u^2_{d}}-1)\ \ \mbox{in}\ Ω,\\ \frac{\partial u_d}{\partialν}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial Ω,\\ \end{cases}\end{equa…
We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant $Q$-curvature. As a byproduct of our method, we also obtain compactness of such metrics.
Motivated by a challenging expectation of Rivière (2011), in the recent interesting work of deLongueville-Gastel (2019), de Longueville and Gastel proposed the following geometrical even order elliptic system \begin{equation*} Δ^{m}u=\sum_{l=0}^{m-1}Δ^{l}\left\langle V_{l},du\right\rangle +\sum_{l=0}^{m-2}Δ^{l}δ\left(w_{l}du\right)\qquad \text{ in } B^{2m}\label{eq: Longue-Gastel system} \end{equation*} which includes polyharmonic mappings as special cases. Under minimal regularity assumptions on the coefficient functions and an additional algebraic antisymmetry assumption on the first order p…
Consider the Milne problem with geometric correction in a 3D convex domain. Via bootstrapping arguments, we establish $W^{1,\infty}$ regularity for its solutions. Combined with a uniform $L^6$ estimate, such regularity leads to the validity of diffusive expansion for the neutron transport equation with diffusive boundary conditions.
In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators under strong absorption conditions. We establish improved geometric regularity along the free boundary, for a sharp value depending only on structural parameters. Non degeneracy among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings.
The H^2-regularity of variational solutions to a two-dimensional transmission problem with geometric constraint is investigated, in particular when part of the interface becomes part of the outer boundary of the domain due to the saturation of the geometric constraint. In such a situation, the domain includes some non-Lipschitz subdomains with cusp points, but it is shown that this feature does not lead to a regularity breakdown. Moreover, continuous dependence of the solutions with respect to the domain is established.
The purpose of this work is to produce a regularity theory for a class of parabolic Isaacs equations. Our techniques are based on approximation methods which allow us to connect our problem with a Bellman parabolic model. An approximation regime for the coefficients, combined with a smallness condition on the source term unlocks new regularity results in Sobolev and Hölder spaces.
We provide a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto's revolutionary method of intrinsic scaling, while the latter is achieved by means of what has been termed geometric tangential analysis. We discuss, in particular, sharp estimates for the parabolic p-Poisson equation, for the porous medium equation and for the doubly nonlinear equation.
Regularity theory for diffusive operators is among the finest treasures of the modern mathematical sciences. It appears in several different fields, such as, differential geometry, topology, numerical analysis, dynamical systems, mathematical physics, economics, etc. Within the general field of Partial Differential Equations, regularity theory places itself in the very core, by bridging the notion of weak solutions (often found by energy methods or probabilistic interpretations) to the classical concept of solutions. In this article we discuss about a new systematic approach, termed geometric …
Several formulations of a local geometric measure-type condition are imposed on super-level sets of mild solutions to the homogeneous incompressible 3D magnetohydrodynamical system with bounded initial data to prevent finite-time singularity formation. Supporting this, results regarding the existence, uniqueness, and real analyticity of mild solutions are established as is a sharp lower bound on the radius of analyticity.
We establish a Liouville type result for a backward global solution to the Navier-Stokes equations in the half plane with the no-slip boundary condition. No assumptions on spatial decay for the vorticity nor the velocity field are imposed. We study the vorticity equations instead of the original Navier-Stokes equations. As an application, we extend the geometric regularity criterion for the Navier-Stokes equations in the three-dimensional half space under the no-slip boundary condition.
In this article we discuss higher Sobolev regularity of convex integration solutions for the geometrically non-linear two-well problem. More precisely, we construct solutions to the differential inclusion $\nabla u\in K$ subject to suitable affine boundary conditions for $ u$ with $$ K:= SO(2)\left[\begin{array}{ ccc } 1 & δ\\ 0 & 1 \end{array}\right] \cup SO(2)\left[\begin{array}{ ccc } 1 & -δ\\ 0 & 1 \end{array}\right] $$ such that the associated deformation gradients $\nabla u$ enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape…
In this paper, we examine the regularity of the solutions to the double-divergence equation. We establish improved Hölder continuity as solutions approach their zero level-sets. In fact, we prove that $α$-Hölder continuous coefficients lead to solutions of class $\mathcal{C}^{1^-}$, locally. Under the assumption of Sobolev differentiable coefficients, we establish regularity in the class $\mathcal{C}^{1,1^-}$. Our results unveil improved continuity along a nonphysical free boundary, where the weak formulation of the problem vanishes. We argue through a geometric set of techniques, implemented …
We establish the regularity of weak solutions for the vorticity equation associated to a family of desingularised models for vortex filament dynamics in 3D incompressible viscous flows. These include and generalise the classical model "of an allowance for the thickness of the vortices" due to Louis Rosenhead in 1930. Our approach is based on an interplay between the geometry of vorticity and analytic inequalities in Sobolev spaces.
We establish sharp geometric $C^{1+α}$ regularity estimates for bounded weak solutions of evolution equations of $p$-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.
This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary Hölder regularity under proper geometric conditions. "Unified" means that our method is applicable for the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the $p-$Laplace equations and the fractional Laplace equations etc. In addition, these geometric conditions are quite general. In particular, for local equations, the meas…
Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove Hölder regularity of weak solutions.
The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge-Ampère type known as generated Jacobian equations. These equations, whose general existence theory has been recently developed by Trudinger go beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under a condition analogous to the A3w condition of Ma, Trudinger and Wang. Estimates of this type have played an important role in the regularity theory for optimal transport maps and were previously unknown in this context, inc…
In this note, we prove the boundary Hölder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the corkscrew domains (including the non-tangentially accessible domains) are special cases. The key idea, following [3], is that the strong maximum principle and the scaling invariance imply the boundary Hölder regularity.
We consider critical points of the geometric obstacle problem on vectorial maps $u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N \backslash \mathbb{B}^N(0)$}. \] Our main result is $C^{1,α}$-regularity for any $α< 1$. Technically, we split the map $u=λv$, where $v: \mathbb{B}^2 \to \mathbb{S}^{N-1}$ is the vectorial component and $λ= |u|$ the scalar component measuring the distance to the origin. While $v$ satisfies a weighted harmonic map equation with weight $λ^2$, $λ$ solves the obstacle problem for \[ \in…
In this paper we study nonnegative minimizers of general degenerate elliptic functionals, $\int F(X,u,Du) dX \to \min$, for variational kernels $F$ that are discontinuous in $u$ with discontinuity of order $\sim χ_{\{u > 0 \}}$. The Euler-Lagrange equation is therefore governed by a non-homogeneous, degenerate elliptic equation with free boundary between the positive and the zero phases of the minimizer. We show optimal gradient estimate and nondegeneracy of minima. We also address weak and strong regularity properties of free boundary. We show the set $\{u > 0 \}$ has locally finite perimeter…
That the weak solutions of degenerate parabolic pdes modelled on the inhomogeneous $p-$Laplace equation $$ u_t - \mathrm{div} \left(|\nabla u|^{p-2} \nabla u \right) = f \in L^{q,r}, \quad p>2 $$ are $C^{0,α}$, for some $α\in (0,1)$, is known for almost 30 years. What was hitherto missing from the literature was a precise and sharp knowledge of the Hölder exponent $α$ in terms of $p, q, r$ and the space dimension $n$. We show in this paper that $$ α= \frac{(pq-n)r-pq}{q[(p-1)r-(p-2)]}, $$ using a method based on the notion of geometric tangential equations and the intrinsic scaling of the $p-$…
We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators; more specifically, we consider viscosity solutions of \[ |D u|^γF(x, D^2u) = f(x)χ_{\{u>ϕ\}} \textrm{ in } B_1 \] with $γ>0$, $ϕ\in C^{1, α}(B_1)$ for some $α\in(0,1]$ and $f\in L^\infty(B_1)$ constrained to satisfy \[ u\geq ϕ\textrm{ in } B_1 \] and prove that they are $C^{1,β}(B_{1/2})$ (and in particular along free boundary points) where $β=\min\left\{α, \frac{1}{γ+1}\right\}$. Moreover, we achieve such a feature by using a recently developed geometric appro…
This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation $u_{t}=Δu^{m}$. Our main objective is to improve the H$\ddot{o}$lder estimate obtained by pioneers and then, to show the geometric characteristic of free boundary of degenerate parabolic equation. To be exact, the present work will show that: (1) the weak solution $u(x,t)\in{}C^{α,\fracα{2}}(\mathbb{R}^{n}\times\mathbb{R}^{+})$, where $α\in(0,1)$ when $m\geq2$ and $α=1$ when $m\in(1,2)$; (2) the surface $ϕ=(u(x,t))^β$ is a complete Riemannian manifold, which is …
The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the inequalities. Then, as main application of our inequalities, we establish new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-Δu=g(u)$ in a bounded domain $Ω$ of $\mathbb{R}^n$. These estimates lead to an $L^{2n/(n-4)}(Ω)$ bound for the extremal solution of $-Δu=λf(u)$ when $n\geq 5$ and the domain is convex. We recall that extremal solutions are known to…
In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \textit{a priori} estimates for semi-stable solutions of $-Δ_p u= g(u)$ in a smooth bounded domain $Ω\subset \mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ when the domain is strictly convex. More precisely, we prove that $u^\star\in L^\infty(Ω)$ if $n\leq p+2$ and $u^\star\in L^{\frac{np}{n-p-2}}(Ω)\…
We prove an abstract result ensuring that one-sided geometric control yields two-sided estimates for functions satisfying general conditions. Our findings resonate in the context of nonlinear elliptic problems, including supersolutions to fully nonlinear elliptic equations and functions in the De Giorgi class. Among the consequences of our abstract results are regularity estimates, and conditions for a continuous function to be in the class of viscosity solutions. We also prove that one-sided geometric control yields $L^pL^\infty$-estimates. It provides a converse to the implication in the De …
This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills equation. We consider topologically trivial solutions in the caloric gauge, which was defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy c…
Let $u$ be a weak solution of the free boundary problem $$\mathcal L u=λ_0 \mathcal H^1\lfloor\partial\{u>0\}, u\ge 0,$$ where $\mathcal L u={\text{div}}(g(\nabla u)\nabla u)$ is a quasilinear elliptic operator and $g(ξ)$ is a given function of $ξ$ satisfying some structural conditions. We prove that the free boundary $\partial\{ u>0\}$ is continuously differentiable in $\mathbb R^2$, provided that $\partial\{ u>0\}$ has locally finite connectivity. Moreover, we show that the free boundaries of weak solutions with finite $\it{Morse \ index}$ must have finite connectivity. The weak solutions ar…
We study geometric and regularity properties of the largest subsolution of a one-phase free boundary problem under a very general free boundary condition in R2. Moreover, we provide density bounds for the positivity set and its complement near the free boundary.
We announce a result on the existence of a unique local solution to a stochastic geometric wave equation on the one dimensional Minkowski space $\mathbb{R}^{1+1}$ with values in an arbitrary compact Riemannian manifold. We consider a rough initial data in the sense that its regularity is lower than the energy critical.
A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a Hölder continuous function, regularity of the resulting solution is in line with the standard parabolic theory.
The aim of this article is to give a rather extensive, and yet nontechnical, account of the birth of the regularity theory for generalized minimal surfaces, of its various ramifications along the decades, of the most recent developments, and of some of the remaining challenges.
We obtain a result about propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension $N\geq2$. In particular, we investigate conservation of striated and conormal regularity, which is a natural way of generalizing the 2-D structure of vortex patches. The results we get are only local in time, even in the dimension N=2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N=2 or 3, we investigate also propagation of Hölder regularity in the interior of a bounded domain.
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.
For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior Hölder regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in $(\frac{32}{15},3)$. The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous $p$-harmonic maps to $S^3$. For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model.
Given an open bounded subset $Ω$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-Δ_{\infty} u = 1$ in $Ω$, subject to the homogeneous boundary condition $u = 0$ on $\partial Ω$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C ^1(Ω)$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $|\nabla u| = a$ on $\partial Ω$; by using a suitable $P$-function we prove that, if $Ω$ satisfies the same assumptions as above a…
We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. As a consequence we deduce that both geometric and analytical mixing have a lower bound of order $t^{-1}$ as $t \to \infty$.
We establish sharp geometric Holder regularity estimates for Gradient for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy. Such regularity estimates simplify and generalize, to some extent, earlier ones via a totally different modus operandi. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. In the end, we present some connections of our findings with a variety of nonlinear geometric free boundary problems and relevant nonlinear problems in t…
We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques we obtain the sharp local $C^{1,α}$ regularity of viscosity solutions to such equations.
We prove geometrically improved version of Prodi-Serrin type blow-up criterion. Let $v$ and $ω$ be the velocity and the vorticity of solutions to the 3D Navier-Stokes equations and denote $\{f\}_+=\max\{f, 0\}$ , $Q_T=\Bbb R^3\times (0, T)$. If $\left\{\left( v \times \fracω{|ω|} \right)\cdot \frac{Λ^βv}{|Λ^βv|}\right\}_+ \in L^{γ, α}_{x,t} (Q_T)$ with $3/γ+2/α\leq 1$ for some $γ>3$ and $1 \leq β\leq 2$, then the local smooth solution $v$ of the Navier-Stokes equations on $(0,T)$ can be continued to $(0, T+δ)$ for some $δ>0$. We also prove localized version of a special case of this. Let $v$ b…
We characterise regular boundary points of the parabolic $p$-Laplacian in terms of a family of barriers, both when $p>2$ and $1<p<2$. Due to the fact that $p\not=2$, it turns out that one can multiply the $p$-Laplace operator by a positive constant, without affecting the regularity of a boundary point. By constructing suitable families of barriers, we give some simple geometric conditions that ensure the regularity of boundary points.
